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On best proximity points for pseudocontractions in the intermediate sense fornon-cyclic and cyclic self-mappings in metric spaces

Fixed Point Theory and Applications2013, 2013:146 doi:10.1186/1687-1812-2013-146

manuel de la Sen ([email protected])

ISSN 1687-1812

Article type Research

Submission date 17 September 2012

Acceptance date 17 May 2013

Publication date 5 June 2013

Article URL http://www.fixedpointtheoryandapplications.com/content/2013/1/146

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On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-

mappings in metric spaces

M De la Sen

Institute of Research and Development of Processes, University of the Basque Country

Campus of Leioa (Bizkaia) P.O. Box 644- Bilbao, 48080- Bilbao, SPAIN, email: [email protected]

Abstract. This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on

the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive

images in the same subsets of its domain. If the space is uniformly convex and the subsets are non-empty, closed and

convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity

points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

1. Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated

in the framework of Hilbert spaces have received a certain attention in the last years concerning their

convergence properties and the existence of fixed points. See, for instance, [1-4] and references therein.

Results about the existence of a fixed point are discussed in those papers. On the other hand, important

attention has been paid during the last decades to the study of the convergence properties of distances in

cyclic contractive self-mappings on p subsets XA i of a metric space ( )d,X , or a Banach space

( ),X . The cyclic self-mappings under study have been of standard contractive or weakly contractive

types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of

the involved sets has been investigated in the last years. See, for instance, [6-21] and references therein. It

has to be noticed that every nonexpansive mapping [22-23] is a 0-strict pseudocontraction and also that

strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [2]. The uniqueness

of the best proximity points to which all the sequences of iterations converge is proven in [7] for the

extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces

(then being strictly convex and reflexive [5]) or in reflexive Banach spaces [14]. The p subsets XA i

of the metric space ( )d,X , or the Banach space ( ),X , where the cyclic self-mappings are defined, are

supposed to be non-empty, convex and closed. If the involved subsets have nonempty intersections, then

all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the

subsets, and framework can be simply given on complete metric spaces. The research in [7] is centred on

the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space.

Those results are extended in [8] for Meir- Keeler cyclic contraction maps and, in general, with the p

( )2 -cyclic self-mapping self- mapping pi ipi i AA:T defined on any number of subsets of

the metric space with { }p,...,,p 21:= . Other recent research which has been performed in the field of

cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M,

as the union of a set of nonempty sets as mi i

MM1== , with respect to an operator MMf : , [15].

Subsequently, cyclic representations have been used in [16] to investigate operators from toM which

are cyclic -contractions, where ++ 00 RR: is a given comparison function, X and ( )dX, is a

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metric space. The above cyclic representation has also been used in [17] to prove the existence of a fixed

point for a self-mapping defined on a complete metric space which satisfies a cyclic weak -contraction.

In [19], a characterization of best proximity points is studied for individual and pairs of non-self-

mappings BATS :, , whereA andB are nonempty subsets of a metric space. The existence of common

fixed points of self-mappings is investigated in [24] for a class of nonlinear integral equations, while fixed

point theory is investigated in locally convex spaces and nonconvex sets in [25-28]. More recently, the

existence and uniqueness of best proximity points of more general cyclic contractions have been

investigated in [29-30] and a study of best proximity points for generalized proximal contractions, a

concept referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study

and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been

performed in [32] and recent contributions on the study of best proximity and proximal points can be

found in [34-39] and references therein. In general, best proximity points do not fulfil the usual best

proximity condition TxSxx == under this framework. However, best proximity points are proven to

jointly globally optimize the mappings from x to the distances ( )Txxd , and ( )Sxxd , . Furthermore, a

class of cyclic -contractions, which contains the cyclic contraction maps as a subclass, has been

proposed in [19] in order to investigate the convergence and existence results of best proximity points in

reflexive Banach spaces completing previous related results in [7]. Also, the existence and uniqueness of

best proximity points of cyclic contractive self-mappings in reflexive Banach spaces have been

investigated in [20]. This paper is devoted to the convergence properties and the existence of fixed points

of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically

pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of

2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the

contraction may be different on each of the subsets and only the product of all the constants is requested

to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on

each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping.

The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial

point in the union of the subsets is proven ifXis a uniformly convex Banach space and all the subsets of

Xare nonempty, convex and closed. Such a limiting sequence is of size pq (with the inequality being

strict if there is at least one iteration with image in the same subset as its domain), wherep of its elements

(all of them if pq = ) are best proximity points between adjacent subsets. In the case that all the subsets

XA i intersect, the above limit sequence reduces to a unique fixed point allocated within the

intersection of all such subsets.

2. Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces

IfH is a real Hilbert space with an inner product .,. and a norm . and A is a nonempty closed

convex subset ofH, then AAT : is said to be an asymptotically -strictly pseudocontractive self-

mapping in the intermediate sense for some [ )10 , if

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( ) ( ) 0222

yTIxTIyxyTxTsupsuplim nnn

nn

Ay,xn

(2.1)

for some sequence { } [ ) ,n 1 , 1n as n [1-5]. Such a concept was firstly introduced in [1].

