Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces

Research ArticleIterative Algorithms for Mixed EquilibriumProblems System of Quasi-Variational Inclusionand Fixed Point Problem in Hilbert Spaces

Poom Kumam12 and Thanyarat Jitpeera3

1 Computational Science and Engineering Research Cluster (CSEC) King Mongkutrsquos University of Technology Thonburi (KMUTT)Bang Mod Thrung Khru Bangkok 10140 Thailand

2Department of Mathematics Faculty of Science King Mongkutrsquos University of Technology Thonburi (KMUTT) Bang ModThrung Khru Bangkok 10140 Thailand

3Department of Mathematics Faculty of Science and Agriculture Rajamangala University of Technology Lanna PhanChiangrai 57120 Thailand

Correspondence should be addressed toThanyarat Jitpeera tjitpeerahotmailcom

Received 27 April 2014 Accepted 26 June 2014 Published 24 July 2014

Academic Editor Xiaolong Qin

Copyright copy 2014 P Kumam and T Jitpeera This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce a new iterative algorithm for approximating a common element of the set of solutions formixed equilibriumproblemsthe set of solutions of a system of quasi-variational inclusion and the set of fixed points of an infinite family of nonexpansivemappings in a real Hilbert space Strong convergence of the proposed iterative algorithm is obtained Our results generalize extendand improve the results of Peng and Yao 2009 Qin et al 2010 and many authors

1 Introduction

Throughout this paper we assume that 119867 is a real Hilbertspace with inner product and norm denoted by ⟨sdot sdot⟩ and sdot respectively Let 119862 be a nonempty closed convex subset of119867A mapping 119879 119862 rarr 119862 is called nonexpansive if 119879119909 minus 119879119910 le

119909 minus 119910 forall119909 119910 isin 119862 They use 119865(119879) to denote the set of fixedpoints of 119879 that is 119865(119879) = 119909 isin 119862 119879119909 = 119909 It is assumedthroughout the paper that 119879 is a nonexpansive mapping suchthat 119865(119879) = 0 Recall that a self-mapping 119891 119862 rarr 119862 isa contraction on 119862 if there exists a constant 120572 isin [0 1) and119909 119910 isin 119862 such that 119891(119909) minus 119891(119910) le 120572119909 minus 119910

Let 120593 119862 rarr Rcup +infin be a proper extended real-valuedfunction and let 119865 be a bifunction of 119862 times 119862 into R where Ris the set of real numbers Ceng and Yao [1] considered thefollowing mixed equilibrium problem for finding 119909 isin 119862 suchthat

119865 (119909 119910) + 120593 (119910) ge 120593 (119909) forall119910 isin 119862 (1)

The set of solutions of (1) is denoted by MEP(119865 120593) We seethat 119909 is a solution of problem (1) which implies that 119909 isin

dom120593 = 119909 isin 119862 | 120593(119909) lt +infin If 120593 equiv 0 then the mixedequilibrium problem (1) becomes the following equilibriumproblem for finding 119909 isin 119862 such that

119865 (119909 119910) ge 0 forall119910 isin 119862 (2)

The set of solutions of (2) is denoted by EP(119865) The mixedequilibrium problems include fixed point problems vari-ational inequality problems optimization problems Nashequilibrium problems and the equilibrium problem as spe-cial cases Numerous problems in physics optimization andeconomics reduce to find a solution of (2) Some methodshave been proposed to solve the equilibriumproblem (see [2ndash14])

Let 119861 119862 rarr 119867 be a mapping The variational inequalityproblem denoted by VI(119862 119861) is for finding 119909 isin 119862 such that

⟨119861119909 119910 minus 119909⟩ ge 0 (3)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 271208 17 pageshttpdxdoiorg1011552014271208

2 Abstract and Applied Analysis

for all 119910 isin 119862 The variational inequality problem has beenextensively studied in the literature See for example [15 16]and the references therein A mapping 119861 of 119862 into119867 is calledmonotone if

⟨119861119909 minus 119861119910 119909 minus 119910⟩ ge 0 (4)

for all 119909 119910 isin 119862 119861 is called 120573-inverse-strongly monotone ifthere exists a positive real number 120573 gt 0 such that for all119909 119910 isin 119862

⟨119861119909 minus 119861119910 119909 minus 119910⟩ ge 1205731003817100381710038171003817119861119909 minus 119861119910

1003817100381710038171003817

2

(5)

We consider a system of quasi-variational inclusion for finding(119909lowast 119910lowast) isin 119867 times 119867 such that

120579 isin 119909lowastminus 119910lowast+ 1205881(1198611119910lowast+1198721119909lowast)

