- Research Article
- Open Access

# A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems

- Phayap Katchang
^{1}and - Poom Kumam
^{1, 2}Email author

**2010**:370197

https://doi.org/10.1155/2010/370197

© P. Katchang and P. Kumam 2010

**Received:**27 October 2009**Accepted:**16 March 2010**Published:**31 May 2010

## Abstract

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.

## Keywords

- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Maximal Monotone
- Real Hilbert Space

## 1. Introduction

*nonexpansive*if for all . We use to denote the set of fixed points of , that is, . It is assumed throughout the paper that is a nonexpansive mapping such that . Recall that a self-mapping is

*contraction*on if there exists a constant and such that . Let be a strongly positive bounded linear operator on : that is, there is a constant with property

*variational inclusion problem*, which is to find a point such that

where is the zero vector in . The set of solutions of problem (1.2) is denoted by .

which is called the *mixed quasivariational inequality* (see [1]).

This problem is called *Hartman-Stampacchia variational problem* (see [2–4]).

A set-valued mapping is called monotone if, for all , and imply that . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies that .

*resolvent operator*that is associate with and as follows:

where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone (see [5, 6]).

*mixed equilibrium problem*for finding such that

*x*is a solution of problem (1.7) implying that . If , then the mixed equilibrium problem (1.7) becomes the following

*equilibrium problem*that is to find such that

The set of solutions of (1.8) is denoted by EP*(F)*. Given a mapping
, let
for all
. Then
if and only if
for all
that is,
is a solution of the variational inequality. The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). Some methods have been proposed to solve the equilibrium problem (see [8–21]).

They proved that if and satisfy appropriate conditions, then converges strongly to , where .

where is a potential function for , that is, , for all .

where
. Such a mapping
is called the *W-mapping* generated by
and
. Nonexpansivity of each
ensures the nonexpansivity of
. Moreover, in [24, Lemma
], it is shown that
.

The concept of -mappings was introduced in [25, 26]. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in [24, 27, 28] and the references cited therein.

where is monotone and Lipschitz continuous mapping and is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences and of parameters satisfy appropriate conditions.

In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in [6], Peng et al. in [22], Peng and Yao in [29], and Plubtieng and Sriprad in [23], we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al. [6], Peng et al. [22], Peng and Yao [29], Plubtieng and Sriprad [23], and some authors.

## 2. Preliminaries

*metric projection*of onto It is well known that is a nonexpansive mapping of onto and satisfies

*inverse-strongly monotone*if there exists a positive real number such that

So, if then is a nonexpansive mapping from into itself.

*H*satisfies the Opial condition [30], that is, for any sequence with , the inequality

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction *F*,
, and the set *C*.

(A2) is monotone, that is, for all

(A4)For each is convex and lower semicontinuous.

(A5)For each is weakly upper semicontinuous.

(B2)*C* is a bounded set.

We need the following lemmas for proving our main result.

Lemma 2.1 (Peng and Yao [31]).

for all . Then, the following hold.

(3) is firmly nonexpansive, that is, for any

Lemma 2.2 (Xu [32]).

where is a sequence in and is a sequence in such that

Lemma 2.3 (Osilike and Igbokwe [33]).

Lemma 2.4 (Colao et al. [28]).

Lemma 2.5 (Suzuki [34]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose that for all integers and Then,

Lemma 2.6 (Marino and Xu [35]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.7 (Brézis [5]).

Let be a maximal monotone mapping and be a Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.

Remark 2.8.

Lemma 2.7 implies that is closed and convex if is a maximal monotone mapping and is a Lipschitz continuous mapping.

Lemma 2.9 (Zhang et al. [6]).

## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping in a real Hilbert space.

Theorem 3.1.

for every , where and satisfy the following conditions:

Proof.

Therefore, is bounded. We also obtain that and , are all bounded.

where is an approximate constant such that , .

Dividing by *t*, we get
. From (A3) and the weakly lower semicontinuity of
, we have
for all
and hence

which is a contradiction. Thus, we obtain .

It follows from the maximal monotonicity of that , that is, . This implies that .

where . By (3.65), we get . Hence by Lemma 2.2 to (3.66), we conclude that . This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2.

for every , where and satisfy the conditions (i)–(iii) in Theorem 3.1. Then converges strongly to

Proof.

Taking for , and , in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

Corollary 3.3.

Proof.

From Theorem 3.1 put ; then . So we have and . The conclusion of Corollary 3.3 can be obtained from Theorem 3.1 immediately.

## 4. Application

where is a convex and lower semicontinuous functional defined convex subset of a Hilbert space . We denote by the set of solutions of (4.1). Let be a bifunction defined by . We consider the equilibrium problem (1.8); it is obvious that . Therefore, from Theorem 3.1, we give the following corollary.

Corollary 4.1.

for every , where and satisfy the following conditions:

Proof.

From Theorem 3.1 put and . The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.

## Declarations

### Acknowledgments

The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.

## Authors’ Affiliations

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