# Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems

- Hong Yu Li
^{1}Email author and - Hong Zhi Li
^{2}

**2009**:362191

**DOI: **10.1155/2009/362191

© H. Li and H. Li. 2009

**Received: **26 August 2008

**Accepted: **9 January 2009

**Published: **28 January 2009

## Abstract

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed points of finite many nonexpansive mappings. We prove strong convergence of the iterative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for the minimization problem.

## 1. Introduction

The set of such solutions is denoted by EP( ). Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].

The set of solutions of variational inequality problem is denoted by . The variational inequality problem has been extensively studied in literatures, see, for example, [4, 5] and references therein.

where *T* is a nonexpansive mapping on
and
is a point on
.

where for all . Such a mapping is called -mapping generated by and . We know that is nonexpansive and , see [6].

They proved that converges strongly to , where is the metric projection from onto .

and proved that converges strongly to a point and also solves the variational inequality (1.8). For equilibrium problems, also see [10, 11].

and established a strong convergence theorem of to an element of .

where is defined by (1.5), is -cocoercive, and is a bounded linear operator. We should show that the sequences converge strongly to an element of . Our result extends the corresponding results of Qin et al. [16] and Colao et al. [9], and many others.

## 2. Preliminaries

Let be a monotone mapping of into , then if and only if . The following result is useful in the rest of this paper.

Lemma 2.1 (see [17]).

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

(1) ,

(2) or .

Then, .

Lemma 2.2 (see [18]).

Let be bounded sequences in Banach space satisfying and . Let be a sequence in with . Then, .

Lemma 2.3.

Lemma 2.4 (see [7]).

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

For solving the equilibrium problem for a bifunction , we assume that satisfies the following conditions:

(A1) for all ;

(A2) is monotone: for all ;

(A3)for all ;

(A4)for all is convex and lower semicontinuous.

The following result is in Blum and Oettli [1].

Lemma 2.5 (see [1]).

We also know the following lemmas.

Lemma 2.6 (see [19]).

for all . Then, the following holds:

(1) is single-valued;

(3) ;

(4) is closed and convex.

It is known that in this case is maximal monotone, and if and only if .

## 3. Strong Convergence Theorem

Theorem 3.1.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , be in and be in , is a constant in . Assume and

(i) , ;

(ii) , ;

(iii) for some with and ;

(iv) .

Proof.

We divide the proof into several steps.

Step 1.

is bounded.

hence is bounded, so is , .

Step 2.

.

Step 3.

for .

Step 4.

.

Step 5.

, where is the unique solution of variational inequality

Since is bounded, without loss of generality, we assume itself converges weakly to a point . We should prove .

with the same argument as used in [14], we can derive , since is maximal monotone, we know .

Divide by in both side yields , let , by (A3) we conclude . Therefore, .

Step 6.

The sequence converges strongly to .

so, by Lemma 2.1, we conclude . This completes the proof.

Putting and for all in Theorem 3.1, we obtain the following corollary.

Corollary 3.2.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself. Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , be a constant with . Let sequences , in and be a constant in . Assume and

(i) , ;

(ii) , for some with and ;

(iii) .

Putting and , , in Theorem 3.1, we obtain the following corollary.

Corollary 3.3.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , in and in . Assume and

(i) , ;

(ii) , .

## Authors’ Affiliations

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