# 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.

## 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;

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

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

(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:

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

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

Lemma 2.10.

Then, the following results hold:

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]).

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.

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

## References

- Kohsaka F, Takahashi W: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces.
*Archiv der Mathematik*2008, 91(2):166–177. 10.1007/s00013-008-2545-8MathSciNetView ArticleMATHGoogle Scholar - Mann WR: Mean value methods in iteration.
*Proceedings of the American Mathematical Society*1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3MathSciNetView ArticleMATHGoogle Scholar - Dotson WG Jr.: On the Mann iterative process.
*Transactions of the American Mathematical Society*1970, 149: 65–73. 10.1090/S0002-9947-1970-0257828-6MathSciNetView ArticleMATHGoogle Scholar - Iemoto S, Takahashi W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71: 2082–2089. 10.1016/j.na.2009.03.064MathSciNetView ArticleMATHGoogle Scholar - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123–145.MathSciNetMATHGoogle Scholar - Moudafi A: From alternating minimization algorithms and systems of variational inequalities to equilibrium problems.
*Communications on Applied Nonlinear Analysis*2009, 16(3):31–35.MathSciNetMATHGoogle Scholar - Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities.
*Computers & Mathematics with Applications*2001, 41(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9MathSciNetView ArticleMATHGoogle Scholar - Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.
*Journal of Computational and Applied Mathematics*2008, 214(1):186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleMATHGoogle Scholar - Yao Y, Liou Y-C, Yao J-C: A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems.
*Fixed Point Theory and Applications*2008, -15.Google Scholar - Ceng L-C, Wang C-Y, Yao J-C: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities.
*Mathematical Methods of Operations Research*2008, 67(3):375–390. 10.1007/s00186-007-0207-4MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations.
*Computers & Mathematics with Applications*2000, 40(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.
*Journal of Optimization Theory and Applications*2003, 118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar - Xu H-K: Viscosity approximation methods for nonexpansive mappings.
*Journal of Mathematical Analysis and Applications*2004, 298(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms.
*Mathematical Programming*1997, 78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.