# Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces

- Somyot Plubtieng
^{1}Email author and - Wanna Sriprad
^{1}

**2010**:962628

**DOI: **10.1155/2010/962628

© The Author(s). 2010

**Received: **1 August 2009

**Accepted: **19 November 2009

**Published: **31 December 2009

## Abstract

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi (2008 and 2009), and many others.

## 1. Introduction

Let be a real Banach space and the dual space of . Let be a nonempty closed convex subset of and a bifunction from to , where denotes the set of real numbers. The equilibrium problem is to find such that

The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all , that is, is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduced to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].

Recall that a mapping is said to be nonexpansive if

We denote by the set of fixed points of . If a Banach space is uniformly convex, is bounded, closed and convex, and is a nonexpansive mapping of into itself, then is nonempty; see [4] for more details. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, [5–13] and the references therein.

A popular method is the hybrid projection method developed by Nakajo and Takahashi [14], Kamimura and Takahashi [15], and Martinez-Yanes and Xu [16]; see also [5, 17–20] and references therein. Recently Takahashi et al. [21] introduced an alterative projection method, which is called the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings. In 2008, Takahashi and Zembayashi [12] introduced two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solution of an equilibrium problem in a Banach space. Then they prove strong and weak convergence of the sequences. Very recently, Takahashi and Zembayashi [13] proved a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using a new hybrid method.

On the other hand, motivated by Nakajo and Takahashi [14], Matsushita and Takahashi [17] reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Very recently, Aoyama et al. [22] introduce a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings. Let and

for all where is a nonempty closed convex subset of a Banach space; is a sequence in and is a sequence of nonexpansive mappings with some condition. They proved that defined by (1.3) converges strongly to a common fixed point of

Motivated and inspired by the research going on in this direction, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of relatively nonexpansive mappings in a Banach space by using the shrinking projection method. Further, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. The result obtained in this paper improves and extends the corresponding result of [13] and many others.

## 2. Preliminaries

Let be a real Banach space with norm and let be the dual of . For all and , we denote the value of at by . The normalized duality mapping from to is defined by

for . By Hahn-Banach theorem, is nonempty; see [4] for more details. We denote the strong convergence and the weak convergence of a sequence to in by and , respectively. We also denote the weak convergence of a sequence to in by A Banach space is said to be strictly convex if for all with and . It is also said to be uniformly convex if for each , there exists such that for with and . A uniformly convex Banach space has the Kadec-Klee property, that is, and imply . Let be the unit sphere of . Then the Banach space is said to be smooth provided that

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well know that if is smooth, strictly convex and reflexive, then the duality mapping is single valued, one-to-one and onto.

Let be a smooth, strictly convex and reflexive Banach space, and let be a nonempty closed convex subset of . Throughout this paper, we denote by the function defined by

It is obvious from the definition of the function that for all . Following Alber [23], the generalized projection from onto is defined by

If is a Hilbert space, then and is the metric projection of onto . We know the following lemmas for generalized projections.

Lemma 2.1 (Alber [23], Kamimura and Takahashi [15]).

Lemma 2.2 (Alber [23], Kamimura and Takahashi [15]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space ; let be a mapping from into itself. We denote by the set of fixed point of . A point is said to be an asymptotic fixed point of if there exists in which converges weakly to and . The set of asymptotic fixed points of will be denoted by . Following Matsushita and Takahashi [17], a mapping from into itself is said to be relatively nonexpansive if is nonempty, and .

The following lemma is according to Matsushita and Takahashi [17].

Lemma 2.3 (Matsushita and Takahashi [17]).

Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space , and let be a relatively nonexpansive mapping from into itself. Then is closed and convex.

We also know the following three lemmas.

Lemma 2.4 (Kamimura and Takahashi [15]).

Let be a uniformly convex and smooth Banach space, and let , be sequences in such that either or is bounded. If , then .

Lemma 2.5 (Xu [24], Z linescu [25, 26]).

Lemma 2.6 (Kamimura and Takahashi [15]).

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

is monotone, that is, for all ;

for each is convex and lower semicontinuous.

The following result is in Blum and Oettlli [1].

Lemma 2.7 (Blum and Oettlli [1]).

We also know the following lemmas.

Lemma 2.8 (Takahashi and Zembayashi [12]).

for all , Then, the following holds:

Lemma 2.9 (Takahashi and Zembayashi [12]).

