Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

Research ArticleHybrid Viscosity Approaches to General Systems ofVariational Inequalities with Hierarchical Fixed Point ProblemConstraints in Banach Spaces

Lu-Chuan Ceng1 Saleh A Al-Mezel2 and Abdul Latif2

1 Department of Mathematics Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai UniversitiesShanghai 200234 China

2Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Saleh A Al-Mezel mathsalehyahoocom

Received 4 August 2013 Revised 4 December 2013 Accepted 20 December 2013 Published 17 February 2014

Academic Editor Hichem Ben-El-Mechaiekh

Copyright copy 2014 Lu-Chuan Ceng et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities(GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banachspaces Here the hybrid viscosity methods are based on Korpelevichrsquos extragradient method viscosity approximation method andhybrid steepest-descent method We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solvingthe GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family ofnonexpansive mappings inX respectivelyWe derive some strong convergence theorems under appropriate conditions Our resultsextend improve supplement and develop the recent results announced by many authors

1 Introduction

Let 119883 be a real Banach space whose dual space is denoted by119883lowast Let 119880 = 119909 isin 119883 119909 = 1 denote the unit sphere of 119883

A Banach space 119883 is said to be uniformly convex if for each120598 isin (0 2] there exists 120575 gt 0 such that for all 119909 119910 isin 119880

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 ge 120598 997904rArr

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2le 1 minus 120575 (1)

It is known that a uniformly convex Banach space is reflexiveand strictly convex The normalized duality mapping 119869

119883 rarr 2119883lowast

is defined by

119869 (119909) = 119909lowast

isin 119883lowast

⟨119909 119909lowast

⟩ = 1199092

=1003817100381710038171003817119909lowast1003817100381710038171003817

2

forall119909 isin 119883

(2)

where ⟨sdot sdot⟩ denotes the generalized duality pairing It is animmediate consequence of the Hahn-Banach theorem that119869(119909) is nonempty for each 119909 isin 119883

Let119862 be a nonempty closed convex subset of a real Banachspace 119883 A mapping 119879 119862 rarr 119862 is said to be 119871-Lipschitzian

if there exists a constant 119871 gt 0 such that 119879119909 minus 119879119910 le

119871119909 minus 119910 for all 119909 119910 isin 119862 In particular if 119871 = 1 then 119879

is said to be nonexpansive The set of fixed points of 119879 isdenoted by Fix(119879) We use the notation to indicate theweak convergence and the one rarr to indicate the strongconvergence A mapping 119860 119862 rarr 119883 is said to be

(i) accretive if for each 119909 119910 isin 119862 there exists 119895(119909 minus 119910) isin

119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ ge 0 (3)

where 119869 is the normalized duality mapping of 119883(ii) 120572-inverse-strongly accretive if for each 119909 119910 isin 119862

there exists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ ge 1205721003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(4)

for some 120572 isin (0 1)(iii) pseudocontractive if for each 119909 119910 isin 119862 there exists

119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(5)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 945985 18 pageshttpdxdoiorg1011552014945985

2 Abstract and Applied Analysis

(iv) 120573-strongly pseudocontractive if for each 119909 119910 isin 119862there exists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ le 1205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2 (6)

for some 120573 isin (0 1)(v) 120582-strictly pseudocontractive if for each 119909 119910 isin 119862



1003817100381710038171003817

2

minus 1205821003817100381710038171003817119909 minus 119910 minus (119860119909 minus 119860119910)

1003817100381710038171003817

2

(7)

for some 120582 isin (0 1)It is worth emphasizing that the definition of the inverse-

strongly accretive mapping is based on that of the inverse-strongly monotone mapping which was studied by so manyauthors see for example [1ndash7]

A Banach space 119883 is said to be smooth if the limit

lim119905rarr0

1003817100381710038171003817119909 + 1199051199101003817100381710038171003817 minus 119909

119905

(8)

exists for all 119909 119910 isin 119883 in this case 119883 is also said tohave a Gateaux differentiable norm Moreover it is said tobe uniformly smooth if this limit is attained uniformly for119909 119910 isin 119880 in this case 119883 is also said to have a uniformly Fre-chet differentiable norm The norm of 119883 is said to be theFrechet differential if for each 119909 isin 119880 this limit is attaineduniformly for 119910 isin 119880 In the meantime we define a function120588 [0infin) rarr [0infin) called the modulus of smoothness of119883as follows

120588 (120591) = sup 1

2(1003817100381710038171003817119909 + 119910

1003817100381710038171003817 +1003817100381710038171003817119909 minus 119910

1003817100381710038171003817) minus 1 119909 119910 isin 119883

119909 = 11003817100381710038171003817119910

1003817100381710038171003817 = 120591

(9)

It is known that 119883 is uniformly smooth if and only iflim120591rarr0

120588(120591)120591 = 0 Let 119902 be a fixed real number with 1 lt 119902 le

2 Then a Banach space119883 is said to be 119902-uniformly smooth ifthere exists a constant 119888 gt 0 such that 120588(120591) le 119888120591

119902 for all 120591 gt 0As pointed out in [8] no Banach space is 119902-uniformly smoothfor 119902 gt 2 In addition it is also known that 119869 is single-valuedif and only if119883 is smooth whereas if119883 is uniformly smooththen the mapping 119869 is norm-to-norm uniformly continuouson bounded subsets of 119883

In a real smooth Banach space119883 we say that an operator119860 is strongly positive (see [9]) if there exists a constant 120574 gt 0

with the property

⟨119860119909 119869 (119909)⟩ ge 1205741199092

119886119868 minus 119887119860 = sup119909le1

|⟨(119886119868 minus 119887119860) 119909 119869 (119909)⟩|

119886 isin [0 1] 119887 isin [minus1 1]

(10)

where 119868 is the identity mapping

Proposition CB (see [9 Lemma 25]) Let 119862 be a nonemptyclosed convex subset of a uniformly smooth Banach space119883 Let

119879 119862 rarr 119862 be a continuous pseudocontractive mapping withFix(119879) = 0 and let 119891 119862 rarr 119862 be a fixed Lipschitzian stronglypseudocontractive mapping with pseudocontractive coefficient120573 isin (0 1) and Lipschitzian constant 119871 gt 0 Let 119860 119862 rarr 119862

be a strongly positive linear bounded operator with coefficient120574 gt 0 Assume that 119862 plusmn 119862 sub 119862 and 0 lt 120573 lt 120574 Let 119909

119905 be

defined by

119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) 119879119909

119905 (11)

Then as 119905 rarr 0 119909119905 converges strongly to some fixed point 119901

of 119879 such that 119901 is the unique solution in Fix(119879) to the VIP

⟨(119860 minus 119891) 119901 119869 (119901 minus 119906)⟩ le 0 forall119906 isin Fix (119879) (12)

On the other hand Cai and Bu [10] considered the follo-wing general system of variational inequalities (GSVI) ina real smooth Banach space 119883 which involves finding(119909lowast

119910lowast

) isin 119862 times 119862 such that

⟨12058311198611119910lowast

+ 119909lowast

minus 119910lowast

119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨12058321198612119909lowast

+ 119910lowast

minus 119909lowast

119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(13)

where 119862 is a nonempty closed and convex subset of 1198831198611 1198612

119862 rarr 119883 are two nonlinear mappings and 1205831and

1205832are two positive constants Here the set of solutions of

GSVI (13) is denoted by GSVI(119862 1198611 1198612) Very recently Cai

and Bu [10] constructed an iterative algorithm for solvingGSVI (13) and a common fixed point problem of an infinitefamily of nonexpansive mappings in a uniformly convex and2-uniformly smooth Banach space They proved the strongconvergence of the proposed algorithm by virtue of thefollowing inequality in a 2-uniformly smooth Banach space119883

Lemma 1 (see [11]) Let 119883 be a 2-uniformly smooth BanachspaceThen there exists a best smooth constant 120581 gt 0 such that

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909)⟩ + 21003817100381710038171003817120581119910

1003817100381710038171003817 forall119909 119910 isin 119883

(14)

where 119869 is the normalized duality mapping from 119883 into 119883lowast

The authors [10] have used the following inequality in areal smooth and uniform convex Banach space 119883

Proposition 2 (see [12]) Let119883 be a real smooth and uniformconvex Banach space and let 119903 gt 0 Then there exists a strictlyincreasing continuous and convex function 119892 [0 2119903] rarr R119892(0) = 0 such that

119892 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817) le 1199092

minus 2 ⟨119909 119869 (119910)⟩ +1003817100381710038171003817119910

1003817100381710038171003817

2

forall119909 119910 isin 119861119903

(15)

where 119861119903= 119909 isin 119883 119909 le 119903

2 Preliminaries

We list some lemmas that will be used in the sequel Lemma 3can be found in [13] Lemma 4 is an immediate consequenceof the subdifferential inequality of the function (12) sdot

2

Abstract and Applied Analysis 3

Lemma 3 Let 119886119899 be a sequence of nonnegative real numbers

such that

119886119899+1

le (1 minus 119887119899) 119886119899+ 119887119899119888119899 forall119899 ge 0 (16)

where 119887119899 and 119888

119899 are sequences of real numbers satisfying the

following conditions

(i) 119887119899 sub [0 1] and sum

infin

119899=0119887119899= infin

(ii) either lim sup119899rarrinfin

119888119899le 0 or suminfin

119899=0|119887119899119888119899| lt infin

Then lim119899rarrinfin

119886119899= 0

Lemma 4 In a smooth Banach space 119883 there holds theinequality

1199092

+ 2 ⟨119910 119869 (119909)⟩ le1003817100381710038171003817119909 + 119910

1003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883

(17)

where 119869 is the normalized duality mapping of 119883

Let 120583 be a mean if 120583 is a continuous linear functional on119897infin satisfying 120583 = 1 = 120583(1) Then we know that 120583 is a meanon N if and only if

inf 119886119899 119899 isin N le 120583 (119886) le sup 119886

119899 119899 isin N (18)

for every 119886 = (1198861 1198862 ) isin 119897

infin According to time and circu-mstances we use 120583

119899(119886119899) instead of 120583(119886) A mean 120583 on N is

called a Banach limit if and only if

120583119899(119886119899) = 120583119899(119886119899+1

) (19)

for every 119886 = (1198861 1198862 ) isin 119897

infin We know that if 120583 is a Banachlimit then

lim inf119899rarrinfin

119886119899le 120583119899(119886119899) le lim sup119899rarrinfin

119886119899 (20)

for every 119886 = (1198861 1198862 ) isin 119897

infin So if 119886 = (1198861 1198862 ) 119887 =

(1198871 1198872 ) isin 119897

infin and 119886119899

rarr 119888 (resp 119886119899minus119887119899

rarr 0) as 119899 rarr infinwe have

120583119899(119886119899) = 120583 (119886) = 119888 (resp 120583

119899(119886119899) = 120583119899(119887119899)) (21)

Further it is well known that there holds the followingresult

Lemma 5 (see [14]) Let119862 be a nonempty closed convex subsetof a uniformly smooth Banach space 119883 Let 119909

119899 be a bounded

sequence of 119883 let 120583 be a mean on N and let 119911 isin 119862 Then

120583119899

1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817

2

= min119910isin119862

120583119899

1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817

2

(22)

if and only if

120583119899⟨119910 minus 119911 119869 (119909

119899minus 119911)⟩ le 0 forall119910 isin 119862 (23)


Lemma 6 (see [9 Lemma 26]) Let 119862 be a nonempty closedconvex subset of a real Banach space 119883 which has uniformlyGateaux differentiable norm Let 119879 119862 rarr 119862 be a continuouspseudocontractive mapping with Fix(119879) = 0 and let 119891 119862 rarr

119862 be a fixed Lipschitzian strongly pseudocontractive mappingwith pseudocontractive coefficient 120573 isin (0 1) and Lipschitzianconstant 119871 gt 0 Let 119860 119862 rarr 119862 be a 120574-strongly positive linearbounded operator with coefficient 120574 gt 0 Assume that 119862 plusmn 119862 sub

