- Research Article
- Open Access

# A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

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

**2009**:374815

https://doi.org/10.1155/2009/374815

© C. Jaiboon and P. Kumam. 2009

**Received:**25 December 2008**Accepted:**4 May 2009**Published:**9 June 2009

## Abstract

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for -inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.

## Keywords

- Unique Solution
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme

## 1. Introduction

*contraction*if there exists a constant such that In addition, let be a nonlinear mapping. Let be the projection of onto . The classical variational inequality which is denoted by is to find such that

One can see that the variational inequality (1.1) is equivalent to a fixed point problem. The variational inequality has been extensively studied in literature; see, for instance, [2–6]. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Recall the following.

Remark 1.1.

- (5)
An operator is strongly positive on if there exists a constant with the property

- (6)
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is 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 . Let be a monotone map of into and let be the normal cone to at , that is, and define

- (7)
Let be a bifunction of into , where is the set of real numbers. The equilibrium problem for is to find such that

The set of solutions of (1.13) is denoted by . Given a mapping let for all . Then, if and only if for all Numerous problems in physics, saddle point problem, fixed point problem, variational inequality problems, optimization, and economics are reduced to find a solution of (1.13). Some methods have been proposed to solve the equilibrium problem; see, for instance, [10–16]. Recently, Combettes and Hirstoaga [17] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem.

In 1976, Korpelevich [18] introduced the following so-called extragradient method:

where is -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 (1.16) converges strongly to some .

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [21–24] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :

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

Furthermore, for finding approximate common fixed points of an infinite countable family of nonexpansive mappings under very mild conditions on the parameters. Wangkeeree [27] introduced an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of a countable family of nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by

where and are sequences in . It is proved that under certain appropriate conditions imposed on and , the sequence generated by (1.22) strongly converges to the unique solution , where which extend and improve the result of Kumam [14].

Definition 1.2 (see [21]).

Such a mapping is nonexpansive from to and it is called the -mapping generated by and .

On the other hand, Colao et al. [28] introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem (1.13) and the set of common fixed points of infinitely many nonexpansive mappings on . Starting with an arbitrary initial , define a sequence recursively by

where is a sequence in . It is proved [28] that under certain appropriate conditions imposed on and , the sequence generated by (1.24) strongly converges to , where is an equilibrium point for and is the unique solution of the variational inequality (1.20), that is, .

In this paper, motivated by Wangkeeree [27], Plubtieng and Punpaeng [26], Marino and Xu [25], and Colao, et al. [28], we introduce a new iterative scheme in a Hilbert space which is mixed bythe iterative schemes of (1.18), (1.19), (1.22), and (1.24) as follows.

where is the sequence generated by (1.23), and and satisfying appropriate conditions. We prove that the sequences , , and generated by the above iterative scheme (1.25) converge strongly to a common element of the set of solutions of the equilibrium problem (1.13), the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality (1.1) for a -inverse-strongly monotone mapping in Hilbert spaces. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree [27], Plubtieng and Punpaeng [26], Marino and Xu [25], Colao, et al. [28], and many others.

## 2. Preliminaries

We now recall some well-known concepts and results.

Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. We denote weak convergence and strong convergence by notations and , respectively.

Lemma 2.1 (see [25]).

That is, is strongly monotone with coefficient .

Lemma 2.2 (see [25]).

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

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

(*A* 1)
for all

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

(*A3*)for each

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

The following lemma appears implicitly in [30].

Lemma 2.3 (see [30]).

The following lemma was also given in [17].

Lemma 2.4 (see [17]).

for all . Then, the following holds:

(1) is single-valued;

(3)

(4) is closed and convex.

For each , let the mapping be defined by (1.23). Then we can have the following crucial conclusions concerning . You can find them in [31]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.5 (see [31]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, for every and , the limit exists.

for every . Such a is called the -mapping generated by and . Throughout this paper, we will assume that for every . Then, we have the following results.

Lemma 2.6 (see [31]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, .

Lemma 2.7 (see [32]).

If is a bounded sequence in , then .

Lemma 2.8 (see [33]).

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

Lemma 2.9 (see [34]).

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

(1)

(2) or

Then

Lemma 2.10.

Let be a real Hilbert space. Then for all

(1) ;

(2) .

## 3. Main Results

In this section, we prove the strong convergence theorem for infinitely many nonexpansive mappings in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), let be an infinitely many nonexpansive of into itself, and let be an -inverse-strongly monotone mapping of into such that . Let be a contraction of into itself with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by (1.25), where is the sequence generated by (1.23), , are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

(v) for some and .

Equivalently, one has

Proof.

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

We will divide the proof into five steps.

Step 1.

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

Step 2.

We claim that .

Observing that and we get

where is a constant such that for all

Combining (3.27) and (3.28), we have

Step 3.

We claim that the following statements hold:

(i)

(ii)

which together with (3.38) gives

Step 4.

We claim that where is the unique solution of the variational inequality

Since is a unique solution of the variational inequality (3.1), to show this inequality, we choose a subsequence of such that

and hence . From (A3), we have for all and hence

Next, we show that By Lemma 2.6, we have . Assume Since and it follows by the Opial's condition that

Step 5.

we also conclude that in norm. This completes the proof.

Corollary 3.2 ([28, Theorem 3.1]).

where is the sequence generated by (1.23), is a sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

Equivalently, one has

Proof.

Put , and in Theorem 3.1., then . The conclusion of Corollary 3.2 can obtain the desired result easily.

Corollary 3.3.

where , and are three sequences in and is a real sequence in satisfying the following conditions:

(i) and

(ii) and

(iii)

(iv) and

Equivalently, one has

Proof.

Put for all and for all . Then for all . The conclusion follows from Theorem 3.1.

Corollary 3.4.

where is the sequences generated by (1.23), and , , are three sequences in satisfying the following conditions:

(i) and

(ii) and

(iii)

(iv) for some and .

Equivalently, one has

Proof.

Put for all and for all in Theorem 3.1. Then, we have . So, by Theorem 3.1, we can conclude the desired conclusion easily.

If and in Theorem 3.1, then we can obtain the following result immediately.

Corollary 3.5.

where is the sequences generated by (1.23), , , are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii) and ;

(iv)

(v) and

(vi) for some and .

Equivalently, one has

Corollary 3.6.

where , , and are three sequences in , and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

Equivalently, one has

Proof.

Put and in Corollary 3.5. then . The conclusion of Corollary 3.6 can obtain the desired result easily.

## 4. Application for Optimization Problem

It is obvious that
, where
denotes the set of solution of equilibrium problem (4.2). In addition, it is easy to see that
satisfies the conditions (*A* 1)–(*A* 4) in Section 1. Therefore, from the Corollary 3.6, we know the following iterative sequence
defined by

where , , and are three sequences in and is a real sequence in satisfying the following conditions:

(i)

(ii) and

(iii)

(iv) and

Then, converges strongly to a point of optimization problem (4.1).

Then, converges strongly to a solution of optimization problem (4.1). In fact, the is the minimum norm point on the .

In fact, the solution of optimization problem (4.4) is named the minimum norm point on the closed convex set . From iterative algorithm (4.4) we obtain the following iterative algorithm (4.5), and is defined by

for any initial guess . Then, converges strongly to a minimum norm point on the closed convex set .

## Declarations

### Acknowledgments

The first author was supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5180034. Moreover, the authors would like to thank Professor Somyot Plubiteng for providing valuable suggestions, and they also would like to thank the referee for the comments.

## Authors’ Affiliations

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