120579 isin 119910lowastminus 119909lowast+ 1205882(1198612119909lowast+1198722119910lowast)

(6)

where 119861119894 119867 rarr 119867 and 119872

119894 119867 rarr 2

119867 are nonlinearmappings for each 119894 = 1 2 The set of solutions of problem(6) is denoted by SQVI(119861

11198721 11986121198722) As special cases of

problem (6) we have the following

(1) If 1198611= 1198612= 119861 and119872

1= 1198722= 119872 then problem (6)

is reduced to (7) for finding (119909lowast 119910lowast) isin 119867 times 119867 such

that

120579 isin 119909lowastminus 119910lowast+ 1205881(119861119910lowast+119872119909

lowast)

120579 isin 119910lowastminus 119909lowast+ 1205882(119861119909lowast+119872119910

lowast)

(7)

(2) Further if 119909lowast = 119910lowast then problem (7) is reduced to (8)

for finding 119909lowast isin 119867 such that

120579 isin 119861119909lowast+119872119909

lowast (8)

where 120579 is the zero vector in 119867 The set of solutionsof problem (8) is denoted by 119868(119861119872) A set-valuedmapping 119872 119867 rarr 2

119867 is called monotone if forall 119909 119910 isin 119867 119891 isin 119872(119909) and 119892 isin 119872(119910) imply⟨119909 minus 119910 119891 minus 119892⟩ ge 0 A monotone mapping 119872 ismaximal if its graph 119866(119872) = (119891 119909) isin 119867 times 119867 119891 isin

119872(119909) of 119872 is not properly contained in the graphof any other monotone mapping It is known that amonotone mapping 119872 is maximal if and only if for(119909 119891) isin 119867times119867 ⟨119909minus119910 119891minus119892⟩ ge 0 for all (119910 119892) isin 119866(119872)

imply 119891 isin 119872(119909) Let 119861 be a monotone mapping of 119862into119867 and let119873

119862119910 be the normal cone to 119862 at 119910 isin 119862

that is119873119862119910 = 119908 isin 119867 ⟨119906 minus 119910 119908⟩ le 0 forall119906 isin 119862 and

define

119872119910 = 119861119910 + 119873

119862119910 119910 isin 119862

0 119910 notin 119862(9)

Then119872 is themaximal monotone and 120579 isin 119872119910 if andonly if 119910 isin VI(119862 119861) see [17]

Let 119872 119867 rarr 2119867 be a set-valued maximal monotone

mapping then the single-valued mapping 119869119872120582

119867 rarr 119867

defined by

119869119872120582

119909lowast= (119868 + 120582119872)

minus1119909lowast 119909lowastisin 119867 (10)

is called the resolvent operator associated with 119872 where 120582

is any positive number and 119868 is the identity mapping Thefollowing characterizes the resolvent operator

(R1) The resolvent operator 119869119872120582

is single-valued andnonexpansive for all 120582 gt 0 that is

1003817100381710038171003817119869119872120582 (119909) minus 119869119872120582

(119910)1003817100381710038171003817 le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119867 forall120582 gt 0

(11)

(R2) The resolvent operator 119869119872120582

is 1-inverse-stronglymonotone see [18] that is

1003817100381710038171003817119869119872120582 (119909) minus 119869119872120582

(119910)1003817100381710038171003817

2

le ⟨119909 minus 119910 119869119872120582

(119909) minus 119869119872120582

(119910)⟩ forall119909 119910 isin 119867

(12)

(R3) The solution of problem (8) is a fixed point of theoperator 119869

119872120582(119868 minus 120582119861) for all 120582 gt 0 see also [19] that

is

119868 (119861119872) = 119865 (119869119872120582

(119868 minus 120582119861)) forall120582 gt 0 (13)

(R4) If 0 lt 120582 le 2120573 then the mapping 119869119872120582

(119868 minus 120582119861) 119867 rarr

119867 is nonexpansive(R5) 119868(119861119872) is closed and convex

Let 119860 be a strongly positive linear bounded operator on119867 that is there exists a constant 120574 gt 0 with property

⟨119860119909 119909⟩ ge 1205741199092 forall119909 isin 119867 (14)

A typical problem is to minimize a quadratic function overthe set of the fixed points of a nonexpansive mapping on areal Hilbert space119867

min119909isin119865(119879)