119862 and that 119909119905 converges strongly to 119901 isin Fix(119879) as 119905 rarr 0

where 119909119905is defined by 119909

119905= 119905119891(119909

119905) + (119868 minus 119905119860)119879119909

119905 Suppose that

119909119899 sub 119862 is bounded and that lim

119899rarrinfin119909119899minus 119879119909119899 = 0 Then

lim sup119899rarrinfin

⟨(119891 minus 119860)119901 119869(119909119899minus 119901)⟩ le 0

Lemma 7 Let 119862 be a nonempty closed convex subset of a realsmooth Banach space 119883 Let 119865 119862 rarr 119883 be an 120572-stronglyaccretive and 120582-strictly pseudocontractive with 120572 + 120582 ge 1Then 119868 minus119865 is nonexpansive and 119865 is Lipschitz continuous withconstant (1+radic(1 minus 120572)120582) Further for any fixed 120591 isin (0 1) 119868minus

120591119865 is contractive with coefficient 1 minus 120591(1 minus radic(1 minus 120572)120582)

Proof From the 120582-strictly pseudocontractivity and 120572-strongly accretivity of 119865 we have for all 119909 119910 isin 119862

1205821003817100381710038171003817(119868 minus 119865) 119909 minus (119868 minus 119865) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

minus ⟨119865 (119909) minus 119865 (119910) 119869 (119909 minus 119910)⟩

le (1 minus 120572)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(24)

which implies that

1003817100381710038171003817(119868 minus 119865) 119909 minus (119868 minus 119865) 1199101003817100381710038171003817 le radic

1 minus 120572

120582

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (25)

Because 120572 + 120582 ge 1 hArr radic(1 minus 120572)120582 le 1 we know that 119868 minus 119865 isnonexpansive Also note that

1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le

1003817100381710038171003817(119868 minus 119865) 119909 minus (119868 minus 119865) 1199101003817100381710038171003817 +

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le (1 + radic1 minus 120572

120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(26)

Now take a fixed 120591 isin (0 1) arbitrarily Observe that for all119909 119910 isin 119862


=1003817100381710038171003817(1 minus 120591) (119909 minus 119910) + 120591 [(119868 minus 119865) 119909 minus (119868 minus 119865) 119910]

1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 1205911003817100381710038171003817(119868 minus 119865) 119909 minus (119868 minus 119865) 119910

1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 120591(radic1 minus 120572

120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

= (1 minus 120591(1 minus radic1 minus 120572

120582))

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(27)

Because 120572 + 120582 gt 1 hArr radic(1 minus 120572)120582 lt 1 we know that 119868 minus 120591119865 iscontractive with coefficient 1 minus 120591(1 minus radic(1 minus 120572)120582)


Let 119863 be a subset of 119862 and let Π be a mapping of 119862 into119863 Then Π is said to be sunny if

Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (28)

whenever Π(119909) + 119905(119909 minus Π(119909)) isin 119862 for 119909 isin 119862 and 119905 ge 0 Amapping Π of 119862 into itself is called a retraction if Π2 = Π Ifa mappingΠ of 119862 into itself is a retraction thenΠ(119911) = 119911 forevery 119911 isin 119877(Π) where 119877(Π) is the range of Π A subset 119863 of119862 is called a sunny nonexpansive retract of 119862 if there exists asunny nonexpansive retraction from119862 onto119863The followinglemma concerns the sunny nonexpansive retraction

Lemma 8 (see [15]) Let119862 be a nonempty closed convex subsetof a real smooth Banach space 119883 Let 119863 be a nonempty subsetof 119862 LetΠ be a retraction of 119862 onto119863 Then the following areequivalent

(i) Π is sunny and nonexpansive(ii) Π(119909) minus Π(119910)

2

le ⟨119909 minus 119910 119869(Π(119909) minus Π(119910))⟩ for all119909 119910 isin 119862

(iii) ⟨119909 minus Π(119909) 119869(119910 minus Π(119909))⟩ le 0 for all 119909 isin 119862 119910 isin 119863

It is well known that if 119883 = 119867 is a Hilbert space thena sunny nonexpansive retraction Π

119862is coincident with the

metric projection from 119883 onto 119862 that is Π119862

= 119875119862 If 119862

is a nonempty closed convex subset of a strictly convex anduniformly smooth Banach space 119883 and if 119879 119862 rarr 119862 isa nonexpansive mapping with the fixed point set Fix(119879) = 0then the set Fix(119879) is a sunny nonexpansive retract of 119862

Lemma 9 Let 119862 be a nonempty closed convex subset of asmooth Banach space 119883 Let Π

119862be a sunny nonexpansive

retraction from119883 onto119862 and let 1198611 1198612 119862 rarr 119883 be nonlinear

mappings For given 119909lowast

119910lowast

isin 119862 (119909lowast

119910lowast

) is a solution ofGSVI (13) if and only if 119909lowast = Π

119862(119910lowast

minus 12058311198611119910lowast

) where 119910lowast

=

Π119862(119909lowast

minus 12058321198612119909lowast

)

Proof We can rewrite GSVI (13) as

⟨119909lowast

minus (119910lowast

minus 12058311198611119910lowast

) 119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨119910lowast

minus (119909lowast

minus 12058321198612119909lowast

) 119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(29)

which is obviously equivalent to

119909lowast

= Π119862(119910lowast

minus 12058311198611119910lowast

)

119910lowast

= Π119862(119909lowast

minus 12058321198612119909lowast

)

(30)

because of Lemma 8 This completes the proof

In terms of Lemma 9 define the mapping 119866 119862 rarr 119862 asfollows

119866 (119909) = Π119862(119868 minus 120583

11198611)Π119862(119868 minus 120583

21198612) 119909 forall119909 isin 119862 (31)

Then we observe that

119909lowast

= Π119862[Π119862(119909lowast

minus 12058321198612119909lowast

) minus 12058311198611Π119862(119909lowast

minus 12058321198612119909lowast

)]

(32)

which implies that 119909lowast is a fixed point of the mapping 119866

Throughout this paper the set of fixed points of the mapping119866 is denoted by Ω

Lemma 10 (see [16]) Let 119862 be a nonempty closed convexsubset of a strictly convex Banach space 119883 Let 119879

119899infin

119899=0

be a sequence of nonexpansive mappings on 119862 Suppose⋂infin

119899=0Fix(119879119899) is nonempty Let 120582

119899 be a sequence of positive

numbers with suminfin

119899=0120582119899

= 1 Then a mapping 119879 on 119862 definedby 119879119909 = sum

infin

119899=0120582119899119879119899119909 for 119909 isin 119862 is well-defined nonexpansive

and Fix(119879) = ⋂infin

119899=0Fix(119879119899) holds

Lemma 11 (see [17]) Let119862 be a nonempty closed convex subsetof a Banach space 119883 Let 119878

1 1198782 be a sequence of mappings

of119862 into itself Suppose thatsuminfin119899=1

sup119878119899+1

119909minus119878119899119909 119909 isin 119862 lt

infinThen for each119910 isin 119862 119878119899119910 converges strongly to some point

of 119862 Moreover let 119878 be a mapping of 119862 into itself defined by119878119910 = lim

119899rarrinfin119878119899119910 for all 119910 isin 119862 Then lim

119899rarrinfinsup119878119909 minus

119878119899119909 119909 isin 119862 = 0

3 GSVI with Hierarchical FixedPoint Problem Constraint fora Nonexpansive Mapping

In this section we introduce our hybrid implicit viscosityscheme for solving theGSVI (13) with hierarchical fixed pointproblem constraint for a nonexpansivemapping and show thestrong convergence theorem First we list several useful andhelpful lemmas

Lemma 12 (see [10 Lemma 28]) Let 119862 be a nonempty closedconvex subset of a real 2-uniformly smooth Banach space 119883Let the mapping 119861

119894 119862 rarr 119883 be 120572

119894-inverse-strongly accretive

Then one has

1003817100381710038171003817(119868 minus 120583119894119861119894) 119909 minus (119868 minus 120583

119894119861119894) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120583119894(1205831198941205812

minus 120572119894)1003817100381710038171003817119861119894119909 minus 119861

1198941199101003817100381710038171003817

2

forall119909 119910 isin 119862

(33)

for 119894 = 1 2 where 120583119894gt 0 In particular if 0 lt 120583

119894le 1205721198941205812

(where 120581 is the best constant of119883 as in Lemma 1) then 119868minus120583119894119861119894

is nonexpansive for 119894 = 1 2

Lemma 13 (see [10 Lemma 29]) Let 119862 be a nonempty closedconvex subset of a real 2-uniformly smooth Banach space 119883LetΠ119862be a sunny nonexpansive retraction from119883 onto 119862 Let

the mapping 119861119894 119862 rarr 119883 be 120572

119894-inverse-strongly accretive for

119894 = 1 2 Let 119866 119862 rarr 119862 be the mapping defined by

119866119909 = Π119862[Π119862(119909 minus 120583

21198612119909) minus 120583

11198611Π119862(119909 minus 120583

21198612119909)]


(34)

If 0 lt 120583119894le 1205721198941205812 for 119894 = 1 2 then 119866 119862 rarr 119862 is none-

xpansive

Lemma 14 (see [18]) Let119883 be a Banach space 119862 a nonemptyclosed and convex subset of 119883 and 119879 119862 rarr 119862 a continuous


and strong pseudocontractionThen119879 has a unique fixed pointin 119862

Lemma 15 (see [19]) Assume that 119860 is a strongly positive lin-ear bounded operator on a smooth Banach space119883with coeffi-cient 120574 gt 0 and 0 lt 120588 le 119860

minus1 Then 119868 minus 1205881198602

le 1 minus 120588120574

We now state and prove our first result

Theorem 16 Let 119862 be a nonempty closed convex subset ofa uniformly convex and 2-uniformly smooth Banach space 119883

such that 119862 plusmn 119862 sub 119862 Let Π119862be a sunny nonexpansive

retraction from 119883 onto 119862 Let the mapping 119861119894 119862 rarr 119883 be

120572119894-inverse-strongly accretive for 119894 = 1 2 Let 119879 119862 rarr 119862

be a nonexpansive mapping such that Λ = Fix(119879) cap Ω = 0

where Ω is the fixed point set of the mapping 119866 = Π119862(119868 minus

12058311198611)Π119862(119868 minus 120583

21198612) with 0 lt 120583

119894lt 1205721198941205812 for 119894 = 1 2 Let

119891 119862 rarr 119862 be a fixed Lipschitzian strongly pseudocontractivemapping with pseudocontractive coefficient 120573 isin (0 1) andLipschitzian constant 119871 gt 0 let 119865 119862 rarr 119862 be 120572-stronglyaccretive and 120582-strictly pseudocontractive with 120572 + 120582 gt 1 andlet119860 119862 rarr 119862 be a 120574-strongly positive linear bounded operatorwith 0 lt 120574 minus 120573 le 1 Let 119909

119905 be defined by

119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (35)

where 120579119905 119905 isin (0 1) sub [0 1) with lim

119905rarr0120579119905119905 = 0 Then as

119905 rarr 0 119909119905 converges strongly to a point 119901 isin Λ which is the

unique solution in Λ to the VIP

⟨(119860 minus 119891) 119901 119869 (119901 minus 119906)⟩ le 0 forall119906 isin Λ (36)

Proof First let us show that the net 119909119905 is defined well As a

matter of fact define the mapping 119878119905 119862 rarr 119862 as follows

119878119905119909 = 119905119891 (119909) + (119868 minus 119905119860) [119866 (119879119909) minus 120579

119905119865119866 (119879119909)] forall119909 isin 119862

(37)

We may assume without loss of generality that 119905 le 119860minus1

Utilizing Lemmas 7 13 and 15 we have

⟨119878119905119909 minus 119878119905119910 119869 (119909 minus 119910)⟩

= 119905 ⟨119891 (119909) minus 119891 (119910) 119869 (119909 minus 119910)⟩

+ ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)]

119869 (119909 minus 119910)⟩

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)

times1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574) (1 minus 120579119905(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909) minus 119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

= (1 minus 119905 (120574 minus 120573))1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(38)

Hence it is known that 119878119905

119862 rarr 119862 is a continuous andstrongly pseudocontractive mapping with pseudocontractivecoefficient 1minus119905(120574minus120573) isin (0 1)Thus by Lemma 14 we deducethat there exists a unique fixed point in 119862 denoted by 119909

119905

which uniquely solves the fixed point equation

119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (39)

