- Research Article
- Open Access

# Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications

- Thanyarat Jitpeera
^{1}, - Uamporn Witthayarat
^{1}and - Poom Kumam
^{1}Email author

**2011**:971479

https://doi.org/10.1155/2011/971479

© Thanyarat Jitpeera et al. 2011

**Received:**5 January 2011**Accepted:**20 February 2011**Published:**13 March 2011

## Abstract

We introduce new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others.

## Keywords

- Variational Inequality
- Convex Function
- Equilibrium Problem
- Nonexpansive Mapping
- Complementarity Problem

## 1. Introduction

*nonexpansive*if , for all . The set of

*fixed points*of denoted by ; that is, . If is bounded, closed, and convex and is a nonexpansive mapping of into itself, then ; see, for instance, [1]. Let be a bifunction of into , where is the set of real numbers, a mapping, and a real-valued function. The

*generalized mixed equilibrium problem*is for finding such that

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problems as special cases [2–7].

*mixed equilibrium problem*[8] for finding such that

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

The set of solutions of (1.4) is denoted by , which this problem was studied by S. Takahashi and W. Takahashi [10].

*equilibrium problem*which is to find such that

*variational inequality problem,*denoted by , is to find such that

*monotone*if

where is the -mapping and , are , -inverse-strongly monotone mappings of into , respectively. He proved that if the sequences , , , , , , and of parameters satisfies appropriate conditions, then generated by (1.11) converges strongly to .

Recently, Shehu [34] motivated Chantarangsi et al. [35] who studied the problem of approximating a common element of the set of fixed points of an infinite family of nonexpansive mapping, the set of common solutions of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert spaces.

In this paper, motivated by the above results, we present a new hybrid iterative scheme for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces. Then, we prove strong convergence theorems under some mild conditions. Finally, we give some applications of our results. The results presented in this paper generalize, extend, and improve the results of Takahashi et al. [32], Kumam [20], Kangtunyakarn [33], and many authors.

## 2. Preliminaries

*metric projection*of onto . It is well known that is a nonexpansive mapping of onto and satisfies

holds for every with .

So, if , then is a nonexpansive mapping of into .

For solving the generalized mixed equilibrium problem, let us assume that the bifunction , the nonlinear mapping is continuous monotone, is convex, and lower semicontinuous satisfies the following conditions:

(A1) for all ,

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

(A4) is convex and lower semicontinuous for each ,

(B2) is a bounded set.

The following lemma appears implicitly in [2]. We need the following lemmas for proving our main result.

Lemma 2.1 (see [2]).

The following lemma was also given in [36].

Lemma 2.2 (see [36]).

for all . Then, the following hold:

(1) is single-valued,

(2) is firmly nonexpansive, that is, for any , ,

(3) ,

(4) is closed and convex.

Lemma 2.3 (see [37]).

for all . Then, the following hold:

(1) is single-valued,

(2) is firmly nonexpansive, that is, for any , ,

(3) ,

(4) is closed and convex.

Lemma 2.4 (see [38]).

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

(1) ,

(2) or .

Then, .

## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces.

Theorem 3.1.

for every , where , and satisfying the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

Then, converges strongly to .

Proof.

Therefore, we obtain and .

Next, we will divide the proof into four steps.

Step 1.

which shows that , for all , for all . So, , for all , for all . Therefore, it follows that , for all . This implies that is well defined.

Step 2.

We claim that and .

Step 3.

We claim that the following statements hold:

(S1) ,

(S2) ,

(S3) .

By (3.46) and (3.48), we have that , for all .

Step 4.

We show that .

which is a contradiction. Thus, we obtain .

This implies that . By the same arguments, we can show that .

Lastly, by the same proof of [39, Theorem 3.1, pages 346-347], we can show that and . Therefore, ; that is, .

By (2.4), again, we conclude that . This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2.

for every , where , , and satisfy the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

Then, converges strongly to .

Proof.

Taking for , to be nonexpansive mappings, in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

where denotes the identity operator on .

Lemma 3.3 (see [41]).

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

Using Theorem 3.1, we obtain the following result.

Theorem 3.4.

for every , where , , and satisfy the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

Then, converges strongly to .

Proof.

From Theorem 3.1, is a finite family of -strict pseudocontraction. By Lemma 3.3, we have that is nonexpansive mappings. The conclusion of Theorem 3.4 can be obtained from Theorem 3.1 immediately.

## 4. Some Applications

### 4.1. Complementarity Problem

*polar*of in to be the set

The set of solution of the complementarity problem is denoted by , . We will assume that , satisfies the following conditions:

(E1) , is a , -inverse-strongly monotone mapping, respectively,

(E2) .

(B2) is a bounded set.

Corollary 4.1.

for every , where , , and satisfy the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

Then, converges strongly to .

Proof.

Using Lemma 7.1.1 of [42], we have that and . Hence, by Corollary 4.1, we can conclude the desired conclusion easily. This completes the proof.

### 4.2. Optimization Problem

*optimization problem*with nonempty set of solutions:

where is a convex and lower semicontinuous functional, and define as a closed convex subset of a real Hilbert space . We denote the set of solutions of (4.5) by and . Let be a bifunction defined by . We consider the equilibrium problem, it is obvious that , . Therefore, from Theorem 3.1, we obtained the following corollary.

Corollary 4.2.

for every , where , , and satisfy the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

Proof.

From Theorem 3.1, put , , and . The conclusion of Corollary 4.2 can be obtained from Theorem 3.1 immediately.

### 4.3. Minimization Problem

In this section, we study the problem for finding a minimizer of a continuously Frèchet differentiable convex functional in a Hilbert space.

First, we use the following lemma in our result.

Lemma 4.3 (see [43]).

Let be a Banach space, let be a continuously Frèchet differentiable convex functional on , and let be the gradient of . If is -Lipschitz continuous, then is an -inverse-strongly monotone.

Let , be functionals on which satisfies the following conditions:

(C1)let, , be a continuously Frèchet differentiable convex functional on , and let, , be , -Lipschitz continuous, respectively,

(C2) and .

Corollary 4.4.

for every , where , , and satisfying the following conditions:

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) ,

(vi) .

Then, converges strongly to .

Proof.

We know form condition (C1) and Lemma 4.3 that , is a , -inverse-strongly monotone operator from in to itself, respectively. The conclusion of Corollary 4.4 can be obtained from Theorem 3.1 immediately.

## Declarations

### Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this work. The authors would like to thank The National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under the NRU-CSEC Project no. 54000267). Furthermore, P. Kumam's research supported by the Commission on Higher Education and the Thailand Research Fund under Grant no. MRG5380044.

## Authors’ Affiliations

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