# Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings

- Wei-You Zeng
^{1}, - Nan-Jing Huang
^{1}Email author and - Chang-Wen Zhao
^{2}

**2008**:714939

**DOI: **10.1155/2008/714939

© Wei-You Zeng et al. 2008

**Received: **17 July 2008

**Accepted: **11 November 2008

**Published: **19 November 2008

## Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.

## 1. Introduction

Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see [1–8]. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.

Such a mapping is called the -mapping generated by and , see [14].

The purpose of this paper is to develop an iterative algorithm for finding a common element of set of solutions of GMEP (1.2) and set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and extends the main result of S. Takahashi and W. Takahashi [12].

## 2. Preliminaries

We denote by the set of fixed points of a self-mapping on , that is, . It is well known that if is nonempty, bounded, closed, and convex and is nonexpansive, then is nonempty; see [15]. Let be a sequence of nonexpansive mappings of into itself, where is a nonempty closed convex subset of a real Hilbert space . Given a sequence in , we define a sequence of self-mappings on by (1.4). Then we have the following lemmas which are important to prove our results.

Lemma 2.1 (see [14]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, for every and the limit exists.

for every . Such a mapping is called the -mapping generated by and Throughout this paper, we will assume that for every . Since is nonexpansive, is also nonexpansive.

Lemma 2.2 (see [14]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that , and let be a sequence in for some . Then, .

If , then is said to be -convex. In particular, if for all , then is said to be strongly convex.

We denote for weak convergence and for strong convergence. A function is called weakly sequentially continuous at , if as for each sequence in converging weakly to . The function is called weakly sequentially continuous on if it is weakly sequentially continuous at each point of .

Lemma 2.3 (see [16]).

Let and . Then for , there must exist a point such that .

Let be a nonempty closed convex subset of a real Hilbert space and a multivalued mapping. For , let . Let be a real-valued function satisfying the following:

for each fixed , is convex and upper semicontinuous;

Let be a differentiable functional with Fréchet derivative at satisfying the following:

is sequentially continuous from the weak topology to the strong topology;

is Lipschitz continuous with Lipschitz constant .

Let be a function satisfying the following:

is affine in the first coordinate variable;

for each fixed , is sequentially continuous from the weak topology to the weak topology.

Let us consider the equilibrium-like function which satisfies the following conditions with respect to the multivalued mapping :

for each fixed , is an upper semicontinuous function from to , that is, and imply ;

for each fixed , is a concave function;

for each fixed , is a convex function.

It is easy to see that if , then is a solution of GMEP(1.2).

Lemma 2.4 (see [6]).

- (a)
the auxiliary problem (2.8) has a unique solution;

We also need the following lemmas for our main results.

Lemma 2.5 (see [17]).

Lemma 2.6.

Proof.

Note that Lemma 2.5 implies that exists. Suppose for some . It is obvious that and so inequality (2.12) implies that , which is a contradiction. Thus, . This completes the proof.

Lemma 2.7 (see [6]).

Lemma 2.8 (see [18]).

## 3. Iterative Algorithm and Convergence Theorem

Let be a nonempty closed convex subset of a real Hilbert space , a multivalued mapping, a contraction mapping with constant , and an -mapping generated by and , where sequence is nonexpansive. Let be a sequence in and a sequence in . We can develop Algorithm 3.1 for finding a common element of a set of fixed points of -mapping and a set of solutions of GMEP(1.2).

Algorithm.

We now prove the strong convergence of iterative sequence , , and generated by Algorithm 3.1.

Theorem 3.2.

Let be a nonempty closed convex bounded subset of a real Hilbert space , a multivalued -Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying conditions and :

Assume that is a Lipschitz function with Lipschitz constant which satisfies the conditions . Let be an -strongly convex function with constant which satisfies conditions and with . Let be an -mapping generated by and and , where sequence is nonexpansive. Let , and be sequences generated by Algorithm 3.1, where is a sequence in and in satisfying the following conditions:

Then the sequences and converge strongly to , and converges strongly to , where .

Proof.

for all and , where , , and . All the conclusions (a)–(e) of Lemma 2.4 hold.