Let us show the uniqueness of the solution of VIP (36)Suppose that both 119901

1isin Λ and 119901

2isin Λ are solutions to VIP

(36) Then we have

⟨(119860 minus 119891) 1199011 119869 (1199011minus 1199012)⟩ le 0

⟨(119860 minus 119891) 1199012 119869 (1199012minus 1199011)⟩ le 0

(40)

Adding up the above two inequalities we obtain

⟨(119860 minus 119891) 1199011minus (119860 minus 119891) 119901

2 119869 (1199011minus 1199012)⟩ le 0 (41)

Note that


2 119869 (1199011minus 1199012)⟩

= ⟨119860 (1199011minus 1199012) 119869 (119901

1minus 1199012)⟩

minus ⟨119891 (1199011) minus 119891 (119901

2) 119869 (119901

1minus 1199012)⟩

ge 12057410038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

minus 12057310038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

= (120574 minus 120573)10038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

ge 0

(42)

Consequently we have1199011= 1199012 and the uniqueness is proved

Next let us show that for some 119886 isin (0 1) 119909119905 119905 isin (0 119886]

is bounded Indeed since 120579119905

119905 isin (0 1) sub [0 1) withlim119905rarr0

(120579119905119905) = 0 there exists some 119886 isin (0 1) such that

0 le 120579119905119905 lt 1 for all 119905 isin (0 119886] Take a fixed 119901 isin Fix(Λ)

arbitrarily Utilizing Lemma 7 we have

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

= ⟨119905 (119891 (119909119905) minus 119891 (119901)) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901]

minus119905 (119860119901 minus 119891 (119901)) 119869 (119909119905minus 119901)⟩

= 119905 ⟨119891 (119909119905) minus 119891 (119901) 119869 (119909

119905minus 119901)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119901] 119869 (119909

119905minus 119901)⟩

minus 119905 ⟨(119860 minus 119891) 119901 119869 (119909119905minus 119901)⟩


le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)

and hence for all 119905 isin (0 119886]

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)

Thus this implies that 119909119905 119905 isin (0 119886] is bounded and so are

119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]

Let us show that 119879119909119905minus 119866(119879119909

119905) rarr 0 as 119905 rarr 0

Indeed for simplicity we put 119902 = Π119862(119868minus12058321198612)119901119909119905= 119879119909119905

119906119905= Π119862(119868minus12058321198612)119909119905 and V

119905= Π119862(119868minus12058311198611)119906119905Then it is clear

that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from

(43) it follows that

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)

From Lemma 12 we have1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)

From the last two inequalities we obtain1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)

which together with (45) implies that1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 minus (1 minus 119905120574)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)

So it immediately follows that

(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)

Since 0 lt 120583119894lt 1205721198941205812 for 119894 = 1 2 we have

lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)

Utilizing Proposition 2 and Lemma 8 we have that thereexists 119892

1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le ⟨119909119905minus 12058321198612119909119905minus (119901 minus 120583

21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)

In the same way we derive that there exists 1198922

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)

Substituting (52) for (54) we get

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)

which together with (45) implies that

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)

Hence from (50) we conclude that

lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)

Utilizing the properties of 1198921and 119892

2 we get

lim119905rarr0

1003817100381710038171003817119909119905 minus 119906119905minus (119901 minus 119902)

1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)

which leads to1003817100381710038171003817119909119905 minus V

119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)

Note that 119909119905 119905 isin (0 119886] is bounded and so are 119891(119909

119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)

as 119905 rarr 0 Also observe that

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)

This together with (61) and (62) implies that

lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)

Utilizing the nonexpansivity of 119866 we obtain

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)

which together with (62) and (64) implies that

lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)

Now let 119905119896 be a sequence in (0 119886] that converges to 0 as 119896 rarr

infin and define a function 119892 on 119862 by

119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)

where 120583 is a Banach limit Define the set

119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping

119882119909 = (1 minus 120579) 119879119909 + 120579119866119909 forall119909 isin 119862 (69)

where 120579 is a constant in (0 1) Then by Lemma 10 we knowthat Fix(119882) = Fix(119879) cap Fix(119866) = Λ We observe that

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)

So from (64) and (66) we obtain

lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)

Since119883 is a uniformly smooth Banach space119870 is a nonemptybounded closed convex subset of119862 for more details see [14]We claim that 119870 is also invariant under the nonexpansivemapping 119882 Indeed noticing (71) we have for 119908 isin 119870

119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)

Since every nonempty closed bounded convex subset of auniformly smooth Banach space 119883 has the fixed point prop-erty for nonexpansive mappings and 119882 is a nonexpansivemapping of 119870 119882 has a fixed point in 119870 say 119901 UtilizingLemma 5 we get

120583119896⟨119909 minus 119901 119869 (119909

119905119896

minus 119901)⟩ le 0 forall119909 isin 119862 (73)

Putting 119909 = (119891 minus 119860)119901 + 119901 isin 119862 we have

120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896

) minus 119901] minus 119905119896(119860 minus 119891)119901 we get

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)

Since lim119896rarrinfin

(120579119905119896

119905119896) = 0 from (74) and the boundedness of

sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)

Therefore for the sequence 119909119905119896

in 119909119905

119905 isin (0 119886]there exists a subsequence which is still denoted by 119909

119905119896

thatconverges strongly to some fixed point 119901 of 119882

Now we claim that such a 119901 is the unique solution inΛ tothe VIP (36)

Indeed from (35) it follows that for all119906 isin Λ = Fix(119879)capΩ

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩ minus 119905 ⟨(119860 minus 119891) 119906 119869 (119909

119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [(1 minus 120579119905(1 minus radic

1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905 (120574 minus 120573))1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 minus 119905 ⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩

le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 minus 119905 ⟨(119860 minus 119891) 119906 119869 (119909

119905minus 119906)⟩

(78)

which hence implies that

⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 forall119906 isin Λ

(79)

Since 119909119905119896

rarr 119901 as 119905119896

rarr 0 and lim119905rarr0

(120579119905119905) = 0 we obtain

from the last inequality that


Utilizing the well-known Minty-type Lemma we get


So 119901 is a solution in Λ to the VIP (36)In order to prove that the net 119909

119905 119905 isin (0 119886] converges

strongly to 119901 as 119905 rarr 0 suppose that there exists anothersubsequence 119909

119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896

rarr 0then we also have 119902 isin Fix(119882) = Fix(119879) cap Ω = Λ due to(71) Repeating the same argument as above we know that119902 is another solution in Λ to the VIP (36) In terms of theuniqueness of solutions inΛ to the VIP (36) we immediatelyget 119901 = 119902 This completes the proof


Remark 17 It is worth emphasizing that in the assertion ofTheorem 16 ldquoas 119905 rarr 0 119909

119905 converges strongly to a point

119901 isin Λrdquo this 119901 depends on no one of the mappings 119891 119860 and119865 Indeed although 119909

119905 is defined by

119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)

in the proof ofTheorem 16 it can be readily seen that 119901 is firstfound out as a fixed point of the nonexpansive self-mapping119882 of119870This shows that119901 depends on no one of themappings119891 119860 and 119865

Remark 18 Theorem 16 improves extends supplements anddevelops Cai and Bu [9 Lemma 25] in the following aspects

(i) The GSVI (13) with hierarchical fixed point problemconstraint for a nonexpansive mapping is more general andmore subtle than the problem in Cai and Bu [9 Lemma 25]because our problem is to find a point 119901 isin Λ = Fix(119879) cap Ωwhich is the unique solution in Λ to the VIP


(ii) The iterative scheme in [9 Lemma 25] is extendedto develop the iterative scheme in Theorem 16 by virtueof hybrid steepest-descent method The iterative scheme inTheorem 16 is more advantageous and more flexible thanthe iterative scheme of [9 Lemma 25] because our iterativescheme involves solving two problems the GSVI (13) and thefixed point problem of a nonexpansive mapping 119879

(iii) The iterative scheme in Theorem 16 is very differentfrom the iterative scheme in [9 Lemma 25] because ouriterative scheme involves hybrid steepest-descent method(namely we add a strongly accretive and strictly pseudocon-tractive mapping 119865 in our iterative scheme) and because themapping 119879 in [9 Lemma 25] is replaced by the compositemapping 119866 ∘ 119879 in the iterative scheme of Theorem 16

(iv) The argument techniques of Theorem 16 are verydifferent from Cai and Bursquos ones of [9 Lemma 25] Becausethe composite mapping 119866 ∘ 119879 appears in the iterativescheme of Theorem 16 the proof of Theorem 16 dependson the argument techniques in [18] the inequality in 2-uniformly smooth Banach spaces (see Lemma 1) the inequal-ity in smooth and uniform convex Banach spaces (seeProposition 2) and the properties of the strongly positivelinear bounded operator (see Lemmas 15) the Banach limit(see Lemma 5) and the strongly accretive and strictly pseu-docontractive mapping (see Lemma 7)

4 GSVI with Hierarchical Fixed PointProblem Constraint for a Countable Familyof Nonexpansive mappings

In this section we propose our hybrid explicit viscosityscheme for solving theGSVI (13) with hierarchical fixed pointproblem constraint for a countable family of nonexpansivemappings and show the strong convergence theorem



retraction from 119883 onto 119862 Let the mapping 119861119894

119862 rarr 119883

be 120572119894-inverse-strongly accretive for 119894 = 1 2 Let 119878

119899infin

119899=0be an

infinite family of nonexpansive mappings of 119862 into itself suchthat Δ = ⋂

infin

119894=0Fix(119878119894) cap Ω = 0 where Ω is the fixed point

set of the mapping 119866 = Π119862(119868 minus 120583

11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894lt 1205721198941205812 for 119894 = 1 2 Let 119891 119862 rarr 119862 be a fixed contra-

ctive map with coefficient 120573 isin (0 1) let 119865 119862 rarr 119862 be 120572-strongly accretive and 120582-strictly pseudocontractive with 120572+120582 gt

1 and let 119860 119862 rarr 119862 be a 120574-strongly positive linear boundedoperator with 0 lt 120574 minus 120573 le 1 Given sequences 120582

119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin

119899=0in (0 1] suppose that there hold

the following conditions

(i) lim119899rarrinfin

120573119899= 0 and sum

infin

119899=0120573119899= infin

(ii) lim119899rarrinfin

(120582119899120583119899)120573119899= 0

(iii) 120572119899 sub [119886 119887] for some 119886 119887 isin (0 1)

(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin

Assume thatsuminfin119899=0

sup119909isin119863

119878119899+1

119909 minus 119878119899119909 lt infin for any bounded

subset 119863 of 119862 and let 119878 be a mapping of 119862 into itself definedby 119878119909 = lim

119899rarrinfin119878119899119909 for all 119909 isin 119862 and suppose that Fix(119878) =

⋂infin

119899=0Fix(119878119899) Then for any given point 119909

0isin 119862 the sequence

119909119899 generated by

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)

converges strongly to 119901 isin Δ which is the unique solution in Δ

to the VIP

⟨(119860 minus 119891) 119901 119869 (119901 minus 119906)⟩ le 0 forall119906 isin Δ (85)

Proof First let us show that 119909119899 is bounded Indeed taking

a fixed 119906 isin Δ arbitrarily we have

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909

119899minus 119906 for all 119899 ge 0 Taking into account

lim119899rarrinfin

(120582119899120583119899)120573119899

= 0 we may assume without loss of


generality that 120582119899120583119899

le 120573119899

le 119860minus1 for all 119899 ge 0 Thus by

Lemma 7 (ii) we have

1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times (1 minus 120582119899120583119899(1 minus radic

1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)

Thus 119909119899 is bounded and so is 119910

119899 Because 119866 and 119878

119899

are nonexpansive for all 119899 ge 0 119891 is contractive and 119865 isLipschitzian 119878

119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)

and 119865119866(119878119899119910119899) are bounded From conditions (i) and (ii) we

have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817 997888rarr 0 2 as 119899 997888rarr infin

(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin (90)

In order to prove (90) we estimate 119909119899+1

minus119909119899 first From (84)

we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)

Simple calculations show that

119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))

times (120572119899minus 120572119899minus1

)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)


In the meantime it follows from (84) that

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)

It follows from Lemma 7 (ii) and (93) that

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899minus1120583119899minus1 minus 120582

