- Research Article
- Open Access

# Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space

- Somyot Plubtieng
^{1}Email author and - Kamonrat Sombut
^{1}

**2010**:246237

https://doi.org/10.1155/2010/246237

© S. Plubtieng and K. Sombut. 2010

**Received:**26 January 2010**Accepted:**15 April 2010**Published:**24 May 2010

## Abstract

We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

## Keywords

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

## 1. Introduction

*nonexpansive*if for all , and a mapping is said to be

*firmly nonexpansive*if for all . Let be a smooth,strictly convex and reflexive Banach space; let be the duality mapping of and a nonempty closed convex subset of . A mapping is said to be

*nonspreading*if

for all
Let
be the set of fixed points of
, and
be nonempty; a mapping
is said to be *quasi-nonexpansive* if
for all
and

Remark 1.1.

In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [1].

where and satisfies the assumptions and and proved that in case is a Banach space, and is closed, and is continuous, then the convergence of to a point implies that . Recently, Dotson [3] proved that a Mann iteration process was applied to the approximation of fixed points of quasi-nonexpansive mappings in Hilbert space and in uniformly convex and strictly convex Banach spaces.

which is called a general system of variational inequalities where and are two constants. Moreover, if we add up the requirement that , then problem (1.10) reduces to the classical variational inequality .

Then, they proved that the iterative sequence converges strongly to a common element of the set of fixed points of a nonexpansive mapping and a general system of variational inequalities with inverse-strongly monotone mappings under some parameters controlling conditions.

In this paper, we introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping.

## 2. Preliminaries

for all Further, for all and , if and only if , for all .

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [11]).

Lemma 2.2 (see [12]).

Let be a Hilbert space, a closed convex subset of , and a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to y, then .

Lemma 2.3 (see [1]).

Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then the following are equivalent:

(1)there exists such that is bounded;

(2) is nonempty.

Lemma 2.4 (see [1]).

Let be a Hilbert space and a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then is closed and convex.

Lemma 2.5 (see [4]).

Let be a Hilbert space, a closed convex subset of , and a nonspreading mapping with . Then is demiclosed, that is, and imply

Lemma 2.6 (see [13]).

for every . Then, converges strongly to some .

Lemma 2.7 (see [14]).

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

(1) ,

(2) or

Then

For solving the mixed equilibrium problems for an equilibrium bifunction , let us assume that satisfies the following conditions:

(A1) for all ;

(A2) is monotone, that is, for all

(A3)for each , is weakly upper semicontinuous;

(A4)for each , is convex, semicontinuous.

The following lemma appears implicitly in [5, 15].

Lemma 2.8 (see [5]).

The following lemma was also given in [15].

Lemma 2.9 (see [15]).

for all . Then, the following hold:

(1) is single-valued;

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

(3)

(4) is closed and convex.

We note that Lemma 2.9 is equivalent to the following lemma.

Lemma 2.10.

Then, the following results hold:

(i)for each , ;

(ii) is single-valued;

(iv)

(v) is closed and convex.

Proof.

Define , for all . Thus, the bifunction satisfies, (A1)–(A4). Hence, by Lemmas 2.8 and 2.9, we have (i)–(v).

Lemma 2.11 (see [6]).

where .

Proof.

By a similar argument as in the proof of Proposition 2.1 in [6], we obtain the desired result.

We note from Lemma 2.11 that the mapping is nonexpansive. Moreover, if is a closed bounded convex subset of , then the solution of problem (1.8) always exists. Throughout this paper, we denote the set of solutions of (1.8) by

## 3. Main Result

In this section, we prove a weak convergence theorem for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the system of mixed equilibrium problems.

Theorem 3.1.

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.8), where

Proof.

Therefore, .

Corollary 3.2.

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.8), where

Setting and in Theorem 3.1, we have following result.

Corollary 3.3.

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.9), where

Setting in Theorem 3.1, we have the following result.

Corollary 3.4.

for all , where for some and satisfy . Then converges weakly to and is a solution of problem (1.7), where

## Declarations

### Acknowledgments

The authors would like to thank the referee for the insightful comments and suggestions. The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.

## Authors’ Affiliations

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