## 3. Main Results

Let be a nonempty closed convex subset of a Banach space let be a family of mappings of into itself with and denotes the set of all weak subsequential limits of a bounded sequence in . is said to satisfy the NST -condition [27] if for every bounded sequence in ,

In this section, by using the NST -condition, we prove two strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of relatively nonexpansive mappings in a Banach space.

Theorem 3.1.

for every , where is the duality mapping on , satisfies and for some Suppose that satisfy the NST -condition. Then converges strongly to , where is the generalized projection of onto .

Proof.

Putting for all , we have from Lemma 2.9 that are relatively nonexpansive.

is convex. So, is a closed convex subset of for all .

Hence . This implies that for all . So, is well defined.

for all Thus is bounded and therefore and are also bounded.

Since has the Kadec-Klee property, it follows that Therefore, converges strongly to

As direct consequences of Theorem 3.1, we can obtain the following corollaries.

Corollary 3.2 (Takahashi and Zembayashi [13]).

for every , where is the duality mapping on , satisfies and for some Then converges strongly to , where is the generalized projection of onto .

Proof.

Put . Let be a bounded sequence in with and let . Then there exists subsequence of such that . It follows directly from the definition of that . Hence satisfies NST -condition, by Theorem 3.1; converges strongly to .

Corollary 3.3 (Takahashi and Zembayashi [12]).

for every , where is the duality mapping on and for some Then, converges strongly to

Proof.

Putting in Theorem 3.1, we obtain Corollary 3.3.

Corollary 3.4.

for every , where is the duality mapping on , satisfies . Suppose that satisfy the NST -condition. Then converges strongly to , where is the generalized projection of onto .

Proof.

Putting for all and in Theorem 3.1, we obtain Corollary 3.4.

Similarly as in the proof of Theorem 3.1, we can prove the following theorem.

Theorem 3.5.

for every , where is the duality mapping on , satisfies and for some Suppose that satisfy the NST -condition. Then converges strongly to , where is the generalized projection of onto .

Proof.

and hence . So, we have and therefore, Hence for all . This implies that is well defined. By the same argument as in proof of Theorem 3.1, we can prove that the sequence converges strongly to .

Setting in Theorem 3.5, we have the following result.

Corollary 3.6 (Takahashi and Zembayashi [12, Theorem ]).

for every , where is the duality mapping on , satisfies and for some Then, converges strongly to where is the generalized projection of onto .

## 4. Applications

In this section, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator in a Banach space by using the shrinking projection method.

Let
be a real Banach space. An operator
is said to be *monotone* if
whenever
. We denote the set
by
A monotone
is said to be *maximal* if its graph
is not properly contained in the graph of any other monotone operator. If
is maximal monotone, then the solution set
is closed and convex.

Let be a smooth, strictly convex and reflexive Banach space, and let be a maximal monotone operator. Then for each and , there corresponds a unique element satisfying

see Barbu [28] or Takahashi [4]. We define the *resolvent* of
by
. In other words,
for all
. We know that
is relatively nonexpansive and
for all
(see [4, 17]), where
denotes the set of all fixed points of
. We can also define, for each
, the *Yosida approximation* of
by
We know that
for all

We now consider the strong convergence theorem for finding a common element of the solution set of an equilibrium problem and the problem of finding a zero of a maximal monotone operator.

Theorem 4.1.

for every , where is the duality mapping on , , satisfy , and for some Then converges strongly to , where is the generalized projection of onto .

Proof.

for all . Letting , we get . Then, the maximality of implies . Hence by Theorem 3.1, converges strongly to

In case . Putting for all and in Theorem 4.1, we obtain the following corollary.

Corollary 4.2.

for every , where is the duality mapping on , , satisfy . Then converges strongly to , where is the generalized projection of onto .

Similarly as in the proof of Theorem 4.1, we can prove the following theorem.

Theorem 4.3.

for every , where is the duality mapping on , , satisfy , and for some Then converges strongly to , where is the generalized projection of onto .

Corollary 4.4.

for every , where is the duality mapping on , , satisfy . Then converges strongly to , where is the generalized projection of onto .

## Declarations

### Acknowledgments

The first author thanks the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand. This work is dedicated to Professor Wataru Takahashi on his retirement.

## Authors’ Affiliations

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