Hence there exists a unique element such that . Noting that and , we get that .

Next, we prove that there exists , such that , , and as , where .

that is, . We conclude that as .

Then, , , and . It follows from Lemma 2.8 that and so . This completes the proof.

Remark 3.3.

Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi [12].

We now give some applications of Theorem 3.2. If the set-valued mapping in Theorem 3.2 is single-valued, then we have the following corollary.

Corollary 3.4.

Let be a nonempty closed convex bounded subset of a real Hilbert space , a Lipschitz continuous mapping with constant , a contraction mapping with constant . Let be a real-valued function satisfying the conditions and let be an equilibrium-like function satisfying the conditions and :

where is a sequence in and in satisfying conditions . Then the sequences and converge strongly to , where .

Corollary 3.5.

where is a sequence in and in satisfying conditions and . Then the sequences and converge strongly to , and converges strongly to , where .

Proof.

Let in Theorem 3.2 for , where is an identity mapping. Then for Thus, the condition is satisfied. Now Corollary 3.5 follows from Theorem 3.2. This completes the proof.

## Declarations

### Acknowledgments

The authors would like to thank the referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005).

## Authors’ Affiliations

## References

- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123-145.MATHMathSciNetGoogle Scholar - Huang N-J, Lan H-Y, Teo KL: On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces.
*Journal of Inequalities and Applications*2007, 2007:-14.Google Scholar - Giannessi F, Maugeri A, Pardalos PM (Eds):
*Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optimization and Its Applications*.*Volume 58*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001.Google Scholar - Flores-Bazán F: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case.
*SIAM Journal on Optimization*2000, 11(3):675-690.MATHMathSciNetView ArticleGoogle Scholar - Mosco U: Implicit variational problems and quasi variational inequalities. In
*Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), Lecture Notes in Mathematics*.*Volume 543*. Springer, Berlin, Germany; 1976:83-156.Google Scholar - Sahu DR, Wong N-C, Yao J-C: On convergence analysis of an iterative algorithm for finding common solution of generalized mixed equilibrium problems and fixed point problemes. to appear in Mathematical Inequalities & ApplicationsGoogle Scholar
- Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.
*Taiwanese Journal of Mathematics*2008, 12(6):1401-1432.MATHMathSciNetGoogle Scholar - Peng J-W, Yao J-C: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. to appear in Taiwanese Journal of MathematicsGoogle Scholar
- Glowinski R, Lions J-L, Tremolieres R:
*Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications*.*Volume 8*. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.Google Scholar - Moudafi A: Viscosity approximation methods for fixed-points problems.
*Journal of Mathematical Analysis and Applications*2000, 241(1):46-55. 10.1006/jmaa.1999.6615MATHMathSciNetView ArticleGoogle Scholar - Hirstoaga SA: Iterative selection methods for common fixed point problems.
*Journal of Mathematical Analysis and Applications*2006, 324(2):1020-1035. 10.1016/j.jmaa.2005.12.064MATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.
*Journal of Mathematical Analysis and Applications*2007, 331(1):506-515. 10.1016/j.jmaa.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Yao Y, Liou Y-C, Yao J-C: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.
*Fixed Point Theory and Applications*2007, 2007:-12.Google Scholar - Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications.
*Taiwanese Journal of Mathematics*2001, 5(2):387-404.MATHMathSciNetGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis. Fixed Point Theory and Its Application*. Yokohama, Yokohama, Japan; 2000:iv+276.Google Scholar - Nadler SB Jr.: Multi-valued contraction mappings.
*Pacific Journal of Mathematics*1969, 30: 475-488.MATHMathSciNetView ArticleGoogle Scholar - Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings.
*Mathematical and Computer Modelling*2000, 32(10):1181-1191. 10.1016/S0895-7177(00)00199-0MATHMathSciNetView ArticleGoogle Scholar - Xu H-K: Iterative algorithms for nonlinear operators.
*Journal of the London Mathematical Society*2002, 66(1):240-256. 10.1112/S0024610702003332MATHMathSciNetView ArticleGoogle Scholar

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