If (2.1) holds for 1= , then AAT : is said to be an asymptotically pseudocontractive self-mapping in

the intermediate sense. Finally, if [ )10 ,n as n , then AAT : is asymptotically -

strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate

sense if 1= . If (2.1) is changed to the stronger condition

( ) ( ) 0222

yTIxTIyxyTxT nnn

nn ; Ay,x , Nn , (2.2)

then the above concepts translate into AAT : being an asymptotically -strictly pseudocontractive

self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one,

respectively. Note that (2.1) is equivalent to

( ) ( ) nnnnnn yTIxTIyxyTxT ++ 222 ; Ay,x , Nn (2.3)

or, equivalently,

( ) ( )

+++ n

nnn

nn yTxTyxyx,yTxT

22 12

1; Ay,x , Nn , (2.4)

where

( ) ( )

=

2220: yTIxTIyxyTxTsup,max nnnnn

Ay,xn ; Nn . (2.5)

Note that the high-right-hand-side term ( ) ( ) 2yTIxTI nn of (2.3) is expanded as follows for any

Ay,x :

yTxTyxyTxTyx nnnn + 222

( ) ( ) 222 yTxT,yxyTy,xTxyTIxTI nnnnnn =

yx,yTxTyTxTyx nnnn ++= 222 yTxT,yxyTxT,yx nnnn =

yx,yTxTyTxTyx nnnn ++ 222 yTxTyxyTxTyx nnnn ++ 222 .

(2.6)

The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts

in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also

to generalize them to the - parameter to eventually be replaced with a sequence { }n in ( )10, . Now,

if instead of a real Hilbert space Hendowed with an inner product .,. and a norm . , we deal with

any generic Banach space ( ).,X , then its norm induces a homogeneous and translation invariant

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metric + 0: RXXd defined by ( ) ( )210 /yx,yxdy,xd == ; Ay,x so that (2.6) takes the

form

( ) ( ) ( ) ( )yT,xTdy,xdyT,xTdy,xd nnnn 222 +

( ) ( ) ( )( ) ( )yTxT,yxd,yTxTyxdyTIxTI nnnnnn == 222 0

( ) (( ( ) (( 2200 yT,xTdy,xd,yTxTd,yxd nnnn +=+

( ) ( ) ( ) ( )yT,xTdy,xdyT,xTdy,xd nnnn 222 ++= ; Ay,x . (2.7)Define

( ) [ ] ( ) ( ) ( ) ( ) ( )( )yT,xTdy,xdyT,xTdy,xdyTxT,yxd,minyx nnnnnnn 2:11:, 222 ++= ; Ay,x , n , (2.8)

which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that

{ } ( ) ( ) ( ) ( ) ( ) ( ) ( )++ yT,xTdy,xdyT,xTdy,xdyTIxTI nnnnnn 2:1 222R .(2.9)

The following result holds related to the discussion (2.7)-(2.9) in metric spaces.

Theorem 2.1. Let ( )dX, be a metric space and consider a self-mapping XXT : . Assume that the

following constraint holds:

( ) ( ) ( ) ( ) yTxTdy,xdy,xyxdy,xyTxTd nnnnnn ,,, 2222 ++

( ) ( ) ( ) ( )y,xyTxTdyxdy,xy,x nnn

nn ++ ,,2 ; Xy,x , Nn (2.10)

with

( )y,xnn =

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) yTxTdyxdy,xy,xy,xdy,xy,xyTxTdy,x,max nnnnnnnn

n ,,2,10:22 +=

0 ; Xy,x as n (2.11)

for some parameterizing bounded real sequences ( ){ }y,xn , ( ){ }y,xn and ( ){ }y,xn of general terms

( )y,xnn = , ( )yxnn , = , ( )yxnn , = satisfying the following constraints:

( ) ( ) ( )( )

( ) ( )( )

( )( )

+


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( )( ) ( )

( )0

2

2,1

y,x

y,xyxy,x

n

nnn

; Xy,x as n (2.13)

if and only if ( ) 112 ++ nnn ; Xy,x as n .

Then the following properties hold:

(i) ) ( )yxdyTxTdlim nnn

,,

for any Xyx , so that XXT : is asymptotically nonexpansive.

(ii) Let ( )dX, be complete, + 0: RXXd be, in addition, a translation-invariant homogeneous norm

and let ( ) ( )dXX ,, , with being the metric-induced norm from + 0: RXXd , be a uniformly

convex Banach space. Assume also that XXT : is continuous. Then any sequence { xTn ; Ax isbounded and convergent to some point ( ) Cxzz xx = , being in general dependent on x, in some

nonempty bounded, closed and convex subset C ofA , where A is any nonempty bounded subset ofX.