119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899minus1 minus 119878119899minus1

119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872

0gt 0 Since it follows from

conditions (i) and (iv) that suminfin119899=0

120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)

applying Lemma 3 to (96) we obtain from the assumption on119878119899 that

lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)

By condition (iii) and (84) we have1003817100381710038171003817119910119899 minus 119909

119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)

This together with (89)-(90) implies that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)

So we obtain1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)

Let 119906 isin Δ Now we show that lim119899rarrinfin

119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0

Indeed for simplicity put V = Π119862(119906 minus 120583

21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is

clear from (84) that1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)

Utilizing Lemma 12 we have

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)

Substituting (106) for (107) we obtain

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)

It immediately follows that

2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910

119899 are bounded and 0 lt 120583

119894lt 1205721198941205812 for 119894 =

1 2 we deduce from (101) and condition (iii) that

lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)

In the same way we derive that there exists 1198922such that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le ⟨119906119899minus 12058311198611119906119899minus (V minus 120583

11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V

119899 are bounded we deduce from

(101) (111) and condition (iii) that

lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119906119899minus (119906 minus V)1003817100381710038171003817 = 0

lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)

which hence yields1003817100381710038171003817119909119899 minus V

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0

as 119899 rarr infin

(121)

That is

lim119899rarrinfin

1003817100381710038171003817119878119899119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = lim119899rarrinfin

1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)

So from (104) and (122) we have

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)

which together with (104) and the assumption on 119878119899 implies

that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0

as 119899 997888rarr infin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping


where 120579 is a constant in (0 1) Then by Lemma 10 we knowthat Fix(119882) = Fix(119878) cap Fix(119866) = Δ We observe that

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)

So from (126) we get

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)

where 119901 is defined below Now we claim that


⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)

Indeed let 119909119905 be defined by

119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)

Then as 119905 rarr 0 119909119905 converges strongly to 119901 isin Fix(119882) = Δ

which by Proposition CB is the unique solution in Δ to theVIP


In terms of Lemma 6we conclude from (129) that (130) holdsIt is clear that


⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)

Finally let us show that 119909119899

rarr 119901 as 119899 rarr infin We observethat

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865) 119901 minus 120582

119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

times ((1 minus 120582119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817

times [2 (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)

Taking into account (133) and conditions (i) and (ii) weobtain that suminfin

119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)

Therefore applying Lemma 3 to (135) we infer that

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)

This completes the proof

Remark 20 It is worth pointing out that the sequences 120582119899

120583119899 and 120573

119899 can be taken which satisfy the conditions in

Theorem 19 As a matter of fact put 120582119899

= (1 + 119899)minus56 120583

119899=

1 and 120573119899

= (1 + 119899)minus23 for all 119899 ge 0 Then there hold the

following statements


120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1

minus 120573119899| lt infin suminfin

119899=0|120582119899+1

minus 120582119899| lt infin and

suminfin

119899=0|120583119899+1

minus 120583119899| lt infin

By the careful analysis of the proof ofTheorem 19 we canobtain the following result Because its proof is much simplerthan that of Theorem 19 we omit its proof




119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894

lt 1205721198941205812 for 119894 = 1 2 Let 119891 119862 rarr 119862 be a fixed

contractive map with coefficient 120573 isin (0 1) let 119865 119862 rarr 119862

be 120572-strongly accretive and 120582-strictly pseudocontractive with120572 + 120582 gt 1 and let 119860 119862 rarr 119862 be a 120574-strongly positive linearbounded operator with 0 lt 120574minus120573 le 1 Given sequences 120582

119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1

minus 120572119899| lt infin and sum

infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)

Remark 22 Theorems 19 and 21 improve extend supplementand develop Cai and Bu [10 Theorem 31] and Cai and Bu [9Theorems 31] in the following aspects

(i) The GSVI (13) with hierarchical fixed point problemconstraint for a countable family of nonexpansive mappingsis more general and more subtle than every problem in Caiand Bu [10 Theorems 31] and Cai and Bu [9 Theorem 31]


because our problem is to find a point119901 isin Δ = ⋂119899Fix(119878119899)capΩ

which is the unique solution in Δ to the VIP


(ii) The iterative scheme in [10 Theorem 31] is extendedto develop the iterative schemes in Theorems 19 and 21by virtue of hybrid steepest-descent method The iterativeschemes in Theorems 19 and 21 are more advantageous andmore flexible than the iterative scheme of [9 Theorem 31]because the iterative scheme of [9 Theorem 31] is implicitand our iterative schemes involve solving two problems theGSVI (13) and the fixed point problem of a countable familyof nonexpansive mappings 119878

119899

(iii) The iterative schemes inTheorems 19 and 21 are verydifferent from everyone in both [10 Theorem 31] and [9Theorem 31] because our iterative schemes involve hybridsteepest-descentmethod (namely we add a strongly accretiveand strictly pseudocontractive mapping 119865 in our iterativeschemes) and because themappings119866 and 119878

119899in [10Theorem

31] and the mapping 119878119899in [9 Theorem 31] are replaced by

the same composite mapping 119866 ∘ 119878119899in the iterative schemes

of Theorems 19 and 21(iv) Cai and Bursquos proof in [10 Theorem 31] depends

on the argument techniques in [20] the inequality in 2-uniformly smooth Banach spaces (see Lemma 1) and theinequality in smooth and uniform convex Banach spaces (seeProposition 2) Because the compositemapping119866∘119878

119899appears

in the iterative schemes in Theorems 19 and 21 the proofof Theorems 19 and 21 depends on the argument techniquesin [20] the inequality in 2-uniformly smooth Banach spaces(see Lemma 1) the inequality in smooth and uniform convexBanach spaces (see Proposition 2) and the properties of thestrongly positive linear bounded operator (see Lemmas 15)the Banach limit (see Lemma 5) and the strongly accretiveand strictly pseudocontractive mapping (see Lemma 7)

Remark 23 Theorems 19 and 21 extend and improveTheorem 16 of Yao et al [21] to a great extent in the followingaspects

(i) the 119906 is replaced by a fixed contractive mapping(ii) one continuous pseudocontractive mapping (includ-

ing nonexpansivemapping) is replaced by a countablefamily of nonexpansive mappings

(iii) we add a strongly positive linear bounded operator 119860and a strongly accretive and strictly pseudocontrac-tive mapping 119865 in our iterative algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This article was funded by theDeanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR for the technical

and financial support This research was partially supportedto first author by the National Science Foundation of China(11071169) Innovation Program of Shanghai Municipal Edu-cation Commission (09ZZ133) and PhD Program Foun-dation of Ministry of Education of China (20123127110002)Finally the authors thank the referees for their valuablecomments and suggestions for improvement of the paper

References

[1] H Iiduka and W Takahashi ldquoStrong convergence theoremsfor nonexpansive mappings and inverse-strongly monotonemappingsrdquo Nonlinear Analysis A vol 61 no 3 pp 341ndash3502005

[2] L-C Ceng Q H Ansari and J-C Yao ldquoAn extragradientmethod for solving split feasibility and fixed point problemsrdquoComputers amp Mathematics with Applications vol 64 no 4 pp633ndash642 2012

[3] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientmethods for finding minimum-norm solutions of the splitfeasibility problemrdquo Nonlinear Analysis A vol 75 no 4 pp2116ndash2125 2012

[4] L-C Ceng Q H Ansari and J-C Yao ldquoRelaxed extragradientiterative methods for variational inequalitiesrdquo Applied Mathe-matics and Computation vol 218 no 3 pp 1112ndash1123 2011

[5] L-C Ceng Q H Ansari and J-C Yao ldquoMann-type steepest-descent and modified hybrid steepest-descent methods forvariational inequalities in Banach spacesrdquoNumerical FunctionalAnalysis and Optimization vol 29 no 9-10 pp 987ndash1033 2008

[6] W Takahashi andM Toyoda ldquoWeak convergence theorems fornonexpansive mappings and monotone mappingsrdquo Journal ofOptimization Theory and Applications vol 118 no 2 pp 417ndash428 2003

[7] H Iiduka W Takahashi and M Toyoda ldquoApproximation ofsolutions of variational inequalities for monotone mappingsrdquoPanamerican Mathematical Journal vol 14 no 2 pp 49ndash612004

[8] Y Takahashi K Hashimoto and M Kato ldquoOn sharp uniformconvexity smoothness and strong type cotype inequalitiesrdquoJournal of Nonlinear and Convex Analysis vol 3 no 2 pp 267ndash281 2002

[9] G Cai and S Bu ldquoApproximation of common fixed pointsof a countable family of continuous pseudocontractions in auniformly smooth Banach spacerdquo Applied Mathematics Lettersvol 24 no 12 pp 1998ndash2004 2011

[10] G Cai and S Bu ldquoConvergence analysis for variational inequal-ity problems and fixed point problems in 2-uniformly smoothand uniformly convex Banach spacesrdquoMathematical and Com-puter Modelling vol 55 no 3-4 pp 538ndash546 2012

[11] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis A vol 16 no 12 pp 1127ndash1138 1991

[12] S Kamimura and W Takahashi ldquoStrong convergence of aproximal-type algorithm in a Banach spacerdquo SIAM Journal onOptimization vol 13 no 3 pp 938ndash945 2002

[13] H K Xu and T H Kim ldquoConvergence of hybrid steepest-descent methods for variational inequalitiesrdquo Journal of Opti-mization Theory and Applications vol 119 no 1 pp 185ndash2012003

[14] W Takahashi Nonlinear Functional AnalysismdashFixed Point The-ory and Its Applications Yokohama Publishers YokohamaJapan 2000 Japanese


[15] S Reich ldquoWeak convergence theorems for nonexpansive map-pings in Banach spacesrdquo Journal of Mathematical Analysis andApplications vol 67 no 2 pp 274ndash276 1979

[16] R E Bruck Jr ldquoProperties of fixed-point sets of nonexpansivemappings in Banach spacesrdquo Transactions of the AmericanMathematical Society vol 179 pp 251ndash262 1973

[17] K Aoyama Y Kimura W Takahashi andM Toyoda ldquoApprox-imation of common fixed points of a countable family ofnonexpansive mappings in a Banach spacerdquo Nonlinear AnalysisA vol 67 no 8 pp 2350ndash2360 2007

[18] L-C Ceng and J-C Yao ldquoAn extragradient-like approximationmethod for variational inequality problems and fixed pointproblemsrdquo Applied Mathematics and Computation vol 190 no1 pp 205ndash215 2007

[19] Y Censor A Gibali and S Reich ldquoTwo extensions of Kor-pelevichrsquos extragradient method for solving the variationalinequality problem in Euclidean spacerdquo Technical Report 2010

[20] L-C Ceng C-y Wang and J-C Yao ldquoStrong convergencetheorems by a relaxed extragradient method for a generalsystem of variational inequalitiesrdquo Mathematical Methods ofOperations Research vol 67 no 3 pp 375ndash390 2008

[21] Y Yao Y-C Liou and R Chen ldquoStrong convergence of aniterative algorithm for pseudocontractive mapping in BanachspacesrdquoNonlinear Analysis A vol 67 no 12 pp 3311ndash3317 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(iv) 120573-strongly pseudocontractive if for each 119909 119910 isin 119862there exists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that

⟨119860119909 minus 119860119910 119895 (119909 minus 119910)⟩ le 1205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2 (6)

for some 120573 isin (0 1)(v) 120582-strictly pseudocontractive if for each 119909 119910 isin 119862



1003817100381710038171003817

2

minus 1205821003817100381710038171003817119909 minus 119910 minus (119860119909 minus 119860119910)

1003817100381710038171003817

2

(7)

for some 120582 isin (0 1)It is worth emphasizing that the definition of the inverse-

strongly accretive mapping is based on that of the inverse-strongly monotone mapping which was studied by so manyauthors see for example [1ndash7]

A Banach space 119883 is said to be smooth if the limit

lim119905rarr0

1003817100381710038171003817119909 + 1199051199101003817100381710038171003817 minus 119909

