- Research Article
- Open Access

# Convergence Theorems on Generalized Equilibrium Problems and Fixed Point Problems with Applications

- Yan Hao
^{1}Email author, - SunYoung Cho
^{2}and - Xiaolong Qin
^{3}

**2010**:189036

https://doi.org/10.1155/2010/189036

© Yan Hao et al. 2010

**Received:**2 October 2009**Accepted:**23 January 2010**Published:**21 February 2010

## Abstract

The purpose of this work is to introduce an iterative method for finding a common element of a solution set of a generalized equilibrium problem, of a solution set solutions of a variational inequality problem and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces.

## Keywords

- Hilbert Space
- Variational Inequality
- Convex Subset
- Nonexpansive Mapping
- Common Element

## 1. Introduction and Preliminaries

Let be a real Hilbert space, a nonempty closed and convex subset of and a nonlinear mapping. Recall the following definitions.

The classical variational inequality problem is to find such that

In this paper, we use to denote the solution set of the problem (1.4). One can easily see that the variational inequality problem is equivalent to a fixed point problem. is a solution to the problem (1.4) if and only if is a fixed point of the mapping , where is a constant and is the identity mapping.

Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of . Recall the following definitions.

For such a case,
is called a
*-strict pseudocontraction*.

Clearly, the class of strict pseudocontractions falls into the one between a class of nonexpansive mappings and a class of pseudocontractions.

Recently, many authors considered the problem of finding a common element of the solution set of the variational inequality (1.4) and of fixed point set of a nonexpansive mapping in Hilbert spaces; see, for examples, [1–5] and the references therein.

In 2005, Iiduka and Takahashi [2] obtained the following theorem in a real Hilbert space.

Theorem 1 IT.

then converges strongly to

Let be an inverse-strongly monotone mapping, and a bifunction of into , where is the set of real numbers. We consider the following equilibrium problem:

In this paper, the set of such is denoted by , that is,

In the case of , the zero mapping, the problem (1.10) is reduced to

In this paper, we use to denote the solution set of the problem (1.12), which was studied by many others; see, for examples, [1, 3, 6–23] and the reference therein. In the case of , the problem (1.10) is reduced to the classical variational inequality (1.4). The problem (1.10) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instances, [15, 24].

To study the problems (1.10) and (1.12), we may assume that the bifunction satisfies the following conditions:

(A1) for all ;

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

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

Recently, S. Takahashi and W. Takahashi [21] considered the problem (1.12) by introducing an iterative method in a Hilbert space. To be more precise, they proved the following theorem.

Theorem 1 TT1.

Then and converge strongly to where

Very recently, S. Takahashi and W. Takahashi [22] further considered the problem (1.10). Strong convergence theorems of common elements are established. More precisely, they obtained the following result.

Theorem 2 TT2.

Then, converges strongly to

In this paper, motivated by Theorem IT, Theorem TT1*,* and Theorem TT2, we introduce a general iterative method for the problem of finding a common element of a solution set of a generalized equilibrium problem (1.10), of a solution set of a variational inequality problem (1.4), and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others.

In order to prove our main results, we need the following lemmas.

The following lemmas can be found in [11, 24].

Lemma 1.1.

for all and Then, the following hold.

(1) is single-valued;

(3) ;

(4) is closed and convex.

Lemma 1.2 (see [25]).

Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each . Then, as , is nonexpansive and .

Lemma 1.3 (see [26]).

Then

The following lemma can be deduced from Bruck [8].

Lemma 1.4.

for is well defined, nonexpansive, and holds.

Lemma 1.5 (see [6]).

Let be a real Hilbert space, a nonempty closed and convex subset of and a nonexpansive mapping. Then is demiclosed at zero.

Lemma 1.6 (see [14]).

Let be a real Hilbert space, a nonempty closed and convex subset of and a -strict pseudocontraction. Then is closed and convex.

Lemma 1.7 (see [27]).

where is a sequence in and is a sequence such that

(a)

(b) or

Then

## 2. Main Results

Theorem 2.1.

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions

(R1)

(R2) , ;

(R3)

(R4) and .

Then the sequence defined by the iterative process (2.1) converges strongly to .

Proof.

The proof is divided into six steps.

Step 1.

Show that is well defined.

From Lemma 1.6, we see that is closed and convex. On the other hand, we see that the mapping , where , is nonexpansive. Indeed, for any , we have that

This shows that is nonexpansive mapping. Similarly, we can prove that , where is nonexpansive. It follows that , is closed and convex. From Lemma 1.1, we see that . Since is nonexpansive, we obtain that is closed and convex. This shows that is well defined.

Step 2.

Show that is bounded.

Put for each In view of Lemma 1.2 and (R4), we obtain that is nonexpansive and . Letting , we obtain that

This proves that the sequences and are bounded, too.

Step 3.

Show that as

Note that

Step 4.

Show that as

From the iterative process (2.1), we have

Step 5.

Show that where

To show that, we can choose a sequence of such that

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

Next, we show that . In fact, define a mapping by

Step 6.

Show that as

Notice that

In view of the restrictions (R2) and (2.30), we from Lemma 1.7 can conclude the desired conclusion easily. This completes the proof.

As corollaries of Theorem 2.1, we have the following results.

Corollary 2.2.

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions:

(R1)

(R2) , ;

(R3)

(R4) and for all .

Then the sequence defined by the iterative process (2.1) converges strongly to .

Proof.

Then, we can obtain the desired conclusion easily from Theorem 2.1. This completes the proof.

Corollary 2.3.

where is a fixed element in , , , , , and are sequences in , is sequence in , and . Assume that the above control sequences satisfy the following restrictions:

(R1)

(R2) , ;

(R3)

(R4) and .

Then the sequence defined by the above iterative process converges strongly to .

Proof.

It is easy to obtain the desired conclusion from Theorem 2.1.

Remark 2.4.

If is a contractive mapping and we replace by in the recursion formula (2.1), we can obtain the so-called viscosity iteration method. We note that all theorems and corollaries of this paper carry over trivially to the so-called viscosity iteration method; see [28] for more details.

## Declarations

### Acknowledgment

This project is supported by the National Natural Science Foundation of China (no. 10901140).

## Authors’ Affiliations

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