Also, ( )xTxTd mnn +, is bounded; N mn , , 0, =+

xTxTdlim mnn

n; Ax , m and

( ) CTzxzz xxx == is a fixed point of the restricted self-mapping CCT : ; Ax . Furthermore,

( (( 0,, 2112 =++

yTxTdyTxTdlim nnnn

n; Ayx , . (2.14)

Proof: Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given

Xyx , and n as follows:

A) ( ( )yxdyTxTd nn ,, for any Xyx , , Nn . Then one gets from (2.10)

( ) ( ) ( ) nnnnnnnnnn yxdyTxTdyxdyTxTd ++++ ,2,,, 2222

( ) ( )n

nan

nn yxdkyTxTd

+

1,, 2 ; (2.15)

Ayx , , Nn , where

( )( )

11

21,

++==

n

nnnanan yxkk

; Xy,x as n , (2.16)

which holds from (2.12)-(2.13) if ( ) 1n

with 0>n

has to be excluded because of the unboundedness or non-negativity of the

second right-hand-side term of (2.15).

B ) ( )yxdyTxTd nn ,, for some Xyx , , Nn . Then one gets from (2.10)

( ) ( ) ( ) ( ) ( ) nnnnnnnnnnnn yTxTdyTxTdyxdyTxTd ++++ ,2,,, 2222

( ) ( )( )nn

nbn

nn yxdkyTxTd

211,, 2

2

++ , (2.18)

where

( ) ( ) 1211, +

+

== nn

nn

bnbn yxkk

as n , (2.19)

which holds from (2.12) and 1bnk if ( ) ( )( )[ ] 1211


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Then any sequence xTn is convergent (otherwise, the above limit would not exist contradicting

Property (i)), and then bounded in C Ax .This also implies xTxTd mnn +, is bounded; Ax ,

N mn , and 0, =+

xTxTdlim mnn

n; Ax , Nm . This implies also ( )xzxT x

n as n

; Ax such that ( ) xx Tzxz = ; Ax which is then a fixed point of CCT : (otherwise, the above

property 0, =+

xTxTdlim mnn

n; Ax , Nm would be contradicted). Hence, Property (ii) is

proven.

First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a

complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced

by such a metric. Alternatively, the property could be established on any uniformly convex Banach space

by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later

parallel results throughout the paper. Note that the proof of Theorem 2.1 (i) has two parts: Case A refers

to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration

steps and Case B refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive

for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1 (ii) that the

given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the

proof is outlined with the existence of a ( )CTFixz for any nonempty, bounded and closed subset A

ofX . Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed

by construction while any taken nonempty set of initial conditions XA is not required to be convex.

However, the property that all the sequences converge to fixed points opens two potential possibilities

depending on particular extra restrictions on the self-mapping CCT : , namely: 1) the fixed point is

not unique so that zzx for any Ax (and any A in X) so that some set ( )CTFix for some

( ) XACC = contains more than one point. In other words, ( ) 0,2 yTxTd nn as n ; Ay,x

has not been proven although it is true that ( (( 0,, 2112 =++

yTxTdyTxTdlim nnnn

n; Ay,x ; 2)

there is only a fixed point in X . The following result extends Theorem 2.1 for a modification of the

asymptotically nonexpansive condition (2.10).

Theorem 2.2. Let ( )dX, be a metric space and consider the self-mapping XXT : . Assume that the

constraint below holds:


( ) ( ) ( )y,xyTxTdy,xy,x nnn

nn ++ ,22 ; Xy,x , Nn (2.22)

with

( )y,xnn =

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )yTxTdy,xy,xy,xdy,xy,xyTxTdy,x,max nnnnnnnnn ,2,10: 222 +=

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0 ; Xy,x as n (2.23)

for some parameterizing real sequences ( )yxnn , = , ( )yxnn , = and ( )yxnn , = satisfying, for

any n ,

( ){ } [ ) ,y,xn 0 , ( ){ }

( )

( )

y,x

y,x

,y,x n

n

n

2

1

1 , ( ){ } [ ]10,y,xn ; Xy,x , Nn . (2.24)

Then the following properties hold:

(i) ) ( )yxdyTxTdlim nnn

,,

so that XXT : is asymptotically nonexpansive, and then

( ( )yxdyTxTdlim nnn

,,

; Xy,x if

( ) ( )

( ) ( )( )1

211

+

+

y,xy,x

y,xy,x

nn

nn

( )

( ) ( )

( )

( )

( )

yx

yx

yx

yxyx

y,x n

n

n

nn

n ,2

,1

,,2

,2,1

; Xyx ,

, Nn

(2.25)

and the following limit exists:

( ) ( ) ( )( ) 1,1,2, ++ yxyxyx nnn ; Xyx , as n . (2.26)

(ii) Property (ii) of Theorem 2.1 if ( )dX, is complete and ( ) ( )dXX ,, is a uniformly convex Banach

space under the metric-induced norm .

Sketch of the proof: Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for

Case B. Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii).

The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the

intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case B in the

proof of Property (i) with the sequence of constants ( ) 1, yxkn ; Xyx , , Nn and ( ) 1, yxkn

; as n , Xyx , . The concepts of an asymptotic pseudocontraction and an asymptotic strict

pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the

asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to

(2.23) are revisited as follows in the context of metric spaces.