119905

(8)

exists for all 119909 119910 isin 119883 in this case 119883 is also said tohave a Gateaux differentiable norm Moreover it is said tobe uniformly smooth if this limit is attained uniformly for119909 119910 isin 119880 in this case 119883 is also said to have a uniformly Fre-chet differentiable norm The norm of 119883 is said to be theFrechet differential if for each 119909 isin 119880 this limit is attaineduniformly for 119910 isin 119880 In the meantime we define a function120588 [0infin) rarr [0infin) called the modulus of smoothness of119883as follows

120588 (120591) = sup 1

2(1003817100381710038171003817119909 + 119910

1003817100381710038171003817 +1003817100381710038171003817119909 minus 119910

1003817100381710038171003817) minus 1 119909 119910 isin 119883

119909 = 11003817100381710038171003817119910

1003817100381710038171003817 = 120591

(9)

It is known that 119883 is uniformly smooth if and only iflim120591rarr0

120588(120591)120591 = 0 Let 119902 be a fixed real number with 1 lt 119902 le

2 Then a Banach space119883 is said to be 119902-uniformly smooth ifthere exists a constant 119888 gt 0 such that 120588(120591) le 119888120591

119902 for all 120591 gt 0As pointed out in [8] no Banach space is 119902-uniformly smoothfor 119902 gt 2 In addition it is also known that 119869 is single-valuedif and only if119883 is smooth whereas if119883 is uniformly smooththen the mapping 119869 is norm-to-norm uniformly continuouson bounded subsets of 119883

In a real smooth Banach space119883 we say that an operator119860 is strongly positive (see [9]) if there exists a constant 120574 gt 0

with the property

⟨119860119909 119869 (119909)⟩ ge 1205741199092

119886119868 minus 119887119860 = sup119909le1

|⟨(119886119868 minus 119887119860) 119909 119869 (119909)⟩|

119886 isin [0 1] 119887 isin [minus1 1]

(10)

where 119868 is the identity mapping

Proposition CB (see [9 Lemma 25]) Let 119862 be a nonemptyclosed convex subset of a uniformly smooth Banach space119883 Let

119879 119862 rarr 119862 be a continuous pseudocontractive mapping withFix(119879) = 0 and let 119891 119862 rarr 119862 be a fixed Lipschitzian stronglypseudocontractive mapping with pseudocontractive coefficient120573 isin (0 1) and Lipschitzian constant 119871 gt 0 Let 119860 119862 rarr 119862

be a strongly positive linear bounded operator with coefficient120574 gt 0 Assume that 119862 plusmn 119862 sub 119862 and 0 lt 120573 lt 120574 Let 119909

119905 be

defined by

119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) 119879119909

119905 (11)

Then as 119905 rarr 0 119909119905 converges strongly to some fixed point 119901

of 119879 such that 119901 is the unique solution in Fix(119879) to the VIP

⟨(119860 minus 119891) 119901 119869 (119901 minus 119906)⟩ le 0 forall119906 isin Fix (119879) (12)

On the other hand Cai and Bu [10] considered the follo-wing general system of variational inequalities (GSVI) ina real smooth Banach space 119883 which involves finding(119909lowast

119910lowast

) isin 119862 times 119862 such that

⟨12058311198611119910lowast

+ 119909lowast

minus 119910lowast

119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨12058321198612119909lowast

+ 119910lowast

minus 119909lowast

119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(13)

where 119862 is a nonempty closed and convex subset of 1198831198611 1198612

119862 rarr 119883 are two nonlinear mappings and 1205831and

1205832are two positive constants Here the set of solutions of

GSVI (13) is denoted by GSVI(119862 1198611 1198612) Very recently Cai

and Bu [10] constructed an iterative algorithm for solvingGSVI (13) and a common fixed point problem of an infinitefamily of nonexpansive mappings in a uniformly convex and2-uniformly smooth Banach space They proved the strongconvergence of the proposed algorithm by virtue of thefollowing inequality in a 2-uniformly smooth Banach space119883

Lemma 1 (see [11]) Let 119883 be a 2-uniformly smooth BanachspaceThen there exists a best smooth constant 120581 gt 0 such that

1003817100381710038171003817119909 + 1199101003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909)⟩ + 21003817100381710038171003817120581119910

1003817100381710038171003817 forall119909 119910 isin 119883

(14)

where 119869 is the normalized duality mapping from 119883 into 119883lowast

The authors [10] have used the following inequality in areal smooth and uniform convex Banach space 119883

Proposition 2 (see [12]) Let119883 be a real smooth and uniformconvex Banach space and let 119903 gt 0 Then there exists a strictlyincreasing continuous and convex function 119892 [0 2119903] rarr R119892(0) = 0 such that

119892 (1003817100381710038171003817119909 minus 119910

1003817100381710038171003817) le 1199092

minus 2 ⟨119909 119869 (119910)⟩ +1003817100381710038171003817119910

1003817100381710038171003817

2

forall119909 119910 isin 119861119903

(15)

where 119861119903= 119909 isin 119883 119909 le 119903

2 Preliminaries

We list some lemmas that will be used in the sequel Lemma 3can be found in [13] Lemma 4 is an immediate consequenceof the subdifferential inequality of the function (12) sdot

2



such that

119886119899+1


where 119887119899 and 119888



(i) 119887119899 sub [0 1] and sum

infin

119899=0119887119899= infin



119899=0|119887119899119888119899| lt infin


119886119899= 0


1199092

+ 2 ⟨119910 119869 (119909)⟩ le1003817100381710038171003817119909 + 119910

1003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883

(17)




119899 119899 isin N (18)

for every 119886 = (1198861 1198862 ) isin 119897




120583119899(119886119899) = 120583119899(119886119899+1

) (19)

for every 119886 = (1198861 1198862 ) isin 119897




119886119899 (20)

for every 119886 = (1198861 1198862 ) isin 119897

infin So if 119886 = (1198861 1198862 ) 119887 =

(1198871 1198872 ) isin 119897

infin and 119886119899

rarr 119888 (resp 119886119899minus119887119899


120583119899(119886119899) = 120583 (119886) = 119888 (resp 120583

119899(119886119899) = 120583119899(119887119899)) (21)



119899 be a bounded


120583119899

1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817

2

= min119910isin119862

120583119899

1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817

2

(22)

if and only if

120583119899⟨119910 minus 119911 119869 (119909







119905= 119905119891(119909

119905) + (119868 minus 119905119860)119879119909

119905 Suppose that




⟨(119891 minus 119860)119901 119869(119909119899minus 119901)⟩ le 0





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


le (1 minus 120572)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(24)

which implies that


1 minus 120572

120582

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (25)


1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le


1003817100381710038171003817119909 minus 1199101003817100381710038171003817


120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(26)




1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910


1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 120591(radic1 minus 120572

120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817


120582))

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(27)




Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (28)




2






= 119875119862 If 119862






119910lowast


119910lowast


119862(119910lowast

minus 12058311198611119910lowast


=

Π119862(119909lowast

minus 12058321198612119909lowast

)


⟨119909lowast

minus (119910lowast

minus 12058311198611119910lowast

) 119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨119910lowast

minus (119909lowast

minus 12058321198612119909lowast

) 119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(29)


119909lowast

= Π119862(119910lowast

minus 12058311198611119910lowast

)

119910lowast

= Π119862(119909lowast

minus 12058321198612119909lowast

)

(30)



119866 (119909) = Π119862(119868 minus 120583

11198611)Π119862(119868 minus 120583

21198612) 119909 forall119909 isin 119862 (31)


119909lowast

= Π119862[Π119862(119909lowast

minus 12058321198612119909lowast

) minus 12058311198611Π119862(119909lowast

minus 12058321198612119909lowast

)]

(32)




119899infin

119899=0





119899=0120582119899


infin



119899=0Fix(119879119899) holds




sup119878119899+1

119909minus119878119899119909 119909 isin 119862 lt





119878119899119909 119909 isin 119862 = 0




119894 119862 rarr 119883 be 120572


Then one has


119894119861119894) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120583119894(1205831198941205812

minus 120572119894)1003817100381710038171003817119861119894119909 minus 119861

1198941199101003817100381710038171003817

2

forall119909 119910 isin 119862

(33)


119894le 1205721198941205812







119866119909 = Π119862[Π119862(119909 minus 120583

21198612119909) minus 120583

11198611Π119862(119909 minus 120583

21198612119909)]


(34)


xpansive





minus1 Then 119868 minus 1205881198602

le 1 minus 120588120574








12058311198611)Π119862(119868 minus 120583

21198612) with 0 lt 120583

119894lt 1205721198941205812 for 119894 = 1 2 Let



119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (35)


119905rarr0120579119905119905 = 0 Then as






119878119905119909 = 119905119891 (119909) + (119868 minus 119905119860) [119866 (119879119909) minus 120579

119905119865119866 (119879119909)] forall119909 isin 119862

(37)



⟨119878119905119909 minus 119878119905119910 119869 (119909 minus 119910)⟩

= 119905 ⟨119891 (119909) minus 119891 (119910) 119869 (119909 minus 119910)⟩


119905119865)119866 (119879119910)]

119869 (119909 minus 119910)⟩

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)

times1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909) minus 119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1003817100381710038171003817

2

(38)



119905


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (39)


1isin Λ and 119901


(36) Then we have

⟨(119860 minus 119891) 1199011 119869 (1199011minus 1199012)⟩ le 0

⟨(119860 minus 119891) 1199012 119869 (1199012minus 1199011)⟩ le 0

(40)



2 119869 (1199011minus 1199012)⟩ le 0 (41)

Note that


2 119869 (1199011minus 1199012)⟩

= ⟨119860 (1199011minus 1199012) 119869 (119901

1minus 1199012)⟩

minus ⟨119891 (1199011) minus 119891 (119901

2) 119869 (119901

1minus 1199012)⟩

ge 12057410038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

minus 12057310038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

= (120574 minus 120573)10038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

ge 0

(42)








1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

= ⟨119905 (119891 (119909119905) minus 119891 (119901)) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901]


= 119905 ⟨119891 (119909119905) minus 119891 (119901) 119869 (119909

119905minus 119901)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119901] 119869 (119909

119905minus 119901)⟩



le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)


119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]


119905) rarr 0 as 119905 rarr 0


119906119905= Π119862(119868minus12058321198612)119909119905 and V


that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)


1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)


1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)


1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











such that

119886119899+1


where 119887119899 and 119888



(i) 119887119899 sub [0 1] and sum

infin

119899=0119887119899= infin



119899=0|119887119899119888119899| lt infin


119886119899= 0


1199092

+ 2 ⟨119910 119869 (119909)⟩ le1003817100381710038171003817119909 + 119910

1003817100381710038171003817

2

le 1199092

+ 2 ⟨119910 119869 (119909 + 119910)⟩ forall119909 119910 isin 119883

(17)




119899 119899 isin N (18)

for every 119886 = (1198861 1198862 ) isin 119897




120583119899(119886119899) = 120583119899(119886119899+1

) (19)

for every 119886 = (1198861 1198862 ) isin 119897




119886119899 (20)

for every 119886 = (1198861 1198862 ) isin 119897

infin So if 119886 = (1198861 1198862 ) 119887 =

(1198871 1198872 ) isin 119897

infin and 119886119899

rarr 119888 (resp 119886119899minus119887119899


120583119899(119886119899) = 120583 (119886) = 119888 (resp 120583

119899(119886119899) = 120583119899(119887119899)) (21)



119899 be a bounded


120583119899

1003817100381710038171003817119909119899 minus 1199111003817100381710038171003817

2

= min119910isin119862

120583119899

1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817

2

(22)

if and only if

120583119899⟨119910 minus 119911 119869 (119909







119905= 119905119891(119909

119905) + (119868 minus 119905119860)119879119909

119905 Suppose that




⟨(119891 minus 119860)119901 119869(119909119899minus 119901)⟩ le 0





1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


le (1 minus 120572)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

(24)

which implies that


1 minus 120572

120582

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (25)


1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le


1003817100381710038171003817119909 minus 1199101003817100381710038171003817


120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(26)




1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910


1003817100381710038171003817

le (1 minus 120591)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 120591(radic1 minus 120572

120582)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817


120582))

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

(27)




Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (28)




2






= 119875119862 If 119862






119910lowast


119910lowast


119862(119910lowast

minus 12058311198611119910lowast


=

Π119862(119909lowast

minus 12058321198612119909lowast

)


⟨119909lowast

minus (119910lowast

minus 12058311198611119910lowast

) 119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨119910lowast

minus (119909lowast

minus 12058321198612119909lowast

) 119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(29)