1

2⟨119860119909 119909⟩ minus ℎ (119909) (15)

where 119860 is a strongly positive linear bounded operator and ℎ

is a potential function for 120574119891 (ie ℎ1015840(119909) = 120574119891(119909) for 119909 isin 119867)In 2007 Plubtieng and Punpaeng [20] proposed the

following iterative algorithm

119865 (119906119899 119910) +

1

119903119899

⟨119910 minus 119906119899 119906119899minus 119909119899⟩ ge 0 forall119910 isin 119867

119909119899+1

= 120598119899120574119891 (119909119899) + (119868 minus 120598

119899119860)119879119906

119899

(16)

They proved that if the sequences 120598119899 and 119903

119899 of parameters

satisfy appropriate conditions then the sequences 119909119899 and

119906119899 both converge to the unique solution 119911 of the variational

inequality

⟨(119860 minus 120574119891) 119911 119909 minus 119911⟩ ge 0 forall119909 isin 119865 (119879) cap EP (119865) (17)

Abstract and Applied Analysis 3

which is the optimality condition for the minimizationproblem

min119909isin119865(119879)capEP(119865)

1

2⟨119860119909 119909⟩ minus ℎ (119909) (18)

where ℎ is a potential function for 120574119891 (ie ℎ1015840(119909) = 120574119891(119909) for119909 isin 119867)

In 2009 Peng and Yao [21] introduced an iterativealgorithm based on extragradient method which solves theproblem for finding a common element of the set of solutionsof a mixed equilibrium problem the set of fixed points of afamily of finitely nonexpansive mappings and the set of thevariational inequality for a monotone Lipschitz continuousmapping in a real Hilbert space The sequences generated byV isin 119862 are

1199091= 119909 isin 119862

119865 (119906119899 119910) + 120593 (119910) minus 120593 (119906

119899) +

1

119903119899

⟨119910 minus 119906119899 119906119899minus 119909119899⟩ ge 0

forall119910 isin 119862

119910119899= 119875119862(119906119899minus 120574119899119861119906119899)

119909119899+1

= 120572119899V + 120573119899119909119899+ (1 minus 120572

119899minus 120573119899)119882119899119875119862(119906119899minus 120582119899119861119910119899)

(19)for all 119899 ge 1 where119882

119899is119882-mappingThey proved the strong

convergence theorems under some mild conditionsIn 2010 Qin et al [22] introduced an iterative method for

finding solutions of a generalized equilibrium problem theset of fixed points of a family of nonexpansive mappings andthe common variational inclusions The sequences generatedby 1199091isin 119862 and 119909

119899 are a sequence generated by

119865 (119906119899 119910) + ⟨119860

3119909119899 119910 minus 119906

119899⟩ +

1

119903119899

⟨119910 minus 119906119899 119906119899minus 119909119899⟩ ge 0

forall119910 isin 119862

119911119899= 119875119862(119906119899minus 1205821198991198602119906119899)

119910119899= 119875119862(119911119899minus 1205781198991198601119911119899)

119909119899+1

= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119882119899119910119899 forall119899 ge 1

(20)where119891 is a contraction and119860

119894is inverse-stronglymonotone

mappings for 119894 = 1 2 3 and 119882119899is called a 119882-mapping gen-

erated by 119878119899 1198781198991

1198781and 120574119899 120574119899minus1

1205741 They proved the

strong convergence theorems under some mild conditionsLiou [23] introduced an algorithm for finding a commonelement of the set of solutions of a mixed equilibriumproblem and the set of variational inclusion in a real Hilbertspace The sequences generated by 119909

0isin 119862 are

119865 (119906119899 119910) + 120593 (119910) minus 120593 (119906

119899)

+1

119903⟨119910 minus 119906

119899 119906119899minus (119909119899minus 119903119876119909

119899)⟩ ge 0 forall119910 isin 119862

119909119899+1

= 119875119862[(1 minus 120572

119899119860) 119869119872120582

(119906119899minus 120582119861119906

119899)]

(21)

for all 119899 ge 1 where 119860 is a strongly positive boundedlinear operator and119861119876 are inverse-stronglymonotoneTheyproved the strong convergence theorems under some suitableconditions

Next Petrot et al [24] introduced the new followingiterative process for finding the set of solutions of quasi-variational inclusion problem and the set of fixed point of anonexpansive mapping The sequence is generated by

1199090isin 119867 chosen arbitrary

119909119899+1

= 120572119899119891 (119909119899) + 120573119899119909119899+ 120574119899119878119911119899

119911119899= 119869119872120582

(119910119899minus 120582119860119910

119899)

119910119899= 119869119872120588

(119909119899minus 120588119860119909

119899)