Definition 2.3. Assume that ( )dX, is a complete metric space with + 0: RXXd being a

homogeneous translation-invariant metric. Thus, AAT : is asymptotically -strictly

pseudocontractive in the intermediate sense if

( )( ) ( ) ( ) ( )( ) 0,211 22 ++

y,xdyTxTdsuplim nnnn

nn

n

; Ay,x (2.27)

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for [ )10,n = ; Nn and some real sequences { }n , { }n being, in general, dependent on the

initial points , i.e., ( )yxnn ,= , ( )yxnn , = and

{ }

2

11,n and { } [ ) ,n 1 ; Nn , 1n and 1n as n ; Ay,x , n .

(2.28)

Definition 2.4. AAT : is asymptotically pseudocontractive in the intermediate sense if (2.30) holds

with{ }

n

nn ,

2

11 , { } [ ]10,n , { } [ ) ,n 1 , 11 nn , , 1n as n and the

remaining conditions as in Definition 2.3 with ( )yxnn ,= , ( )yxnn , = and ( )yxnn , = .

Definition 2.5. AAT : is asymptotically -strictly contractive in the intermediate sense if

[ ) ,n 0 , [ )1,0= n , N

n,n ;2

11

+

11

2

11, ,min,n ,

[ )10,n as n , in Definition 2.3 with ( )yxnn ,= , ( )yxnn , = .

Definition 2.6. AAT : is asymptotically contractive in the intermediate sense if [ ) ,n 0 ,

{ } [ )10,n ,

n

nn ,

2

11

+

2

11,;

,n nN , [ )10,n , and 1= n as

n in Definition 2.3 with ( )yxnn ,= , ( )yxnn , = and ( )yxnn , = .

Remarks 2.7: Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence

properties under the constraint (2.22), subject to (2.23), by noting the following:

1) If AAT : is asymptotically - strictly pseudocontractive in the intermediate sense(Definition 2.3), then the real sequence { }nk1 of asymptotically nonexpansive constants has a

general term( )

[ )

+

+

+

+= ,,k

/n

/

n

nn 1

1211:

2121

1

; Nn , and it

converges to a limit 11 =k since 0n and 1n as n ; Ay,x from (2.22) since

1n from (2.27). Then AAT : is trivially asymptotically nonexpansive as expected.

2) If AAT : is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), thenthe sequence { }nk2 of asymptotically nonexpansive constants has the general term:

( )[ )

+

+

+

+= ,,k

/

/

n

nn

nn

nnn 1

1211:

21

21

2

; Nn , and it converges to a

limit 12 =k since 1,1 nn as n . Then AAT : is also trivially asymptotically

nonexpansive as expected. Since 1n , note that nnn kk 12 >> and

FPTA_274_edited 9


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nnn kk 12


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The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically

contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that

( ){ } [ ) ,y,xn 1 .

Theorem 2.8. Let ( )dX, be a complete metric space endowed with a homogeneous translation-

invariant metric + 0: RXXd and consider the self-mapping XXT : . Assume that

( ) ( )dXX ,, is a uniformly convex Banach space endowed with a metric-induced norm from the

metric + 0: RXXd . Assume that the asymptotically nonexpansive condition (2.22), subject to

(2.23), holds for some parameterizing real sequences ( )yxnn , = , ( )yxnn , = and ( )yxnn , =

satisfying, for any n ,

( ){ } [ ) ,y,xn 1 , ( ){ } ( )( )

y,x

y,x,y,x

n

nn

2

11 , ( ){ } [ ) [ ]100 ,,y,xn (2.29)

; Xy,x , n . Then ( ) ( )yxdyTxTdlim nnn

,,

for any Xyx , satisfying the conditions

( ) ( )

( ) ( ) ( )( )1

211

+

+

y,xdy,xy,x

y,xy,x

nn

nn

; ( ) ( ) ( )( ) 1,1,2, ++ yxyxyx nnn ; Xyx , as n .

(2.30)

Furthermore, the following properties hold:

(i) CCT : is asymptotically -strictly pseudocontractive in the intermediate sense for some

nonempty, bounded, closed and convex set ( ) XACC = and any given nonempty, bounded and closed

subset XA of initial conditions if (2.29) hold with 10


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(iii) If (2.29) hold with [ ) ,n 0 , [ )10 ,n = ,

2

1,1n

2

21,1, n

Nn; and [ )1,0n as n , then XXT : is asymptotically - strictly contractive in the

intermediate sense. Also, XXT : has a unique fixed point.

(iv) If (2.29) hold with [ ) ,n 0 , { } [ )10 ,n ,

n

nn

2

1,1

+

2

1,1,

n , Nn;

, 1n and [ )10,n as n , then XXT : is asymptotically strictly contractive in the

intermediate sense. Also, XXT : has a unique fixed point.

Proof: (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular

nonexpansive condition (2.22), subject to (2.23), so that AAT : is asymptotically nonexpansive (see

Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)).

Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that theasymptotically nonexpansive self-mapping XXT : is also a strict contraction, then continuous with a

unique fixed point, since ( ) 112


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satisfying nknk Pa and ( )( )ppp

knnk xPP= RRN: both being potentially dependent on the state

as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the

matrix is symmetric, then nk can be taken exactly as the spectral radius of nkF in such a case, i.e. , it

equalizes the absolute value of its dominant eigenvalue. We have to check the condition

( ) ( ) ( ) ( )( ) 0,21 22 ++

kknknkkn

kn

nknkn

y,xdyTxTdsuplim ; k (2.32)

provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then

both being coincident, and provided also that we take the metric space ( d,pR which holds, in particular,if

a) { } + 0Rnk , { } [ ) ,nk 1 , [ )10,nk = , { }

2

11,nk ; N kn, , 1nk and

0nk , 1nk , as n ; k . This implies that ( )1

21121 +

+=nknk

nknkk

; N kn,

and 11 nkk as n ; Nk . Thus,ppT RR : is asymptotically nonexpansive being also an

asymptotic strict -pseudocontraction in the intermediate sense. This also implies that (2.31) is globally

stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence uL of

nkx which is unbounded, and then there is also an infinite subsequence uaL which is strictly

increasing. Since ( ) 0= knnk x and ( ) 111 = knnk xkk as n ; k , one has that for

11 Ax , any given Nk and some sufficiently large ( ) ( ) N== kk xmmxmm 02020101 , ,

( ) ( ) ++ == RR 02220111 , mm such that ( ) 11 11 += kmm xkk and ( ) 22 km x ;

011 mm , 022 mm . Now, take ( )02010 m,mmaxm = and ( )21 ,max= . Then

kkmk x/x/x +++ 1 ; ( ) 0mm N and any given Nk . If 0kx , then stability holds

trivially. Assume not, and there are unbounded solutions. Thus, take ( ) uak Lxx +mk,0 such that

Mx

x

k

mk + for any given +RM , ( ) mm N and some ( ) 0mMmm = . Note that since uaL is a

strictly increasing real sequence ( ){ }mM implying ( ) mM as m , which leads to a contradiction

to the inequality

++

kxM

111 for

+ 0 for some sufficiently large m , then for some

sufficiently largeM, if such a strictly increasing sequence uaL exists. Hence, there is no such sequence,

and then no unbounded sequence uL for any initial condition in 0A . As a result, for any initial condition in

any given subset 1A ofp

R (even if it is unbounded), any solution sequence of (2.31) is bounded, and

then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable

nonempty bounded closed sets { }NR iA pi : , which are not necessarily connected, such that kk Ax

FPTA_274_edited 13


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14

; Nk and any given 00 Ax . Assume that the set 0A of initial conditions is bounded, convex and

closed and consider the collection of convex envelopes { NR i:Aconvex pi , define constructively

the closure convex set ( ) ))== 10 i iAconvexconvexclAC which is trivially bounded, convex and

closed. Note that it is not guaranteed that =1i iAconvex is either open or closed since there is a union of

infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the

collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily

convex (so that=1i iAconvex is not guaranteed to be convex while A is convex ). Consider now the

self-mapping CCT : which defines exactly the same solution as ppT RR : for initial conditions

in 1A so that T is identified with the restricted self-mapping CCT p: R from a nonempty bounded,

convex and closed set to itself. Note that ( dp ,R for the Euclidean distance is a convex metric space

which is also complete since it is finite dimensional. Then pppnkF RRA: and

ppnk RR A: are both continuous, then CCT : is also continuous and has a fixed point in A

from Theorem 2.8 (i).

b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above

conclusions still hold with the modification( )

( ) 11211

22 +

+=

nknk

nknknkk

and

1 nknknk as n ; Nk . From Remark 2.7(2), nknknk kk 12 >> and

nknknk kk 12 nk ; n with the

remaining parameters and parametrical sequences being identical in both cases. If

pppnkF

RRA: and ppnk RR A: ; N kn , are both continuous, then AAT : is

continuous and has a fixed point in A from Theorem 2.8 (ii).

c) If XXT : is asymptotically -strictly contractive in the intermediate sense, then

( )( )( ) [ )1,0,

21133 =+

+=

maxokk

nknk

nknknk ; k so that it is asymptotically strictly

contractive and has a unique fixed point from Theorem 2.8(iii).

d) If XXT : is asymptotically contractive in the intermediate sense,

( ) 4

21

4211

kk

/

nknk

nknknk

+

+=

( )

2

1

2

1+ o ; Nk . Thus, XXT : is an asymptotic

strict contraction and has a unique fixed point from Theorem 2.8 (iv).

FPTA_274_edited 14


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15

Remark 2.10: Note that conditions like (2.32) can be tested on dynamic systems being different from

(2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from

given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the

convergence of the solution to fixed points, then the system stability, etc. in a unified way for different

dynamic systems. Close considerations can be discussed for different dynamic systems and convergenceof the solutions generated by the different cyclic self-mappings defined on the union of several subsets to

the best proximity points of each of the involved subsets.

3. Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate

sense

Let XBA , be nonempty subsets ofX. BABAT : is a cyclic self-mapping if ( ) BAT and

( ) ABT . Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified

as follows:


( ) ( ) ( ) ( ) ( ) 2,2 Dy,xy,xyTxTdy,xdy,xy,x nnnn

nn +++ ; By,Ax , Nn (3.1)

( )y,xnn =

( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) yTxTdyxdy,xy,xy,xdy,xy,xyTxTdy,x,max nnnnnnnn

n ,,2,10:22 +=

; By,Ax , n (3.2)

with ( ) 0, 2 Dyxnn ; Xy,x as n , and that the asymptotically nonexpansive condition

(2.22), subject to (2.23) , is modified as follows:

( ( ) ( ) ( ) ( ) (( yTxTdy,xdy,xyxdy,xyTxTd nnnnnn ,,, 2222 ++

( ) ( ) ( ( ) ( ) 22 ,2 Dy,xy,xyTxTdy,xy,x nnnnnn +++ ; By,Ax , Nn (3.3)( )y,xnn =

( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )yTxTdy,xy,xy,xdy,xy,xyTxTdy,x,max nnnnnnnnn ,2,10: 22 +=

; Xy,x as n (3.4)

with ( ) 0, 2 Dyxnn ; Xy,x as n , where ( ){ } [ ) ,y,xn 0 and ( ) 0, = BAdistD . If

BA , then 0=D and Theorems 2.1, 2.2 and 2.8 hold with the replacement BAA . Then if

A and B are closed and convex, then there is a unique fixed point of BABAT : in BA . In

the following, we consider the case that =BA so that 0>D . The subsequent result based on

Theorems 2.1, 2.2 and 2.8 holds.

FPTA_274_edited 15


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16

Theorem 3.1. Let ( )dX, be a metric space and let BABAT : be a cyclic self-mapping, i.e. ,

( ) BAT and ( ) ABT , where A and B are nonempty subsets of X . Define the sequence

{ } [ ) ,k nn 0N of asymptotically nonexpansive iteration-dependent constants as follows:

( )

( ) ( ) ( )

( )( ) ( )

+

+

++

==(3.5b),if1

211

(3.5a),if11

21

:,

yxdyT,xTd

yxdyT,xTd

yxkknn

nn

nn

nn

n

nnn

nn

; ( ) ( ) ( )ABBAyx , , Nn provided that BABAT :

satisfies the constraint (3.1), subject to (3.2), and

( ) ( )( ) ( ) ( ) ( ) ( )[ ],,,01, N== nABBAyxyxdyT,xTd nnnn (3.6)and

( )( )

1211

, +

+==

nn

nnnn yxkk

(3.7)

; n for ( )ByAx and for ( )AyBx provided that BABAT : satisfies the

constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences ( ){ }y,xn ,

( ){ }y,xn , ( ){ }y,xn and ( ){ }y,xn of general terms ( )y,xnn = , ( )yxnn , = and

( )yxnn , = fulfil the following constraints:

( ) ( ) ( )( )

( ) ( )( ) ( )

( )( )

+


18/29

17

so that BABAT : is a cyclic asymptotically nonexpansive self-mapping. If Ax is a best

proximity point of A and By is a best proximity point of B , then DyTxTdlim nnn

=

, and

( )xzzxT xn =2 and ( )yzzyT y

n =2 , which are best proximity points of A and B (not being

necessarily identical to x and y ), respectively if BABAT : is continuous.

(ii) Property (i) also holds if BABAT : satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and

(3.5b) provided that ( ) ( )yxdyT,xTd nn , ; ( ) ( ) ( )ABBAyx , .

Proof: The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)

( ) ( ) ( ) ( )[ ]yxdDyTxTdlimDyxdkyTxTd nnnn

nnbn

nn ,,,1

,,2

22

++

; ( ) ( ) ( )ABBAyx

, , (3.10)and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)

( ) ( )( )

( ) ( )[ ]yxdDyTxTdlimDyxdkyTxTd nnnnn

nnbn

nn ,,,211

,,2

22 +

++

; ( ) ( ) ( )ABBAyx , (3.11)

since BAxTxT nn , ; n since ( ) BAT and ( ) ABT , and 1nk and

( ) ( ) 01 nn ky,x as n ; ( ) ( ) ( )ABBAyx , . Note that (3.8) implies that there is no

division by zero in (3.11). Now, assume that (3.10) holds with 1=n . From (3.8) and (3.2),

+ 0,

2

1 nn

, equivalently ,

2

1 nn

+ and ( ) ( ) ( )yxdyxdyTxTd

n

nnn ,,2

1, 22

+>

, which

contradicts (3.5a) if 0>n so that 1=n in (3.5a) under (3.7) implies that 0=n and , since 0=n

from (3.6) , there is no division by zero on the right-hand side of (3.10) if 1=n .

Also, if BABAT : is continuous, then ( ) DyTlimxTlimdyTxTdlim nn

n

n

nn

n=

=

2222 ,, so

that AxT n 2 ; n , AclxTlim nn

2 , ByT n 2 and BclxTlim nn

2 since ( ) BAT and

( ) ABT . This proves Properties (i)-(ii).

Remark 3.2 . Note that Theorem 3.1 does not guarantee the convergence of { xT n2 and { yT n2 to bestproximity points if the initial points for the iterations Ax and By are not best proximity points if

BABAT : is not contractive.