119909lowast

= Π119862(119910lowast

minus 12058311198611119910lowast

)

119910lowast

= Π119862(119909lowast

minus 12058321198612119909lowast

)

(30)



119866 (119909) = Π119862(119868 minus 120583

11198611)Π119862(119868 minus 120583

21198612) 119909 forall119909 isin 119862 (31)


119909lowast

= Π119862[Π119862(119909lowast

minus 12058321198612119909lowast

) minus 12058311198611Π119862(119909lowast

minus 12058321198612119909lowast

)]

(32)




119899infin

119899=0





119899=0120582119899


infin



119899=0Fix(119879119899) holds




sup119878119899+1

119909minus119878119899119909 119909 isin 119862 lt





119878119899119909 119909 isin 119862 = 0




119894 119862 rarr 119883 be 120572


Then one has


119894119861119894) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120583119894(1205831198941205812

minus 120572119894)1003817100381710038171003817119861119894119909 minus 119861

1198941199101003817100381710038171003817

2

forall119909 119910 isin 119862

(33)


119894le 1205721198941205812







119866119909 = Π119862[Π119862(119909 minus 120583

21198612119909) minus 120583

11198611Π119862(119909 minus 120583

21198612119909)]


(34)


xpansive





minus1 Then 119868 minus 1205881198602

le 1 minus 120588120574








12058311198611)Π119862(119868 minus 120583

21198612) with 0 lt 120583

119894lt 1205721198941205812 for 119894 = 1 2 Let



119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (35)


119905rarr0120579119905119905 = 0 Then as






119878119905119909 = 119905119891 (119909) + (119868 minus 119905119860) [119866 (119879119909) minus 120579

119905119865119866 (119879119909)] forall119909 isin 119862

(37)



⟨119878119905119909 minus 119878119905119910 119869 (119909 minus 119910)⟩

= 119905 ⟨119891 (119909) minus 119891 (119910) 119869 (119909 minus 119910)⟩


119905119865)119866 (119879119910)]

119869 (119909 minus 119910)⟩

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)

times1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909) minus 119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1003817100381710038171003817

2

(38)



119905


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (39)


1isin Λ and 119901


(36) Then we have

⟨(119860 minus 119891) 1199011 119869 (1199011minus 1199012)⟩ le 0

⟨(119860 minus 119891) 1199012 119869 (1199012minus 1199011)⟩ le 0

(40)



2 119869 (1199011minus 1199012)⟩ le 0 (41)

Note that


2 119869 (1199011minus 1199012)⟩

= ⟨119860 (1199011minus 1199012) 119869 (119901

1minus 1199012)⟩

minus ⟨119891 (1199011) minus 119891 (119901

2) 119869 (119901

1minus 1199012)⟩

ge 12057410038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

minus 12057310038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

= (120574 minus 120573)10038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

ge 0

(42)








1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

= ⟨119905 (119891 (119909119905) minus 119891 (119901)) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901]


= 119905 ⟨119891 (119909119905) minus 119891 (119901) 119869 (119909

119905minus 119901)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119901] 119869 (119909

119905minus 119901)⟩



le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)


119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]


119905) rarr 0 as 119905 rarr 0


119906119905= Π119862(119868minus12058321198612)119909119905 and V


that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)


1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)


1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)


1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Π [Π (119909) + 119905 (119909 minus Π (119909))] = Π (119909) (28)




2






= 119875119862 If 119862






119910lowast


119910lowast


119862(119910lowast

minus 12058311198611119910lowast


=

Π119862(119909lowast

minus 12058321198612119909lowast

)


⟨119909lowast

minus (119910lowast

minus 12058311198611119910lowast

) 119869 (119909 minus 119909lowast

)⟩ ge 0 forall119909 isin 119862

⟨119910lowast

minus (119909lowast

minus 12058321198612119909lowast

) 119869 (119909 minus 119910lowast

)⟩ ge 0 forall119909 isin 119862

(29)


119909lowast

= Π119862(119910lowast

minus 12058311198611119910lowast

)

119910lowast

= Π119862(119909lowast

minus 12058321198612119909lowast

)

(30)



119866 (119909) = Π119862(119868 minus 120583

11198611)Π119862(119868 minus 120583

21198612) 119909 forall119909 isin 119862 (31)


119909lowast

= Π119862[Π119862(119909lowast

minus 12058321198612119909lowast

) minus 12058311198611Π119862(119909lowast

minus 12058321198612119909lowast

)]

(32)




119899infin

119899=0





119899=0120582119899


infin



119899=0Fix(119879119899) holds




sup119878119899+1

119909minus119878119899119909 119909 isin 119862 lt





119878119899119909 119909 isin 119862 = 0




119894 119862 rarr 119883 be 120572


Then one has


119894119861119894) 119910

1003817100381710038171003817

2

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ 2120583119894(1205831198941205812

minus 120572119894)1003817100381710038171003817119861119894119909 minus 119861

1198941199101003817100381710038171003817

2

forall119909 119910 isin 119862

(33)


119894le 1205721198941205812







119866119909 = Π119862[Π119862(119909 minus 120583

21198612119909) minus 120583

11198611Π119862(119909 minus 120583

21198612119909)]


(34)


xpansive





minus1 Then 119868 minus 1205881198602

le 1 minus 120588120574








12058311198611)Π119862(119868 minus 120583

21198612) with 0 lt 120583

119894lt 1205721198941205812 for 119894 = 1 2 Let



119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (35)


119905rarr0120579119905119905 = 0 Then as






119878119905119909 = 119905119891 (119909) + (119868 minus 119905119860) [119866 (119879119909) minus 120579

119905119865119866 (119879119909)] forall119909 isin 119862

(37)



⟨119878119905119909 minus 119878119905119910 119869 (119909 minus 119910)⟩

= 119905 ⟨119891 (119909) minus 119891 (119910) 119869 (119909 minus 119910)⟩


119905119865)119866 (119879119910)]

119869 (119909 minus 119910)⟩

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)

times1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909) minus 119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1003817100381710038171003817

2

(38)



119905


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (39)


1isin Λ and 119901


(36) Then we have

⟨(119860 minus 119891) 1199011 119869 (1199011minus 1199012)⟩ le 0

⟨(119860 minus 119891) 1199012 119869 (1199012minus 1199011)⟩ le 0

(40)



2 119869 (1199011minus 1199012)⟩ le 0 (41)

Note that


2 119869 (1199011minus 1199012)⟩

= ⟨119860 (1199011minus 1199012) 119869 (119901

1minus 1199012)⟩

minus ⟨119891 (1199011) minus 119891 (119901

2) 119869 (119901

1minus 1199012)⟩

ge 12057410038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

minus 12057310038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

= (120574 minus 120573)10038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

ge 0

(42)








1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

= ⟨119905 (119891 (119909119905) minus 119891 (119901)) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901]


= 119905 ⟨119891 (119909119905) minus 119891 (119901) 119869 (119909

119905minus 119901)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119901] 119869 (119909

119905minus 119901)⟩



le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)


119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]


119905) rarr 0 as 119905 rarr 0


119906119905= Π119862(119868minus12058321198612)119909119905 and V


that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)


1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)


1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)


1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus1 Then 119868 minus 1205881198602

le 1 minus 120588120574








12058311198611)Π119862(119868 minus 120583

21198612) with 0 lt 120583

119894lt 1205721198941205812 for 119894 = 1 2 Let



119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (35)


119905rarr0120579119905119905 = 0 Then as






119878119905119909 = 119905119891 (119909) + (119868 minus 119905119860) [119866 (119879119909) minus 120579

119905119865119866 (119879119909)] forall119909 isin 119862

(37)



⟨119878119905119909 minus 119878119905119910 119869 (119909 minus 119910)⟩

= 119905 ⟨119891 (119909) minus 119891 (119910) 119869 (119909 minus 119910)⟩


119905119865)119866 (119879119910)]

119869 (119909 minus 119910)⟩

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)

times1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909) minus (119868 minus 120579

119905119865)119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909) minus 119866 (119879119910)

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

le 1199051205731003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2


1003817100381710038171003817

2

(38)



119905


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] (39)


1isin Λ and 119901


(36) Then we have

⟨(119860 minus 119891) 1199011 119869 (1199011minus 1199012)⟩ le 0

⟨(119860 minus 119891) 1199012 119869 (1199012minus 1199011)⟩ le 0

(40)



2 119869 (1199011minus 1199012)⟩ le 0 (41)

Note that


2 119869 (1199011minus 1199012)⟩

= ⟨119860 (1199011minus 1199012) 119869 (119901

1minus 1199012)⟩

minus ⟨119891 (1199011) minus 119891 (119901

2) 119869 (119901

1minus 1199012)⟩

ge 12057410038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

minus 12057310038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

= (120574 minus 120573)10038171003817100381710038171199011 minus 119901

2

1003817100381710038171003817

2

ge 0

(42)








1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

= ⟨119905 (119891 (119909119905) minus 119891 (119901)) + (119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901]


= 119905 ⟨119891 (119909119905) minus 119891 (119901) 119869 (119909

119905minus 119901)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119901] 119869 (119909

119905minus 119901)⟩



le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)


119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]


119905) rarr 0 as 119905 rarr 0


119906119905= Π119862(119868minus12058321198612)119909119905 and V


that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)


1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)


1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)


1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905) minus 119901

1003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)

times [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865)119866 (119879119901)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119901) minus 119901

1003817100381710038171003817]1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119879119909119905 minus 119879119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(43)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 le

1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +120579119905

119905

10038171003817100381710038171198651199011003817100381710038171003817)

le1

120574 minus 120573(1003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817 +1003817100381710038171003817119865119901

1003817100381710038171003817)

(44)


119891(119909119905) 119905 isin (0 119886] 119879119909

119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886]


119905) rarr 0 as 119905 rarr 0


119906119905= Π119862(119868minus12058321198612)119909119905 and V


that119901 = Π119862(119868minus12058311198611)119902 and V

119905= 119866(119909

119905) = 119866(119879119909

119905) Hence from


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2


1 minus 120572

120582))

times1003817100381710038171003817119866 (119879119909

119905) minus 119866 (119879119901)

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574) 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 119905

1003817100381710038171003817(119860 minus 119891) 1199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(45)


1003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901 minus 120583

2(1198612119909119905minus 1198612119901)

1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862(119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902 minus 120583

1(1198611119906119905minus 1198611119902)

1003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(46)


1003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

(47)


1003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

+ (1 minus 119905120574)1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

] + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(48)


(1 minus 119905120574) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119905 minus 119861

21199011003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119905 minus 119861

11199021003817100381710038171003817

2

]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(49)


lim119905rarr0

10038171003817100381710038171198612119909119905 minus 11986121199011003817100381710038171003817 = 0 lim

119905rarr0

10038171003817100381710038171198611119906119905 minus 11986111199021003817100381710038171003817 = 0 (50)


1such that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119909119905 minus 120583

21198612119909119905) minus Π119862(119901 minus 120583

21198612119901)

1003817100381710038171003817

2


21198612119901) 119869 (119906

119905minus 119902)⟩

= ⟨119909119905minus 119901 119869 (119906

119905minus 119902)⟩ + 120583

2⟨1198612119901 minus 1198612119909119905 119869 (119906119905minus 119902)⟩

le1

2[1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(51)

which implies that

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

(52)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119905 minus 120583

11198611119906119905) minus Π119862(119902 minus 120583

11198611119902)

1003817100381710038171003817

2


11198611119902) 119869 (V

119905minus 119901)⟩

= ⟨119906119905minus 119902 119869 (V

119905minus 119901)⟩ + 120583

1⟨1198611119902 minus 1198611119906119905 119869 (V119905minus 119901)⟩

le1

2[1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) ]

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(53)

which implies that

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119906119905 minus 119902

1003817100381710038171003817

2

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(54)


1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

2

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

(55)


1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

2

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ (1 minus 119905120574)1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

times1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817 + 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

2(1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817V119905 minus 119901

1003817100381710038171003817

2

) + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le 1199051205731003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + (1 minus 119905120574)

times1

21003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 21205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817

+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817


1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

minus (1 minus 119905120574)

times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+ 1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)] + (1 minus 119905120574)

times [1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817]


