- Research Article
- Open Access

# Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities

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

**2010**:728028

https://doi.org/10.1155/2010/728028

© C. Jaiboon and P. Kumam. 2010

**Received:**31 October 2009**Accepted:**1 February 2010**Published:**14 February 2010

## Abstract

We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxed -cocoercive and -Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence of a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of the previously known results.

## Keywords

- Variational Inequality
- Positive Real Number
- Nonexpansive Mapping
- Contraction Mapping
- Maximal Monotone

## 1. Introduction

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences. It is well known that the variational inequalities are equivalent to the fixed point problems. This alternative equivalent formulation has been used to suggest and analyze in variational inequalities. In particular, the solution of the variational inequalities can be computed using the iterative projection methods. It is well known that the convergence of a projection method requires the operator to be strongly monotone and Lipschitz continuous. Gabay [2] has shown that the convergence of a projection method can be proved for cocoercive operators. Note that cocoercivity is a weaker condition than strong monotonicity.

Equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization which has been extended and generalized in many directions using novel and innovative technique; see [3, 4]. Related to the equilibrium problems, we also have the problem of finding the fixed points of the nonexpansive mappings. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of infinitely (finitely) many nonexpansive mappings; see [5–7] and the references therein. In this paper, we suggest and analyze a new iterative method for finding a common element of a set of the solutions of generalized equilibrium problems and a set of fixed points of an infinite family of nonexpansive mappings and the set solution of the variational inequality problems for a relaxed -cocoercive mapping in a real Hilbert space.

Let be a real Hilbert space and let be a nonempty closed convex subset of and is the metric projection of onto Recall that a mapping is contraction on if there exists a constant such that for all A mapping of into itself is called nonexpansive if for all We denote by the set of fixed points of , that is, . If is nonempty, bounded, closed, and convex and is a nonexpansive mapping of into itself, then is nonempty; see, for example, [8]. We recalled some definitions as follows.

Definition 1.1.

Let be a mapping. Then one has the following.

(1)
is called*monotone* if
for all

If we say that is firmly nonexpansive. It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

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

(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 mapping of
into
and let
be the *normal cone* to
at
, that is,

Define

Then is the maximal monotone and if and only if ; see [11, 12]

In addition, let be a inverse-strongly monotone mapping. Let be a bifunction of into , where is the set of real numbers. The generalized equilibrium problem for is to find such that

The set of such is denoted by that is,

**Special Cases**

The generalized equilibrium problem (1.7) is very general in the sense that it includes, as special case, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others (see, e.g., [4, 13]). Some methods have been proposed to solve the equilibrium problem and the generalized equilibrium problem; see, for instance, [5, 14–28]. Recently, Combettes and Hirstoaga [29] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Very recently, Moudafi [24] introduced an itertive method for finding an element of , where is an inverse-strongly monotone mapping and then proved a weak convergence theorem.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem for an -inverse-strongly monotone, Takahashi and Toyoda [30] introduced the following iterative scheme:

where is an -inverse-strongly monotone mapping, 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 . On the other hand, Shang et al. [31] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a relaxed -cocoercive mapping in a real Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial defined sequences recursively by

They proved that under certain appropriate conditions imposed on , , and , the sequence converges strongly to , where

In 2008, S. Takahashi and W. Takahashi [27] introduced the following iterative scheme for finding an element of under some mild conditions. Let be a nonempty closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that Suppose and let , , and by sequences generated by

where , and satisfy some parameters controlling conditions. They proved that the sequence defined by (1.15) converges strongly to a common element of .

On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [32–35] 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 a nonexpansive mapping in a real Hilbert space :

where
is the fixed point set of a nonexpansive mapping
on
and
is a given point in
. Assume that
is a *strongly positive bounded linear operator* on
; that is, there exists a constant
such that

In 2006, Marino and Xu [36] considered the following iterative method:

They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.18) converges strongly to the unique of the variational inequality

which is the optimality condition for the minimization problem

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

In 2008, Qin et al. [26] proposed the following iterative algorithm:

where is a strongly positive linear bounded operator and is a relaxed cocoercive mapping of into . They prove that if the sequences , and of parameters satisfy appropriate condition, then the sequences and both converge to the unique solution of the variational inequality

which is the optimality condition for the minimization problem

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

Furthermore, for finding approximate common fixed points of an infinite family of nonexpansive mappings under very mild conditions on the parameters, we need the following definition.

Definition 1.2 (see [37]).

Such a mappings is called the -mapping generated by and . It is obvious that is nonexpansive, and if then .

On the other hand, Yao et al. [38] introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite family of nonexpansive mappings on . Starting with an arbitrary initial , define sequences and recursively by

where is a sequence in . It is proved [38] that under certain appropriate conditions imposed on and , the sequence generated by (1.25) converges strongly to . Very recently, Qin et al. [6] introduced an iterative scheme for finding a common fixed points of a finite family of nonexpansive mappings, the set of solutions of the variational inequality problem for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems in a real Hilbert space. Starting with an arbitrary initial , define sequences and recursively by

where is a relaxed -cocoercive mapping and is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on and , the sequences and generated by (1.26) converge strongly to some point , which is a unique solution of the variation inequality:

and is also the optimality for some minimization problems.