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with

( )1

1

21 ++++++

yT,xTdyT,xTdsuplim nnnnpnjpnjpnjpnj

n

,

but the above expression is equivalent, for xTx np

pn = and yTy np

pn = which are in BA , but not

jointly in eitherA orB , to

( ) ( )( ) 0,, 2112 > ++++

pnpnj

pnpnj

pnnj

pnnj

n

yTxTdyTxTdsuplim nn ,

which contradicts ( ) ( ) 02112 ++

yT,xTdyT,xTdsuplim njnjnjnj

n

since both sequences { xT jn

and { }yTjn are bounded; ( ) ( ) ( )ABBAyx , . Then there is no infinite oscillating sequence

({ yT,xTd njnj for some ( ) ( ) ( )ABBAyx , so that there is a finite limit ( ) Dyxgg = , of

({ yT,xTd njnj , ( ) ( ) ( )ABBAyx , . Now, proceed by contradiction by assuming the existence

of some ( ) ( ) ( )ABBAyx , such that ( ) ( ) 0, >+== DyxggyT,xTd njnj as n ;Nj . Thus, for any N0,nj , there is some ( ) N 0nn such that there are two consecutive nonzero

elements of a nonzero real sequence { }n , which can depend on x and y , which satisfy nn +1 and

nnjnj

nnjnj DyT,xTdDyT,xTd +=+= +1 (3.15)

; Nj . Otherwise, if nn


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20

( ) ( ) jnjnjnjnjnjn

jn

jn

jnDkn

k

+

=

21

1

1N as n ; Nj . The relations (3.16)

contradict ( ) ( ) 01 22 >+++

DDsuplim jnjnnn

since { }n is positive Nn (and it does not

converge to zero) and 0jn , 0jn as n . Thus, one concludes that { }n converges to zero,

and then DyTxTdlim nn

n=

, ; ( ) ( ) ( )ABBAyx , ; ( ) ( ) ( )ABBAyx , . This leads to

DxTxTdlim nn

n=+

1, ; BAx by taking Txy = with By if Ax and Ay if Bx .

Property (i) has been proven.

Now, Property (ii) is proven. It is first proven that ( ( 0,, 3212222 == ++

+

xTxTdlimxTxTdlim nn

n

nn

n;

BAx if the metric is translation-invariant and homogeneous so that it induces a norm ifA and

B are nonempty, closed and convex subsets ofX and ( ) ( )d,XX is a uniformly convex Banach

space. Assume not and take such a norm to yield ( 0, 222 >+ xTxTd nn . Then ifA is nonempty, closed

and convex and B is nonempty and closed and Ax , then { xT n2 , { AxT n +22 . It is known that

xTxddxTxTd xnn 2222 ,, + from Theorem 3.1 (i) for xTy 2= . Since ( ) ( )dXX ,, is a

uniformly convex Banach space for the metric-induced norm (being equivalent to the translation-

invariant homogeneous metric), we have the following property for the sequences { xT n2 , { AxT n +22

and { } Xpn satisfying for some strictly increasing nonnegative sequence of functions

+0

2

20: Rn

nn

R

r, and any nonnegative sequences { }nr2 and { }nR2 satisfying nn Rr 22 and any

sequence { } Xp n 2 ; Nn that

( ) ( )( ) nnnnnnnnn RpxTpxTmaxpxTdpxTdmax 22222222222 ,,,, = ++ (3.17)

( ) xTxTxTxTxTxTdr nnnnnnn 2222222222 , += +++ (3.18)

nnnnnnnnn RR

rpxTxTpxTxT 2

222222222 1222

++++

(3.19)

; BAx , Nn , which implies that

+

+

+ nn

n

nnnn

n

nnn pR

R

rxTxTR

R

rprmax 22

2

22222

2

222 1212, ,

(3.20)

which has to be valid for( )

( )( )( ) nnnn

n pppp

R 2222

2 221

2121

=+

=

; Nn . Now, for

( ) 02 Xp n and 022 >= nn pR ; n , it follows that ( ) ( ) 02 == s ; [ ]2,0s , which is a

FPTA_274_edited 20


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21

contradiction to [ ] + 020: R,n being strictly increasing, then contradicting ( ),X being a uniformly

convex Banach space, unless 02 22 nn Rr as n so that

+0

2

20: Rn

nn

R

r, converges to

( ) 00 = .Taking Ax , AxTp nn

= +222

; Nn , (3.15) for 022

nn

Rr as n implies the

existence of the first zero limit in (3.13).The existence of the second zero limit in (3.13) is proven in the

same way since BTx . Since those limits are zero, { }xT n2 , { }xT n 22 + are Cauchy sequences in A converging to a best proximity point Az for Ax . Note that Az is necessarily the unique best

proximity of BABA:T in A since{ }xT n2 and ( ){ }xTT n 22 converge to the same point.Otherwise, the first limit of (3.13) would not exist if the sequences do not converge, then a contradiction

holds to a proven result, and also Property (i) would not be true, since (3.12) would not hold, if the limit

of the sequence would not be a best proximity point in A , then a contradiction holds to another proven

result. In the same way, { }xT n 12+

, { }xT n 32+

converge to a unique best proximity point Bz 1 of

BABA:T for any Ax . Now, Tzz =1 . Assume not. Then since { } zxT n 2 , { } 112 zxT n +

and ( ) DxT,xTd nn +122 , one has ( ) Dz,zd =1 . Assume that Tzz 1 so that since A and B are convex,

( ) DzTzTzzzz

zzdD >

+

===

22222

222, 111 ,

which is a contradiction. Then Tzz =1 is the unique best proximity of B . If = BA , then

Tzzz ==1 is the unique fixed point of BABA:T which coincides with the unique best

proximity point in A and B .