+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119901

1003817100381710038171003817

2

+ 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817


times1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(56)


(1 minus 119905120574)1

2[1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817)]

le 1199051003817100381710038171003817(119860 minus 119891) 119901

1003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817 + 1205832

10038171003817100381710038171198612119901 minus 1198612119909119905

1003817100381710038171003817

1003817100381710038171003817119906119905 minus 1199021003817100381710038171003817

+ 1205831

10038171003817100381710038171198611119902 minus 1198611119906119905

1003817100381710038171003817

1003817100381710038171003817V119905 minus 1199011003817100381710038171003817 + 120579119905

10038171003817100381710038171198651199011003817100381710038171003817

1003817100381710038171003817119909119905 minus 1199011003817100381710038171003817

(57)


lim119905rarr0

1198921(1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817) = 0

lim119905rarr0

1198922(1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817) = 0

(58)


2 we get

lim119905rarr0


1003817100381710038171003817 = 0

lim119905rarr0

1003817100381710038171003817119906119905 minus V119905+ (119901 minus 119902)

1003817100381710038171003817 = 0

(59)


119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119906

119905minus (119901 minus 119902)

1003817100381710038171003817

+1003817100381710038171003817119906119905 minus V

119905+ (119901 minus 119902)

1003817100381710038171003817 997888rarr 0 as 119905 997888rarr 0

(60)

That is

lim119905rarr0

1003817100381710038171003817119879119909119905 minus 119866 (119879119909119905)1003817100381710038171003817 = lim119905rarr0

1003817100381710038171003817119909119905 minus V119905

1003817100381710038171003817 = 0 (61)


119905)

119905 isin (0 119886] 119879119909119905 119905 isin (0 119886] and 119866(119879119909

119905) 119905 isin (0 119886] Hence

we have1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817

= 119905

10038171003817100381710038171003817100381710038171003817

119891 (119909119905) minus 119860119866 (119879119909

119905) minus

120579119905

119905(119868 minus 119905119860) 119865119866 (119879119909

119905)

10038171003817100381710038171003817100381710038171003817

997888rarr 0

(62)


1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119879119909

119905

1003817100381710038171003817 (63)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119879119909119905

1003817100381710038171003817 = 0 (64)


1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119866 (119879119909119905) minus 119866119909

119905

1003817100381710038171003817

le1003817100381710038171003817119909119905 minus 119866 (119879119909

119905)1003817100381710038171003817 +

1003817100381710038171003817119879119909119905 minus 119909119905

1003817100381710038171003817

(65)


lim119905rarr0

1003817100381710038171003817119909119905 minus 119866119909119905

1003817100381710038171003817 = 0 (66)



119892 (119909) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990910038171003817100381710038171003817

2

forall119909 isin 119862 (67)


119870 = 119908 isin 119862 119892 (119908) = min 119892 (119910) 119910 isin 119862 (68)

and the mapping



1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119905minus 119879119909119905) + 120579 (119909

119905minus 119866119909119905)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119905 minus 119879119909

119905

1003817100381710038171003817 + 1205791003817100381710038171003817119909119905 minus 119866119909

119905

1003817100381710038171003817

(70)


lim119899rarrinfin

1003817100381710038171003817119909119905 minus 119882119909119905

1003817100381710038171003817 = 0 (71)


119892 (119882119908) = 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11988211990810038171003817100381710038171003817

2

= 120583119896

1

2

10038171003817100381710038171003817119882119909119905119896

minus 11988211990810038171003817100381710038171003817

2

le 120583119896

1

2

10038171003817100381710038171003817119909119905119896

minus 11990810038171003817100381710038171003817

2

= 119892 (119908)

(72)


120583119896⟨119909 minus 119901 119869 (119909

119905119896



120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896



Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 119909119905119896

minus 119901 = 119905119896(119891(119909119905119896

) minus 119891(119901)) + (119868 minus 119905119896119860)[119866(119879119909

119905119896

) minus 120579119905119896

119865119866(119879119909119905119896


10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

= 119905119896⟨119891 (119909

119905119896

) minus 119891 (119901) 119869 (119909119905119896

minus 119901)⟩

+ ⟨(119868 minus 119905119896119860) (119866 (119879119909

119905119896

) minus 119901) 119869 (119909119905119896

minus 119901)⟩

minus 120579119905119896

⟨(119868 minus 119905119896119860)119865119866 (119879119909

119905119896

) 119869 (119909119905119896

minus 119901)⟩

minus 119905119896⟨(119860 minus 119891) 119901 119869 (119909

119905119896

minus 119901)⟩

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119866 (119879119909

119905119896

) minus 11990110038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

+ (1 minus 119905119896120574) 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

le 11990511989612057310038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 119905119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+ (1 minus 119905119896120574)

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

+ 120579119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

(75)

It follows that

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817

2

le1

120574 minus 120573[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

(76)


(120579119905119896


sequences 119865119866(119879119909119905119896

) 119909119905119896

it follows that

120583119896

10038171003817100381710038171003817119909119905119896

minus11990110038171003817100381710038171003817

2

le1

120574 minus 120573120583119896[⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817]

=1

120574 minus 120573[120583119896⟨(119891 minus 119860) 119901 119869 (119909

119905119896

minus 119901)⟩

+120583119896(

120579119905119896

119905119896

10038171003817100381710038171003817119865119866 (119879119909

119905119896

)10038171003817100381710038171003817

10038171003817100381710038171003817119909119905119896

minus 11990110038171003817100381710038171003817)] le 0

(77)


in 119909119905


119905119896




1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

2

= 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909

119905minus 119906)⟩

+ ⟨(119868 minus 119905119860) [119866 (119879119909119905) minus 120579119905119865119866 (119879119909

119905) minus 119906] 119869 (119909

119905minus 119906)⟩


= ⟨(119868 minus 119905119860) [(119868 minus 120579119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

+ (119868 minus 120579119905119865) 119906 minus 119906] 119869 (119909

119905minus 119906)⟩

+ 119905 ⟨119891 (119909119905) minus 119891 (119906) 119869 (119909


119905minus 119906)⟩

le (1 minus 119905120574) [1003817100381710038171003817(119868 minus 120579

119905119865)119866 (119879119909

119905) minus (119868 minus 120579

119905119865) 119906

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120579

119905119865) 119906 minus 119906

1003817100381710038171003817]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1 minus 120572

120582))

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817 + 120579119905119865119906]

times1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2


le (1 minus 119905120574) [1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817 + 120579119905119865119906]

1003817100381710038171003817119909119905 minus 1199061003817100381710038171003817

+ 1199051205731003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2



1003817100381710038171003817

2

+ 120579119905119865119906

times1003817100381710038171003817119909119905 minus 119906


le1003817100381710038171003817119909119905 minus 119906

1003817100381710038171003817

2

+ 120579119905119865119906


119905minus 119906)⟩

(78)


⟨(119860 minus 119891) 119906 119869 (119909119905minus 119906)⟩ le

120579119905

119905119865119906


(79)

Since 119909119905119896

rarr 119901 as 119905119896


(120579119905119905) = 0 we obtain






119905 119905 isin (0 119886] converges


119904119896

sub 119909119905 such that 119909

119904119896

rarr 119902 as 119904119896







119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119909119905= 119905119891 (119909

119905)+(119868 minus 119905119860) [119866 (119879119909

119905) minus 120579119905119865119866 (119879119909

119905)] forall119905 isin (0 1)

(82)













119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with




119899infin

119899=0 120583119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0


(iv) suminfin

119899=0(|120572119899+1

minus120572119899|+|120573119899+1

minus120573119899|+|120582119899+1

minus120582119899|+|120583119899+1

minus120583119899|) lt

infin


sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

forall119899 ge 0

(84)


to the VIP




1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817 =

1003817100381710038171003817120572119899119909119899 + (1 minus 120572119899) 119866 (119878

119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119878119899119909119899 minus 119906

1003817100381710038171003817

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817 + (1 minus 120572

119899)1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 =1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

(86)

So 119910119899minus 119906 le 119909


lim119899rarrinfin

(120582119899120583119899)120573119899



generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










generality that 120582119899120583119899

le 120573119899



1003817100381710038171003817119909119899+1 minus 1199061003817100381710038171003817

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119906)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119906] minus 120573

119899(119860 minus 119891) 119906

1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus119906]

1003817100381710038171003817

+ 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

times [1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865)119866 (119878

119899119906)

1003817100381710038171003817

+1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119906) minus 119906

1003817100381710038171003817] + 120573119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899119906)

1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119878119899119910119899 minus 119878

1198991199061003817100381710038171003817 + 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

+ 120582119899120583119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

le 1205731198991205731003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899119865119906 + 120573

119899

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

+ 120573119899(120574 minus 120573)

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

le max1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573

(87)

By induction

1003817100381710038171003817119909119899minus1199061003817100381710038171003817 le max

10038171003817100381710038171199090 minus 1199061003817100381710038171003817

1003817100381710038171003817(119860 minus 119891) 1199061003817100381710038171003817 + 119865119906

120574 minus 120573 forall119899 ge 0

(88)


119899 Because 119866 and 119878

119899


119899119909119899 119878119899119910119899 119866(119878

119899119909119899) 119866(119878

119899119910119899) 119891(119909

119899)


have

1003817100381710038171003817119909119899+1 minus 119866 (119878119899119910119899)1003817100381710038171003817

= 120573119899

1003817100381710038171003817(119891 (119909119899) minus 119860119866 (119878

119899119910119899)) + (119868 minus 120573

119899119860)

times (119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119866 (119878

119899119910119899))1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ (1 minus 120573119899120574) 120582119899120583119899

1003817100381710038171003817119865119866 (119878119899119910119899)1003817100381710038171003817

le 120573119899

1003817100381710038171003817119891 (119909119899) minus 119860119866 (119878

119899119910119899)1003817100381710038171003817

+ 120582119899120583119899


(89)

Now we claim that

1003817100381710038171003817119909119899+1 minus 119909119899




we have

119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119910119899minus1

= 120572119899minus1

119909119899minus1

+ (1 minus 120572119899minus1

) 119866 (119878119899minus1

119909119899minus1

)

(91)


119910119899minus 119910119899minus1

= (1 minus 120572119899) (119866 (119878

119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

))

+ 120572119899(119909119899minus 119909119899minus1

) + (119909119899minus1

minus 119866 (119878119899minus1

119909119899minus1

))


)

(92)

It follows that

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817

le (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899)1003817100381710038171003817119878119899119909119899 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119878119899119909119899 minus 119878119899119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le (1 minus 120572119899) (

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817)

+ 120572119899

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

le1003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

(93)



119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119866 (119878

119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899)]

119909119899= 120573119899minus1

119891 (119909119899minus1

) + (119868 minus 120573119899minus1

119860)

times [119866 (119878119899minus1

119910119899minus1

) minus 120582119899minus1

120583119899minus1

119865119866 (119878119899minus1

119910119899minus1

)]

(94)


119909119899+1

minus 119909119899

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)]

= (120573119899minus 120573119899minus1

) 119891 (119909119899minus1

) + 120573119899(119891 (119909119899) minus 119891 (119909

119899minus1))

+ (120573119899minus1

minus 120573119899) 119860 (119868 minus 120582

119899minus1120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)

+ (119868 minus 120573119899119860) [(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)

+ (120582119899minus1

120583119899minus1

minus 120582119899120583119899) 119865119866 (119878

119899minus1119910119899minus1

)]

(95)


1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119909

119899minus1)1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817(119868 minus 120582119899120583119899119865)119866 (119878

119899119910119899)

minus (119868 minus 120582119899120583119899119865)119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817


119899120583119899

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [(1 minus 120582

119899120583119899(1 minus radic

1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817 ]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899minus1

119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119878119899119910119899 minus 119878119899119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119910119899 minus 119910119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 + 1205731198991205731003817100381710038171003817119909119899 minus 119909

119899minus1

1003817100381710038171003817

+1003816100381610038161003816120573119899minus1 minus 120573

119899

1003816100381610038161003816

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817

+ (1 minus 120573119899120574) [

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

times1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

1003817100381710038171003817119865119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817]

