# Some Iterative Methods for Solving Equilibrium Problems and Optimization Problems

- Huimin He
^{1}Email author, - Sanyang Liu
^{1}and - Qinwei Fan
^{2}

**2010**:943275

https://doi.org/10.1155/2010/943275

© Huimin He et al. 2010

**Received: **3 September 2010

**Accepted: **29 October 2010

**Published: **31 October 2010

## Abstract

We introduce a new iterative scheme for finding a common element of the set of solutions of the equilibrium problems, the set of solutions of variational inequality for a relaxed cocoercive mapping, and the set of fixed points of a nonexpansive mapping. The results presented in this paper extend and improve some recent results of Ceng and Yao (2008), Yao (2007), S. Takahashi and W. Takahashi (2007), Marino and Xu (2006), Iiduka and Takahashi (2005), Su et al. (2008), and many others.

## Keywords

## 1. Introduction

We denote by the set of fixed points of the mapping .

which is called the variational inequality. For the recent applications, sensitivity analysis, dynamical systems, numerical methods, and physical formulations of the variational inequalities, see [1–24] and the references therein.

if and only if , where is the projection of the Hilbert space onto the closed convex set .

Moreover, is characterized by the properties and for all .

which is also known as the modified double-projection method. For the convergence analysis and applications of this method, see the works of Noor [3] and Y. Yao and J.-C. Yao [16].

for approximating a common element of the set of fixed points of a nonexpansive nonself mapping and the set of solutions of the equilibrium problem and obtained a strong convergence theorem in a real Hilbert space.

where is the fixed point set of a nonexpansive mapping and a potential function for (i.e., for ).

*α*-cocoercive map, Takahashi and Toyoda [13] introduced the following iterative process:

*α*-cocoercive, , is a sequence in (0,1), and is a sequence in . They showed that, if is nonempty, then the sequence generated by (1.14) converges weakly to some . Recently, Iiduka and Takahashi [14] proposed another iterative scheme as follows:

for every
, where
is *α*-cocoercive,
,
is a sequence in (0,1), and
is a sequence in
. They proved that the sequence
converges strongly to
.

and also obtained a strong convergence theorem by viscosity approximation method.

Inspired and motivated by the ideas and techniques of Noor [2, 3] and Y. Yao and J.-C. Yao [16] introduce the following iterative scheme.

*α*-inverse strongly monotone mapping of into , and let be a nonexpansive mapping of into itself such that . Suppose that and , are given by

where
,
, and
are the sequences in
and
is a sequence in
. They proved that the sequence
defined by (1.17) converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for *α*-inverse-strongly monotone mappings under some parameters controlling conditions.

where and is also the optimality condition for the minimization problem , where is a potential function for (i.e., for ). The results obtained in this paper improve and extend the recent ones announced by S. Takahashi and W. Takahashi [6], Iiduka and Takahashi [14], Marino and Xu [8], Chen et al. [15], Y. Yao and J.-C. Yao [16], Ceng and Yao [22], Su et al. [17], and many others.

## 2. Preliminaries

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

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

(A4) for each , is convex and lower semicontinuous.

for , is -strongly monotone. This class of maps are more general than the class of strongly monotone maps. It is easy to see that we have the following implication: -strongly monotonicity relaxed -cocoercivity.

We will give the practical example of the relaxed -cocoercivity and Lipschitz continuous operator.

Example 2.1.

Let , for all , for a constant ; then, is relaxed -cocoercive and Lipschitz continuous. Especially, is -strong monotone.

Then, is Lipschitz continuous.

Obviously, is -strong monotone.

- (5)

Then is the maximal monotone and if and only if ; see [1].

Related to the variational inequality problem (1.2), we consider the equilibrium problem, which was introduced by Blum and Oettli [19] and Noor and Oettli [20] in 1994. To be more precise, let be a bifunction of into , where is the set of real numbers.

which is known as the equilibrium problem. The set of solutions of (2.12) is denoted by . Given a mapping , let for all . Then if and only if for all , that is, is a solution of the variational inequality. That is to say, the variational inequality problem is included by equilibrium problem, and the variational inequality problem is the special case of equilibrium problem.

where is defined as in Lemma 2.7. Once we have the solutions of the equation (2.19), then it simultaneously solves the fixed points problems, equilibrium points problems, and variational inequalities problems. Therefore, the constrained set is very important and applicable.

where is a sequence in (0,1) and is a sequence such that

Lemma 2.3.

Lemma 2.4 (Marino and Xu [8]).

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

Lemma 2.5 (see [21]).

Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.6 (Blum and Oettli [19]).

Lemma 2.7 (Combettes and Hirstoaga [4]).

for all . Then, the following hold:

## 3. Main Results

Theorem 3.1.

Proof.

which implies that the mapping is nonexpansive.

Hence, is bounded, so are , , and .

where is an appropriate constant such that .

Hence, by Lemma 2.5, we obtain as .

Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say , that is, .

Correspondingly, there exists a subsequence of . Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that .

which is a contradiction. Thus, we have .

which implies that , We have and hence . That is, .

That is, (3.47) holds.

By (C1), (3.47), and (3.67), we get . Now applying Lemma 2.2 to (3.65) concludes that .

This completes the proof.

Remark 3.2.

Take , the method (3.1) will be changed as (3.68).

Remark 3.3.

The computational possibility of the resolvent, , of in Lemma 2.7 and Theorem 3.1 is well defined mathematically, but, in general, the computation of is very difficult in large-scale finite spaces and infinite spaces.

## 4. Applications

Theorem 4.1.

Proof.

Taking in Theorem 3.1, we can get the desired conclusion easily.

Theorem 4.2.

Proof.

Put for all and for all in Theorem 3.1. Then we have . we can obtain the desired conclusion easily.

## Declarations

### Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006 and National Science Foundation of China, no. 60974082.

## Authors’ Affiliations

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