Remark 3.4. Theorem 3.2 is known for strictly contractive cyclic self-mappings[21] satisfying the

contractive condition (3.1) in the case that 0n and 1


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22

( )

( ) ( ) ( )

( )( ) ( )

+

+


24/29

23

(iii) Assume that BABAT : is asymptotically -strictly pseudocontractive in the intermediate

sense so that (3.21) holds with 10


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24

10


26/29

25

discrete-time dynamic systems under time-varying sampling periods or under a time-varying

parameterization in general [33]. Assume that the suitable controlled solution (3.28) is of the form

( )( ) ( )( ) ( )( )knkn

knknkn

xx

xxxF

211 +

+= ; Nk . Then

( )knn xFF = ( )( ) ( )( ) ( )( )

( )( ) ( )( )[ ]( ) = ++=++

=== 20 11122

211

nj kjkknk

knkn

knknknn xxFxxF

xx

xxxkk

; Nk

( ) ( )( ) ( )( )

( )( )[ ]

+

+= = +

++

++ 20

21

1111

1111

211

nj kjk

nknk

nknk xxkxx

xx

; Nk (3.29)

( ) ( )[ ] ( ) ( )( ) ( ) ( )[ ] = ++++++= + ++=+= 2011110 nj jkkkkjknknkknknj kkkjk xxGxuxxGxuxxGxu

; Nk . (3.30)

The identities (3.30) allow the feedback generation of the control sequence (3.26) from its previous values

and previous solution values as follows:

( )( ) ( )

( ) ( )( )( ) ( )[ ]( ) ( ) 111201 211 ++

= ++++ ++

+= nknk

nj jkkkkjk

knkn

knknknk xxGxxGxu

xx

xxxu

( ) ( ) 1111 +++ = nknknk xxGxF ; Nk (3.31)

for given parameterizing scalar sequences which can be dependent on the state kx (see Example 2.9).

We are now defining a cyclic self-map [ ) ( ] [ ) ( ]2,,22,,2: /D/D/D/DT so that the

solution belongs alternately to positive (respectively, nonnegative) and negative (respectively,

nonpositive) real intervals [ ),2/D and ( ]2, /D if [ ) ( ]( ) 02,,2: >= /D,/DdistD (respectively,

if 0=D ), that is, [ )( ) ( ]2,,2 /D/DT and ( ]( ) [ ) ,22, /D/DT . For such an objective,

consider the scalar bounded sequences ( ){ }knk x , ( )knk x0 and ( )knk x

0 such that ( )knknk x = ,

( )knknk x00 = and ( )knknk x

00 = ; 0Nk , Nn which satisfy

0nknnk = ;

0nknnk = ;

( )D

nknk

nknknk

100

2112

1

+

+

, 0

00 >+ nknkmin ; N n,k (3.32a)

( )

=

+

+=

100

2112

1

nknk

nknk

nnk

nlimDlim ; Nk . (3.32b)

Note that by using the Euclidean distance and norm on R , it is possible to apply the theoretical

formalism to the expressions ( )002 ,xdF,xTdx/D knkn

nk = + ; N n,k to prove

convergence to the best proximity points 2/D of the sequences { 12 xT n and { 112 xT n+ , respectivelyif [ ) ,/Dx 21 and conversely if ( ]21 /D,x . Assume that:

1) The constraints (3.32) hold;

FPTA_274_edited 25


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26

2) The parametrical constraints of the various parts (a) to (d) of Example 2.9 hold with the replacements

and its appropriate replacements of the constraints nk(Example 2.9) 0nk (Example 3.7), nk

(Example 2.9) 0nk (Example 3.7);

3) { }nk and { }nk are redefined for this example from0

nk and0

nk , respectively, from (3.32).

From Theorem 3.5, the various properties of Example 2.9 hold also for this example if 0=D so that the

cyclic self-map is such that it alternates the values of the solution sequence between +0R and 0R . The

unique fixed point to which the solution converges is { }0 . If 0>D , then the corresponding results are

modified by convergence to each of the unique best proximity points of the sequences { 12 xT n and

{ 112 xT n+ ; [ ) ( ]2,,21 /D/Dx .

Acknowledgements

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651,

and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-

PE12UN015. He is also grateful to the University of Basque Country for its financial support through Grant UFI

2011/07 and to the referees for their useful comments.

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[36] E. Karapinar, Best proximity points of Kannan type cyclic weak phi- contractions in ordered metric spaces, Anale Stiintifice

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[37] E. Karapinar, Best proximity points of cyclic mappings,Applied Mathematics Letters, 25, (2012), No: 11, 1761-1766.

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