- Research Article
- Open Access

# Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

- Poom Kumam
^{1, 2}and - Chaichana Jaiboon
^{2, 3}Email author

**2011**:347204

https://doi.org/10.1155/2011/347204

© Poom Kumam and Chaichana Jaiboon. 2011

**Received:**3 October 2010**Accepted:**5 March 2011**Published:**14 March 2011

## Abstract

We introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

## Keywords

- Variational Inequality
- Equilibrium Problem
- Monotone Mapping
- Nonexpansive Mapping
- Iterative Scheme

## 1. Introduction

*-contraction*on if there exists a constant such that for all . Let be a mapping. In the sequel, we will use to denote the set of

*fixed points*of ; that is, . In addition, let a mapping be called

*nonexpansive*, if , for all . It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive self-mapping on , then is nonempty; see, for example, [1]. Recall that a mapping is called

*strictly pseudo-contraction*if there exists a constant such that

where is a strict pseudo-contraction. Under appropriate restrictions on , it is proved that the mapping is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudo-contractions.

We see that is a solution of a problem (1.3) that implies that .

- (1)

- (2)

The set of solutions of (1.6) is denoted by . The variational inequality has been extensively studied in the literature. See, for example, [6–8] and the references therein.

The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3). Some authors have proposed some useful methods for solving the and ; see, for instance [5, 9–27]. In 1997, Combettes and Hirstoaga [10] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Next, we recall some definitions.

Definition 1.1.

Let be nonlinear mappings. Then is called

Remark 1.2.

It is obvious that any -inverse strongly monotone mappings is monotone and -Lipschitz continuous.

- (5)
A set-valued mapping is called a

*monotone*if, for all , and imply . - (6)
A monotone mapping is a

*maximal*if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies .

*normal cone*to when , that is,

Then is the maximal monotone and if and only if ; see [28].

where is an -inverse strongly monotone, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.13) converges weakly to some .

where is an -inverse strongly monotone, , , are three sequences in [0, 1], and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .

where is a potential function for (i.e., for ).

where is a -strict pseudo-contraction mapping and , are sequences in [0, 1]. They proved that under certain appropriate conditions over , , and , the sequences and converge strongly to some , which solves some variational inequality problems.

- (H)
is -strongly convex with constant and its derivative is sequentially continuous from weak topology to strong topology. We note that the condition (H) for the function is a very strong condition. We also note that the condition (H) does not cover the case and for each . Very recently, R. Wangkeeree and R. Wangkeeree [34] introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a -strict pseudo-contraction mapping, and the set of solutions of the variational inequality for an inverse strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition (H) for the sequences generated by these processes.

They proved that under certain appropriate conditions imposed on , , and , the sequence generated by (1.22) converges strongly to , where .

In the present paper, motivated and inspired by Qin et al. [35], Plubtieng and Punpaeng [32], Peng and Yao [17], R. Wangkeeree and R. Wangkeeree [34], and Y. Yao and J.-C. Yao [30], we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. Moreover, we apply our results for solving convex feasibility problems in Hilbert spaces. The results in this paper extend and improve some well-known results in [17, 30, 32, 34, 35].

## 2. Preliminaries

for all and .

The mapping
is called the *metric projection* of
onto
.

Further, for any and , if and only if , for all .

for all .

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [36]).

Lemma 2.2 (see [31]).

Assume that is a strongly positive linear bounded operator on with coefficient and . Then .

Lemma 2.3 (see [4]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudo-contraction with a fixed point. Then is closed and convex. Define by for each . Then is nonexpansive such that .

Lemma 2.4 (see [37]).

Let be a uniformly convex Banach spaces, be a nonempty closed convex subset of and be a nonexpansive mapping. Then is demi-closed at zero.

Lemma 2.5 (see [38]).

for is well defined, nonexpansive and holds.

In order to solve the mixed equilibrium problem, the following assumptions are given for the bifunction , and the set :

(A1) for all ;

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

(A3) for each , ;

(A4) for each is convex and lower semicontinuous;

(A5) for each is weakly upper semicontinuous;

(B2) is a bounded set.

Lemma 2.6 (see [39]).

for all . Then, the following holds:

(i)for each , ;

(ii) is single-valued;

(iv) ;

(v) is closed and convex.

Remark 2.7.

If , then is rewritten as .

Remark 2.8.

We remark that Lemma 2.6 is not a consequence of Lemma 3.1 in [5], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.9 (see [40]).

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

Lemma 2.10 (see [41]).

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

(1) ,

(2) or .

Then .

Lemma 2.11.

## 3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem and the set of a common solutions of the variational inequalities with inverse strongly monotone mappings in a real Hilbert space.

Theorem 3.1.

where , , , , and are sequences in and , are positive sequences. Assume that the control sequences satisfy the following restrictions:

(C1) ,

(C2) and ,

(C3) ,

(C4) ,

(C5) , , where are two positive constants,

(C6) , and , for some .

or equivalent , where is a metric projection mapping form onto .

Proof.

We divide the proof into seven steps.

Step 1.

We claim that the mapping where has a unique fixed point.

Since , it follows that is a contraction of into itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique element such that .

Step 2.

We claim that is nonexpansive.

where , for all implies that the mapping is nonexpansive and so is, .

Step 3.

We claim that is bounded.

Hence, is bounded, so are , , , , , , , and .

Step 4.

We claim that .

where is an appropriate constant such that , , , , .

Step 5.

We claim that the following statements hold:

(s1) ;

(s2) ;

(s3) ;

(s4) .

Step 6.

We claim that , where is the unique solution of the variational inequality , for all .

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

(a1) First, we prove that .

Assume also that and .

By Lemma 2.4, we have , that is, .

(a2) Now, we prove that .

Step 7.

We claim that .

We have . Applying Lemma 2.10 to (3.77), we conclude that converges strongly to in norm. This completes the proof.

If the mapping is nonexpansive, then . We can obtain the following result from Theorem 3.1 immediately.

Corollary 3.2.

where , , , , and are sequences in and , are positive sequences. Assume that the control sequences satisfy the following restrictions:

(C1) ,

(C2) and ,

(C3) ,

(C4) ,

(C5) , , where are two positive constants,

(C6) , and , for some .

or equivalent , where is a metric projection mapping form onto .

where is an integer and each is assumed to be the of solutions of equilibrium problem with the bifunction and the solution set of the variational inequality problem. There is a considerable investigation on in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [42, 43], computer tomography [44], and radiation therapy treatment planning [45].

The following result can be concluded from Theorem 3.1 easily.

Theorem 3.3.

where such that , are positive sequences and , are sequences in . Assume that the control sequences satisfy the following restrictions:

(C1) and ,

(C2) ,

(C3) , for each ,

(C4) , where is some positive constant for each ,

(C5) , for each .

or equivalent , where is a metric projection mapping form onto .

## Declarations

### Acknowledgments

The authors would like to thank the referees for their valuable suggestions to improve this paper. This work was supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand.

## Authors’ Affiliations

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