# A New Strong Convergence Theorem for Equilibrium Problems and Fixed Point Problems in Banach Spaces

- Weerayuth Nilsrakoo
^{1}Email author

**2011**:572156

https://doi.org/10.1155/2011/572156

© Weerayuth Nilsrakoo. 2011

**Received: **5 June 2010

**Accepted: **20 January 2011

**Published: **6 February 2011

## Abstract

We introduce a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.

## Keywords

## 1. Introduction

*equilibrium problem*is to find such that

The set of solutions of (1.1) is denoted by . The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases.

Using this functional, Matsushita and Takahashi [2, 3] studied and investigated the following mappings in Banach spaces. A mapping
is *relatively nonexpansive* if the following properties are satisfied:

where and denote the set of fixed points of and the set of asymptotic fixed points of , respectively. It is known that satisfies condition (R3) if and only if is demiclosed at zero, where is the identity mapping; that is, whenever a sequence in converges weakly to and converges strongly to 0, it follows that . In a Hilbert space , the duality mapping is an identity mapping and for all . Hence, if is nonexpansive (i.e., for all ), then it is relatively nonexpansive.

for every
,
is relatively nonexpansive,
is an appropriate sequence in
, and
is an appropriate positive real sequence. They proved that if
is weakly sequentially continuous, then
converges *weakly* to some element in
.

Motivated by S. Takahashi and W. Takahashi [17] and Takahashi and Zembayashi [19], we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly smooth and uniformly convex Banach space.

## 2. Preliminaries

*strictly convex*if the following implication holds for :

It is known that if
is a uniformly convex Banach space, then
is reflexive and strictly convex. We say that
is *uniformly smooth* if the dual space
of
is uniformly convex. A Banach space
is *smooth* if the limit
exists for all norm one elements
and
in
. It is not hard to show that if
is reflexive, then
is smooth if and only if
is strictly convex.

for all and . The following lemma is an analogue of Xu's inequality [22, Theorem 2] with respect to .

Lemma 2.1.

It is also easy to see that if and are bounded sequences of a smooth Banach space , then implies that .

Lemma 2.2 (see [23, Proposition 2]).

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

Remark 2.3.

Following Alber [1], we denote such an element
by
. The mapping
is called the *generalized projection* from
onto
. It is easy to see that in a Hilbert space, the mapping
coincides with the metric projection
. Concerning the generalized projection, the following are well known.

Lemma 2.4 (see [23, Propositions 4 and 5]).

Let be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space , , and . Then,

Remark 2.5.

The generalized projection mapping above is relatively nonexpansive and .

for all and . Obviously, for all and . We know the following lemma (see [1] and [24, Lemma 3.2]).

Lemma 2.6.

Lemma 2.7 (see [25, Lemma 2.1]).

for all , where the sequences in and in satisfy conditions: , , and . Then, .

Lemma 2.8 (see [26, Lemma 3.1]).

For solving the equilibrium problem, we usually assume that a bifunction satisfies the following conditions:

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

(A4)for all , is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.9 (see [19, Lemma 2.8 ]).

for all . Then, the following hold:

(ii) is a firmly nonexpansive-type mapping [27], that is, for all

Lemma 2.10 (see [4, Lemma 2.3]).

Let be a nonempty closed convex subset of a Banach space , a bifunction from satisfying conditions (A1)–(A4) and . Then, if and only if for all .

Remark 2.11 (see [27]).

for all and . In particular, satisfies condition (R2).

Lemma 2.12 (see [3, Proposition 2.4]).

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

## 3. Main Results

In this section, we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1.

for all , where satisfying and , , and . Then, and converge strongly to .

Proof.

where for all . Notice that satisfying and .

The rest of the proof will be divided into two parts.

Case 1.

It follows from Lemma 2.7 and (3.8) that . Then, and so .

Case 2.

But for all , we conclude that , and .

From two cases, we can conclude that and converge strongly to and the proof is finished.

Applying Theorem 3.1 and [28, Theorem 3.2], we have the following result.

Theorem 3.2.

Then, and converge strongly to .

Setting and in Theorem 3.1, we have the following result.

Corollary 3.3.

for all , where satisfying and , . Then, and converge strongly to .

Letting in Corollary 3.3, we have the following result.

Corollary 3.4.

for all , where satisfying and , . Then converges strongly to .

Let be the identity mapping in Theorem 3.1, we also have the following result.

Corollary 3.5.

for all , where satisfying and , , and . Then, and converge strongly to .

## 4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, every nonexpansive mappings are relatively nonexpansive, and is the identity operator. We obtain the following result.

Theorem 4.1.

for all , where is the resolvent of , satisfying and , , and . Then, converges strongly to .

Remark 4.2.

In Theorem 4.1, we have the same conclusion if the mapping is only quasinonexpansive (i.e., and for all and ) such that is demiclosed at zero.

Letting in Theorem 4.1, we have the following result.

Corollary 4.3.

for all , where satisfying , , and . Then, converges strongly to .

Let be the identity mapping in Theorem 4.1, we have the following result.

Corollary 4.4.

for all , where is the resolvent of , satisfying , , and . Then converges strongly to .

Proof.

, and . Applying Theorem 4.1, converges strongly to .

Remark 4.5.

Corollary 4.4 improves and extends [29, Corollary 5.3]. More precisely, the conditions and are removed.

Applying Corollary 4.4 and [30, Theorem 8], we have the following result.

Corollary 4.6.

for all , where is the resolvent of , satisfying and and . Then, converges strongly to .

Remark 4.7.

Corollary 4.6 improves and extends [16, Corollary 3.4]. More precisely, the conditions and are removed.

## Declarations

### Acknowledgment

The author would like to thank the referees for their comments and helpful suggestions.

## Authors’ Affiliations

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