(22)

for all 119899 isin N cup 0 where 120572119899 120573119899 120574119899 are three sequences

in [0 1] and 120582 isin (0 2120572] They proved that 119909119899 generated by

(22) converges strongly to 1199110which is the unique solution in

119865(119878) cap 119868(119860119872)In 2011 Jitpeera and Kumam [25] introduced a shrinking

projection method for finding the common element of thecommon fixed points of nonexpansive semigroups the set ofcommon fixed point for an infinite family the set of solutionsof a system of mixed equilibrium problems and the set ofsolution of the variational inclusion problem Let 119909

119899 119910119899

V119899 119911119899 and 119906

119899 be sequences generated by 119909

0isin 119862119862

1= 119862

1199091= 1198751198621

1199090 119906119899isin 119862 and

1199090= 119909 isin 119862 chosen arbitrary

119906119899= 119870119865119873

119903119873119899

119870119865119873minus1

119903119873minus1119899

119870119865119873minus2

119903119873minus2119899

sdot sdot sdot 1198701198652

1199032119899

1198701198651

1199031119899

119909119899

119910119899= 1198691198722120575119899

(119906119899minus 120575119899119861119906119899)

V119899= 1198691198721120582119899

(119910119899minus 120582119899119860119910119899)

119911119899= 120572119899V119899+ (1 minus 120572

119899)1

119905119899

int

119905119899

0

119878 (119904)119882119899V119899119889119904

119862119899+1

= 119911 isin 1198621198991003817100381710038171003817119911119899 minus 119911

1003817100381710038171003817

2

le1003817100381710038171003817119909119899 minus 119911

1003817100381710038171003817

2

minus 120572119899(1 minus 120572

119899)

times

10038171003817100381710038171003817100381710038171003817

V119899minus

1

119905119899

int

119905119899

0

119878 (119904)119882119899V119899119889119904

10038171003817100381710038171003817100381710038171003817

2

119909119899+1

= 119875119862119899+1

1199090 119899 isin N

(23)

where 119870119865119896119903119896

119862 rarr 119862 119896 = 1 2 119873 We proved the strongconvergence theorem under certain appropriate conditions

In this papermotivated by the above results we introducea new iterative method for finding a common element ofthe set of solutions for mixed equilibrium problems the setof solutions of a system of quasi-variational inclusions andthe set of fixed points of an infinite family of nonexpansivemappings in a real Hilbert space Then we prove strongconvergence theorems which are connected with [5 26ndash29]Our results extend and improve the corresponding results of

4 Abstract and Applied Analysis

Jitpeera and Kumam [25] Liou [23] Plubtieng and Punpaeng[20] Petrot et al [24] Peng and Yao [21] Qin et al [22] andsome authors

2 Preliminaries

Let 119867 be a real Hilbert space with inner product ⟨sdot sdot⟩ andnorm sdot and let 119862 be a nonempty closed convex subset of119867 Then

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

= 1199092minus10038171003817100381710038171199101003817100381710038171003817

2

minus 2 ⟨119909 minus 119910 119910⟩

1003817100381710038171003817120582119909 + (1 minus 120582) 1199101003817100381710038171003817

2

= 1205821199092+ (1 minus 120582)

10038171003817100381710038171199101003817100381710038171003817

2

minus 120582 (1 minus 120582)

times1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

2

forall119909 119910 isin 119867 120582 isin [0 1]

(24)

For every point 119909 isin 119867 there exists a unique nearest point in119862 denoted by 119875

119862119909 such that

1003817100381710038171003817119909 minus 1198751198621199091003817100381710038171003817 le

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119910 isin 119862 (25)

119875119862is called themetric projection of119867 onto119862 It is well known

that 119875119862is a nonexpansive mapping of119867 onto 119862 and satisfies

⟨119909 minus 119910 119875119862119909 minus 119875119862119910⟩ ge

1003817100381710038171003817119875119862119909 minus 1198751198621199101003817100381710038171003817

2

forall119909 119910 isin 119867 (26)

Moreover 119875119862119909 is characterized by the following properties

119875119862119909 isin 119862 and

⟨119909 minus 119875119862119909 119910 minus 119875

119862119909⟩ le 0 (27)

1003817100381710038171003817119909 minus 1199101003817100381710038171003817

2

ge1003817100381710038171003817119909 minus 119875

1198621199091003817100381710038171003817

2

+1003817100381710038171003817119910 minus 119875

1198621199091003817100381710038171003817

2

forall119909 isin 119867 119910 isin 119862

(28)

Let 119861 be a monotone mapping of 119862 into 119867 In the contextof the variational inequality problem the characterization ofprojection (27) implies the following