In this paper, motivated by iterative schemes considered in (1.15), (1.25), and (1.26) we will introduce a new iterative process (3.4) below for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the generalized equilibrium problem, and the set of solutions of variational inequality problem for a relaxed -cocoercive mapping in a real Hilbert space. The results obtained in this paper improve and extend the recent ones announced by Yao et al. [38], S. Takahashi and W. Takahashi [27], and Qin et al. [6] and many others.

## 2. Preliminaries

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of We denote weak convergence and strong convergence by notations and , respectively. Recall that the (nearest point) projection from to assigns each the unique point in satisfying the property

The following characterizes the projection .

We need some facts tools in a real Hilbert space which are listed as follows.

Lemma 2.1.

Lemma 2.2 (see [39]).

where is the metric projection of onto .

It is clear from Lemma 2.2 that variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Lemma 2.3 (see [40]).

holds for each with .

Lemma 2.4 (see [36]).

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:

, for all

is monotone, that is, , for all

, for all

for each is convex and lower semicontinuous.

The following lemma appears implicitly in [4].

Lemma 2.5 (see [4]).

The following lemma was also given in [5].

Lemma 2.6 (see [5]).

for all . Then, the following holds:

(1) is single-valued;

(3)

(4) is closed and convex.

Remark 2.7.

Lemma 2.8 (see [41]).

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.

Using Lemma 2.8, one can define a mapping of into itself as follows:

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 2.9 (see [41]).

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

Lemma 2.10 (see [7]).

If is a bounded sequence in , then .

Lemma 2.11 (see [42]).

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

Lemma 2.12.

Let be a real Hilbert space. Then the following inequality holds:

(1) ,

(2)

for all .

Lemma 2.13 (see [43]).

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

(1) ,

(2) or

Then

## 3. Main Results

In this section, we prove a strong convergence theorem of a new iterative method (3.4) for an infinite family of nonexpansive mappings and relaxed -cocoercive mappings in a real Hilbert space.

We first prove the following lemmas.

Lemma 3.1.

Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be -inverse-strongly monotone. It , then is a nonexpansive mapping in .

Proof.

So, is a nonexpansive mapping of into .

Lemma 3.2.

Let be a real Hilbert space, let be a nonempty closed convex subset of and let be a relaxed -cocoercive and -Lipschitz continuous. If , , then is a nonexpansive mapping in .

Proof.

For any and , .

for all . Thus .

So, is a nonexpansive mapping of into .

Now, we prove the following main theorem.

Theorem 3.3.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction satisfying (A1)–(A4). Let

(1) be an infinite family of nonexpansive mappings of into ;

(2) be an -inverse strongly monotone mappings of into ;

(3) be relaxed -cocoercive and -Lipschitz continuous mappings of into .

where is the sequence generated by (1.24) and , , and are sequences in satisfy the following conditions:

and

,

, , for some with , ,

for some with .

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

Proof.

We will divide the proof of Theorem 3.3 into six steps.

Step 1.

We prove that there exists such that .

Therefore, is a contraction mapping of into itself. Therefore by the Banach Contraction Mapping Principle guarantee that has a unique fixed point, say . That is, .

Step 2.

We prove that is bounded.

From Lemma 2.6, we have for all .

Therefore, is bounded. We also obtain that , , , , , , , , and are all bounded.

Step 3.

We claim that and .

where .

where is a constant such that for all

where is an appropriate constant such that .

where is an appropriate constant such that .

where is an appropriate constant such that .

where is an appropriate constant such that .

where is an appropriate constant such that .

Step 4.

We claim that the following statements hold:

(i) ;

(ii) ;

(iii) .

Step 5.

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

- (a)
First, we prove .

Since is Lipschitz continuous, from (3.97), we have as .

- (b)
Next, we show that

Step 6.

Finally, we show that and converge strongly to .

Using (C1), (3.121), and (3.123), we get , and . Applying Lemma 2.13 to (3.126), we conclude that in norm. Finally, noticing we also conclude that in norm. This completes the proof.

Corollary 3.4.

where is the sequence generated by (1.24) and , , , and are sequences in and is a real sequence in satisfying the following conditions:

,

, , for some with , .

Then, and converge strongly to a point , where .

Proof.

Let be a sequence satisfying the restriction: , where . Then we can obtain the desired conclusion easily from Theorem 3.3.

Corollary 3.5.

where is the sequence generated by (1.24) and , , and are sequences in satisfying the following conditions:

,

, , for some with , .

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

Proof.

Put , for all and for all in Theorem 3.3. Then, we have . So, by Theorem 3.3, we can conclude the desired conclusion easily.

## Declarations

### Acknowledgments

The authors would like to express their thanks to the Faculty of Science KMUTT Research Fund for their financial support. 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 are extremely grateful to the referees for their helpful suggestions that improved the content of the paper.

## Authors’ Affiliations

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