- Research Article
- Open Access

# An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems

- Yonghong Yao
^{1}, - Yeong-Cheng Liou
^{2}Email author and - Yuh-Jenn Wu
^{3}

**2009**:632819

https://doi.org/10.1155/2009/632819

© Yonghong Yao et al. 2009

**Received:**2 November 2008**Accepted:**23 May 2009**Published:**29 June 2009

## Abstract

The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First, we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions.

## Keywords

- Iterative Algorithm
- Equilibrium Problem
- Nonexpansive Mapping
- Strong Convergence
- Real Hilbert Space

## 1. Introduction

Denote the set of solutions of (MEP) by and the set of solutions of (EP) by . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, [1–5]. Some methods have been proposed to solve the equilibrium problems, see, for example, [5–21].

- S.
Takahashi and W. Takahashi proved that the sequences and defined by (TT) converge strongly to with the following restrictions on algorithm parameters and :

(i) and ;

(ii) ;

(iii)(A1): ; and (R1): .

Subsequently, some iterative algorithms for equilibrium problems and fixed point problems have further developed by some authors. In particular, Zeng and Yao [16] introduced a new hybrid iterative algorithm for mixed equilibrium problems and fixed point problems and Mainge and Moudafi [22] introduced an iterative algorithm for equilibrium problems and fixed point problems.

where the parameters and satisfy the following conditions:

(i) for some ;

(ii) and .

Then, the sequences and generated by (CAY) converge weakly to an element of .

At this point, we should point out that all of the above results are interesting and valuable. At the same time, these results also bring us the following conjectures.

Questions

(1)Could we weaken or remove the control condition (iii) on algorithm parameters in S. Takahashi and W. Takahashi [8]?

(2)Could we construct an iterative algorithm for -strict pseudocontractions such that the strong convergence of the presented algorithm is guaranteed?

(3)Could we give some proof methods which are different from those in [8, 12, 16, 24].

It is our purpose in this paper that we introduce a general iterative algorithm for approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions. Our results give positive answers to the above questions.

## 2. Preliminaries

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of .

Let be a mapping. We use to denote the set of the fixed points of . Recall what follows.

For such case, we also say that is a -demicontractive mapping.

for all .

for all and .

for all .

It is worth noting that the class of demicontractive mappings includes the class of the nonexpansive mappings, the quasi-nonexpansive mappings and the strictly pseudo-contractive mappings as special cases.

It is well-known that the nonexpansive mappings, strictly pseudo-contractive mappings are all demiclosed. See, for example, [25–27].

for all .

In this paper, for solving problem (2.8) with an equilibrium bifunction , we assume that satisfies the following conditions:

(H1) is monotone, that is, for all ;

(H2) for each fixed , is concave and upper semicontinuous;

(H3) for each , is convex.

A differentiable function on a convex set is called

where is the Frechet derivative of at ;

We need the following important and interesting result for proving our main results.

Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Assume what follows.

(i) is Lipschitz continuous with constant such that

(a) ,

(b) is affine in the first variable,

(c)for each fixed is sequentially continuous from the weak topology to the weak topology.

(ii) is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology.

Then there hold the following:

(i) is single-valued;

where for ;

(iii) ;

(iv) is closed and convex.

## 3. Main Results

Let be a real Hilbert space, be a lower semicontinuous and convex real-valued function, be an equilibrium bifunction. Let be a mapping and be a mapping. In this section, we first introduce the following new iterative algorithm.

Algorithm 3.1.

Now we give a strong convergence result concerning Algorithm 3.1 as follows.

Theorem 3.2.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a demiclosed and -demicontractive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that

(a) ,

(b) is affine in the first variable,

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

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iv) for some , and .

Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.

Proof.

where .

This implies that is bounded, so are and .

Now we divide two cases to prove that converges strongly to .

Case 1.

Assume that the sequence is a monotone sequence. Then is convergent. Setting .

(i)If , then the desired conclusion is obtained.

Since is demiclosed, then we obtain .

for all and hence .

Since and is bounded, hence the last inequality is a contraction. Therefore, , that is to say, .

Case 2.

Hence , that is, converges strongly to . Consequently, it easy to prove that and converge strongly to . This completes the proof.

Remark 3.3.

The advantages of these results in this paper are that less restrictions on the parameters are imposed.

As direct consequence of Theorem 3.2, we obtain the following.

Corollary 3.4.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a nonexpansive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that;

(a) ,

(b) is affine in the first variable,

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

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iv) for some , and .

Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.

Corollary 3.5.

Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a strictly pseudo-contractive mapping such that . Assume what follows.

(i) is Lipschitz continuous with constant such that

(a) ,

(b) is affine in the first variable,

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

(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .

(iv) for some , and .

Then the sequences , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.

## Declarations

### Acknowledgment

The authors are extremely grateful to the anonymous referee for his/her useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant 10771050. The second author was partially supposed by the Grant NSC 97-2221-E-230-017.

## Authors’ Affiliations

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