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

times (1003817100381710038171003817119891 (119909119899minus1

)1003817100381710038171003817 +

1003817100381710038171003817119860 (119868 minus 120582119899minus1

120583119899minus1

119865)119866 (119878119899minus1

119910119899minus1

)1003817100381710038171003817)

+1003817100381710038171003817119909119899minus1 minus 119866 (119878

119899minus1119909119899minus1

)1003817100381710038171003817

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816

times1003817100381710038171003817119865119866 (119878

119899minus1119910119899minus1

)1003817100381710038171003817 +


119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817 +1003816100381610038161003816120573119899 minus 120573

119899minus1

10038161003816100381610038161198720

+ 1198720

1003816100381610038161003816120572119899 minus 120572119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

10038161003816100381610038161198720 +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899120583119899 minus 120582

119899minus1120583119899minus1

1003816100381610038161003816)

+1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817 +1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 119909119899minus1

1003817100381710038171003817

+ 1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) +1003817100381710038171003817119878119899119909119899minus1 minus 119878

119899minus1119909119899minus1

1003817100381710038171003817

+1003817100381710038171003817119878119899119910119899minus1 minus 119878

119899minus1119910119899minus1

1003817100381710038171003817

(96)


where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where sup119899ge0

119891(119909119899)+119860(119868minus120582

119899120583119899119865)119866(119878

119899119910119899)+119865119866(119878

119899119910119899)+

119909119899minus119866(119878119899119909119899) le 119872

0for some119872



120573119899(120574 minus 120573) = infin and

infin

sum

119899=0

1198720(1003816100381610038161003816120572119899 minus 120572

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120573119899 minus 120573

119899minus1

1003816100381610038161003816

+1003816100381610038161003816120582119899 minus 120582

119899minus1

1003816100381610038161003816 +1003816100381610038161003816120583119899 minus 120583

119899minus1

1003816100381610038161003816) lt infin

(97)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 119909119899

1003817100381710038171003817 = 0 (98)


119899

1003817100381710038171003817 = (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119909119899

1003817100381710038171003817

le (1 minus 119886) (1003817100381710038171003817119866 (119878119899119909119899) minus 119866 (119878

119899119910119899)1003817100381710038171003817

+1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

le (1 minus 119886) (1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 +1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817

+1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(99)

which implies that

1003817100381710038171003817119910119899 minus 119909119899

1003817100381710038171003817 le1 minus 119886

119886(1003817100381710038171003817119866 (119878119899119910119899) minus 119909119899+1

1003817100381710038171003817 +1003817100381710038171003817119909119899+1 minus 119909

119899

1003817100381710038171003817)

(100)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 = 0 (101)


119899119909119899)1003817100381710038171003817 le

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 120572119899

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817 + 1198871003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817

(102)

which implies that

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 le

1

1 minus 119887

1003817100381710038171003817119909119899 minus 119910119899

1003817100381710038171003817 (103)

and hence

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866 (119878119899119909119899)1003817100381710038171003817 = 0 (104)


119909119899minus119878119909119899 = 0 and

lim119899rarrinfin

119909119899minus 119866119909119899 = 0


21198612119906) 119909119899= 119878119899119909119899

119906119899

= Π119862(119909119899minus 12058321198612119909119899) and V

119899= Π119862(119906119899minus 12058311198611119906119899) Then

119906 = Π119862(V minus 120583

11198611V) and V

119899= 119866119909119899= 119866(119878

119899119909119899) for all 119899 ge 0 It is


1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119906

1003817100381710038171003817

2

= 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

(105)


1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906 minus 120583

2(1198612119909119899minus 1198612119906)

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

(106)

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V minus 120583

1(1198611119906119899minus 1198611V)10038171003817100381710038172

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(107)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus21205831(1205721minus 1205812

1205831)

times10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

(108)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 21205832(1205722minus 1205812

1205832)

times10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

minus 21205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 2 (1 minus 120572119899)

times [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

(109)


2 (1 minus 120572119899) [1205832(1205722minus 1205812

1205832)10038171003817100381710038171198612119909119899 minus 119861

21199061003817100381710038171003817

2

+ 1205831(1205721minus 1205812

1205831)10038171003817100381710038171198611119906119899 minus 119861

1V10038171003817100381710038172

]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

(110)


Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 119909119899 and 119910


119894lt 1205721198941205812 for 119894 =


lim119899rarrinfin

10038171003817100381710038171198612119909119899 minus 11986121199061003817100381710038171003817 = 0 lim

119899rarrinfin

10038171003817100381710038171198611119906119899 minus 1198611V1003817100381710038171003817 = 0

(111)


1such that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

=1003817100381710038171003817Π119862 (119909119899 minus 120583

21198612119909119899) minus Π119862(119906 minus 120583

21198612119906)

1003817100381710038171003817

2


21198612119906) 119869 (119906

119899minus V)⟩

= ⟨119909119899minus 119906 119869 (119906

119899minus V)⟩

+ 1205832⟨1198612119906 minus 1198612119909119899 119869 (119906119899minus V)⟩

le1

2[1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

+1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)]

+ 1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(112)

which implies that

1003817100381710038171003817119906119899 minus V10038171003817100381710038172

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

(113)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

=1003817100381710038171003817Π119862 (119906119899 minus 120583

11198611119906119899) minus Π119862(V minus 120583

11198611V)10038171003817100381710038172


11198611V) 119869 (V

119899minus 119906)⟩

= ⟨119906119899minus V 119869 (V

119899minus 119906)⟩ + 120583

1⟨1198611V minus 1198611119906119899 119869 (V119899minus 119906)⟩

le1

2[1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

+1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

+ 1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(114)

which implies that

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119906119899 minus V1003817100381710038171003817

2

minus1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(115)


1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(116)


1003817100381710038171003817119910119899 minus 1199061003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817V119899 minus 119906

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199061003817100381710038171003817

2

+ (1 minus 120572119899)

times [1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus 1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

minus 1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817]

=1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus (1 minus 120572119899)

times [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)] + 2 (1 minus 120572

119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

(117)


(1 minus 120572119899) [1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817)

+1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817)]

le1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817

2

minus1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817

2

+ 2 (1 minus 120572119899)

times (1205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+1205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817)

le (1003817100381710038171003817119909119899 minus 119906

1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119906

1003817100381710038171003817)1003817100381710038171003817119909119899 minus 119910

119899

1003817100381710038171003817

+ 21205832

10038171003817100381710038171198612119906 minus 1198612119909119899

1003817100381710038171003817

1003817100381710038171003817119906119899 minus V1003817100381710038171003817

+ 21205831

10038171003817100381710038171198611V minus 1198611119906119899

1003817100381710038171003817

1003817100381710038171003817V119899 minus 1199061003817100381710038171003817

(118)


Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 119909119899 119910119899 119906119899 and V



lim119899rarrinfin

1198921(1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817) = 0

lim119899rarrinfin

1198922(1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817) = 0

(119)


2 we get

lim119899rarrinfin


lim119899rarrinfin

1003817100381710038171003817119906119899 minus V119899+ (119906 minus V)1003817100381710038171003817 = 0

(120)


119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119906

119899minus (119906 minus V)1003817100381710038171003817

+1003817100381710038171003817119906119899 minus V

119899+ (119906 minus V)1003817100381710038171003817 997888rarr 0


(121)

That is

lim119899rarrinfin


1003817100381710038171003817119909119899 minus V119899

1003817100381710038171003817 = 0 (122)

Note that1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119878119899119909119899

1003817100381710038171003817 (123)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119899119909119899

1003817100381710038171003817 = 0 (124)


that1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119878

119899119909119899

1003817100381710038171003817 +1003817100381710038171003817119878119899119909119899 minus 119878119909

119899

1003817100381710038171003817 997888rarr 0


1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119866 (119878119899119909119899) minus 119866119909

119899

1003817100381710038171003817

le1003817100381710038171003817119909119899 minus 119866 (119878

119899119909119899)1003817100381710038171003817 +

1003817100381710038171003817119878119899119909119899 minus 119909119899

1003817100381710038171003817 997888rarr 0


(125)

That is

lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119878119909119899

1003817100381710038171003817 = 0 lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119866119909119899

1003817100381710038171003817 = 0 (126)

Define a mapping



1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 =1003817100381710038171003817(1 minus 120579) (119909

119899minus 119878119909119899) + 120579 (119909

119899minus 119866119909119899)1003817100381710038171003817

le (1 minus 120579)1003817100381710038171003817119909119899 minus 119878119909

119899

1003817100381710038171003817 +1003817100381710038171003817120579119909119899 minus 119866119909

119899

1003817100381710038171003817

(128)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 119882119909119899

1003817100381710038171003817 = 0 (129)



⟨(119891 minus 119860) 119901 119869 (119909119899minus 119901)⟩ le 0 (130)


119909119905= 119905119891 (119909

119905) + (119868 minus 119905119860)119882119909

119905 (131)






⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ le 0 (133)



1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119866 (119878119899119909119899) minus 119901

1003817100381710038171003817

2

le 120572119899

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ (1 minus 120572119899)1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

=1003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

2

(134)

and hence

1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817

2

=1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901] + 120573

119899(119891 minus 119860) 119901

1003817100381710038171003817

2

le1003817100381710038171003817120573119899 (119891 (119909

119899) minus 119891 (119901)) + (119868 minus 120573

119899119860)

times [119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901]

1003817100381710038171003817

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817119866 (119878119899119910119899) minus 120582119899120583119899119865119866 (119878

119899119910119899) minus 119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [120573119899

1003817100381710038171003817119891 (119909119899) minus 119891 (119901)

1003817100381710038171003817 + (1 minus 120573119899120574)

times1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899)


119899120583119899119865119901

1003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [120573119899

1003817100381710038171003817120573119909119899 minus 1199011003817100381710038171003817 + (1 minus 120573

119899120574)

times (1003817100381710038171003817(119868 minus 120582

119899120583119899119865)119866 (119878

119899119910119899) minus (119868 minus 120582

119899120583119899119865) 119901

1003817100381710038171003817

+120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩


le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)


1 minus 120572

120582))

times1003817100381710038171003817119866 (119878119899119910119899) minus 119901

1003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 )]

2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817

+ (1 minus 120573119899120574) (

1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le [1205731198991205731003817100381710038171003817119909119899 minus 119901

1003817100381710038171003817 + (1 minus 120573119899120574)

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= [(1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]2

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

21003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817


1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817]

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

le (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+ 2120573119899⟨(119891 minus 119860) 119901 119869 (119909

119899+1minus 119901)⟩

= (1 minus 120573119899(120574 minus 120573))

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817

2

+ 120573119899(120574 minus 120573) sdot

1

120574 minus 120573

times [120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ]

(135)


119899=0(120574 minus 120573)120573

119899= infin and


1

120574 minus 120573[120582119899120583119899

120573119899

10038171003817100381710038171198651199011003817100381710038171003817 (2

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 + 120582119899120583119899

10038171003817100381710038171198651199011003817100381710038171003817)

+2 ⟨(119891 minus 119860) 119901 119869 (119909119899+1

minus 119901)⟩ ] le 0

(136)


lim119899rarrinfin

1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (137)



120583119899 and 120573



= (1 + 119899)minus56 120583

119899=

1 and 120573119899




120573119899= 0 and sum

infin

119899=0120573119899= infin


(120582119899120583119899)120573119899= 0

(iii) suminfin

119899=0|120573119899+1


119899=0|120582119899+1


suminfin

119899=0|120583119899+1






119862 rarr 119883


119899infin

119899=0be an


infin



11198611)Π119862(119868 minus 120583

21198612) with

0 lt 120583119894




119899infin

119899=0

in [0 1] and 120572119899infin

119899=0 120573119899infin




120573119899= 0 and sum

infin

119899=0120573119899= infin


120582119899120573119899= 0 and sum

infin

119899=0|120582119899+1



(iv) suminfin

119899=0|120572119899+1


infin

119899=0|120573119899+1



sup119909isin119863

119878119899+1




⋂infin




119910119899= 120572119899119909119899+ (1 minus 120572

119899) 119866 (119878

119899119909119899)

119909119899+1

= 120573119899119891 (119909119899) + (119868 minus 120573

119899119860) [119910119899minus 120582119899119865 (119910119899)] forall119899 ge 0

(138)


to the VIP (85)








119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119899


119899in [10Theorem





119899appears








Acknowledgment



References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

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