119906 isin VI (119862 119861) lArrrArr 119906 = 119875119862(119906 minus 120582119861119906) 120582 gt 0 (29)

It is also known that119867 satisfies the Opial condition [30] thatis for any sequence 119909

119899 sub 119867 with 119909

119899 119909 the inequality

lim inf119899rarrinfin

1003817100381710038171003817119909119899 minus 1199091003817100381710038171003817 lt lim inf119899rarrinfin

1003817100381710038171003817119909119899 minus 1199101003817100381710038171003817 (30)

holds for every 119910 isin 119867 with 119909 = 119910For the infinite family of nonexpansive mappings of

1198791 1198792 and sequence 120582

119894infin

119894=1in [0 1) see [31] we define

the mapping119882119899of 119862 into itself as follows

1198801198990

= 119868

1198801198991

= 120582111987911198801198990

+ (1 minus 1205821) 1198801198990

1198801198992

= 120582211987921198801198991

+ (1 minus 1205822) 1198801198991

119880119899119873minus1

= 120582119873minus1

119879119873minus1

119880119899119873minus2

+ (1 minus 120582119873minus1

) 119880119899119873minus2

119882119899= 119880119899119873

= 120582119873119879119873119880119899119873minus1

+ (1 minus 120582119873) 119880119899119873minus1

(31)

Lemma 1 (Shimoji and Takahashi [32]) Let 119862 be a nonemptyclosed convex subset of a real Hilbert space119867 LetT = 119879

119894119873

119894=1

be a family of infinitely nonexpanxive mappings with 119865(T) =

⋂infin

119894=1119865(119879119894) = 0 and let 120582

119894 be a real sequence such that 0 lt

120582119894le 119887 lt 1 for every 119894 ge 1 Then

(1) 119882119899is nonexpansive and 119865(119882

119899) = ⋂

119899

119894=1119865(119879119894) for each

119899 ge 1(2) for each 119909 isin 119862 and for each positive integer 119896 the limit

lim119899rarrinfin

119880119899119896119909 exists

(3) the mapping 119882 119862 rarr 119862 defined by 119882119909 =

lim119899rarrinfin

119882119899119909 = lim

119899rarrinfin1198801198991119909 is a nonexpansive

mapping satisfying 119865(119882) = 119865(T) and it is called the119882-mapping generated by 119879

1 1198792 and 120582

1 1205822

(4) if 119870 is any bounded subset of 119862 thenlim119899rarrinfin

sup119909isin119870

119882119909 minus119882119899119909 = 0

For solving themixed equilibriumproblem let us give thefollowing assumptions for a bifunction 119865 119862 times 119862 rarr R anda proper extended real-valued function 120593 119862 rarr R cup +infin

satisfies the following conditions

(A1) 119865(119909 119909) = 0 for all 119909 isin 119862(A2) 119865 is monotone that is 119865(119909 119910) + 119865(119910 119909) le 0 for all

119909 119910 isin 119862(A3) for each 119909 119910 119911 isin 119862 lim

119905rarr0119865(119905119911 + (1 minus 119905)119909 119910) le

119865(119909 119910)(A4) for each 119909 isin 119862 119910 997891rarr 119865(119909 119910) is convex and lower

semicontinuous(A5) for each 119910 isin 119862 119909 997891rarr 119865(119909 119910) is weakly upper

semicontinuous(B1) for each119909 isin 119867 and 119903 gt 0 there exist a bounded subset

119863119909sube 119862 and 119910

119909isin 119862 such that for any 119911 isin 119862 119863

119909

119865 (119911 119910119909) + 120593 (119910

119909) +

1

119903⟨119910119909minus 119911 119911 minus 119909⟩ lt 120593 (119911) (32)

(B2) 119862 is a bounded set

We need the following lemmas for proving our mainresults

Lemma 2 (Peng and Yao [21]) Let 119862 be a nonempty closedconvex subset of 119867 Let 119865 119862 times 119862 rarr R be a bifunction thatsatisfies (A1)ndash(A5) and let 120593 119862 rarr R cup +infin be a properlower semicontinuous and convex function Assume that either(B1) or (B2) holds For 119903 gt 0 and 119909 isin 119867 define a mapping119879119903 119867 rarr 119862 as follows

119879119903(119909) = 119911 isin 119862 119865 (119911 119910) + 120593 (119910)

+1

119903⟨119910 minus 119911 119911 minus 119909⟩ ge 120593 (119911) forall119910 isin 119862

(33)

for all 119909 isin 119867 Then the following hold

(1) for each 119909 isin 119867 119879119903(119909) = 0

(2) 119879119903is single-valued