# A Hybrid-Extragradient Scheme for System of Equilibrium Problems, Nonexpansive Mappings, and Monotone Mappings

- Jian-Wen Peng
^{1}Email author, - Soon-Yi Wu
^{2}and - Gang-Lun Fan
^{2}

**2011**:232163

https://doi.org/10.1155/2011/232163

© Jian-Wen Peng et al. 2011

**Received: **21 October 2010

**Accepted: **24 November 2010

**Published: **1 December 2010

## Abstract

We introduce a new iterative scheme based on both hybrid method and extragradient method for finding a common element of the solutions set of a system of equilibrium problems, the fixed points set of a nonexpansive mapping, and the solutions set of a variational inequality problems for a monotone and -Lipschitz continuous mapping in a Hilbert space. Some convergence results for the iterative sequences generated by these processes are obtained. The results in this paper extend and improve some known results in the literature.

## 1. Introduction

In this paper, we always assume that is a real Hilbert space with inner product and induced norm and is a nonempty closed convex subset of , is a nonexpansive mapping; that is, for all , denotes the metric projection of onto , and denotes the fixed points set of .

The set of solutions of (1.2) is denoted by . And it is easy to see that the set of solutions of (1.1) can be written as .

The set of solutions of (1.3) is denoted by .

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, [1–4].

where . They proved that the sequence generated by (1.5) converges strongly to .

for all , where , is monotone and -Lipschitz continuous mapping of into . She proved that, if is nonempty, the sequences and , generated by (1.6), converge to the same point .

They proved that under certain appropriate conditions imposed on and , the sequences and converge strongly to , where .

for every , where is the duality napping on , for all , and for all . They proved that the sequence generated by (1.8) converges strongly to if satisfies and for some .

On the other hand, Combettes and Hirstoaga [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [4] introduced a new viscosity approximation scheme based on the extragradient method for finding a common element of the set of solutions of problem (1.1), the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions to the variational inequality for a monotone, Lipschitz continuous mapping in a Hilbert space and obtained two strong convergence theorems. Colao et al. [3] introduced an implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem. Peng et al. [12] introduced a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed points set of a family of infinitely nonexpansive mappings, and the solution set of a variational inequality for a relaxed coercive mapping in a Hilbert space and obtained a strong convergence theorem.

In this paper, motivated by the above results, we introduce a new hybrid extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and -Lipschitz continuous mappings in a Hilbert space and obtain some strong convergence theorems. Our results unify, extend, and improve those corresponding results in [8, 11] and the references therein.

## 2. Preliminaries

A mapping of into is called monotone if for all . A mapping is called -Lipschitz continuous if there exists a positive real number such that for all .

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions which were imposed in [2]:

(A2) is monotone; that is, for any ;

(A4)for each is convex and lower semicontinuous.

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1 (See [2]).

Lemma 2.2 (See [1]).

for all . Then, the following statements hold:

## 3. Main Results

In this section, we will introduce a new algorithm based on hybrid and extragradient method to find a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of a nonexpansive mapping, and the set of solutions of the variational inequality for monotone and -Lipschitz continuous mappings in a Hilbert space and show that the sequences generated by the processes converge strongly to a same point.

Theorem 3.1.

for each . If for some for some , and satisfies for each , then , , , and generated by (3.1) converge strongly to .

Proof.

we also have that is convex for each . Thus, , , , and are welldefined. By taking for and , for each , where is the identity mapping on . Then, it is easy to see that . We divide the proof into several steps.

Step 1.

and hence . This implies that for all .

Step 2.

Step 3.

It follows from and that (3.17) holds.

Step 4.

Step 5.

As is bounded, there exists a subsequence which converges weakly to . From , we obtain that . It follows from , , and that , , and .

and hence, for each , . From (A3), we have, for each , . Thus, .

which is a contradiction. Thus, we obtain .

From and the Kadec-Klee property of , we have , and hence . This implies that . It is easy to see that , , and . The proof is now complete.

By Theorem 3.1, we can easily obtain some new results as follows.

Corollary 3.2.

for each . If for some for some , and satisfies , then , , , and converge strongly to .

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.2.

Corollary 3.3.

for each . If for some and satisfies for each , then , , and converge strongly to .

Proof.

Let in Theorem 3.1, then complete the proof.

Corollary 3.4.

for each . If for some for some , then , , and converge strongly to .

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.4.

Remark 3.5.

Letting in Corollary 3.3, we obtain the Hilbert space version of Theorem 3.1 in [11]. Letting in Corollary 3.4, we recover Theorem 4.1 in [8]. Hence, Theorem 3.1 unifies, generalizes, and extends the corresponding results in [8, 11] and the references therein.

## Declarations

### Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grants 10771228 and 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), and the Research Project of Chongqing Normal University (Grant 08XLZ05). The authors are grateful to the referees for the detailed comments and helpful suggestions, which have improved the presentation of this paper.

## Authors’ Affiliations

## References

- Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MATHMathSciNetGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MATHMathSciNetGoogle Scholar - Colao V, Acedo GL, Marino G:
**An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(7–8):2708–2715. 10.1016/j.na.2009.01.115MATHMathSciNetView ArticleGoogle Scholar - Peng J-W, Yao J-C:
**A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(12):6001–6010. 10.1016/j.na.2009.05.028MATHMathSciNetView ArticleGoogle Scholar - Mann WR:
**Mean value methods in iteration.***Proceedings of the American Mathematical Society*1953,**4:**506–510. 10.1090/S0002-9939-1953-0054846-3MATHMathSciNetView ArticleGoogle Scholar - Genel A, Lindenstrauss J:
**An example concerning fixed points.***Israel Journal of Mathematics*1975,**22**(1):81–86. 10.1007/BF02757276MATHMathSciNetView ArticleGoogle Scholar - Reich S:
**Weak convergence theorems for nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*1979,**67**(2):274–276. 10.1016/0022-247X(79)90024-6MATHMathSciNetView ArticleGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MATHMathSciNetView ArticleGoogle Scholar - Korpelevič GM:
**An extragradient method for finding saddle points and for other problems.***Èkonomika i Matematicheskie Metody*1976,**12**(4):747–756.MathSciNetGoogle 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 - Takahashi W, Zembayashi K:
**Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-11.Google Scholar - Peng J-W, Wu S-Y, Yao J-C:
**A new iterative method for finding common solutions of a system of equilibrium problems, fixed-point problems, and variational inequalities.***Abstract and Applied Analysis*2010,**2010:**-27.Google Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MATHMathSciNetView ArticleGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MATHMathSciNetView ArticleGoogle 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.