- Research Article
- Open Access

# Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces

- Somyot Plubtieng
^{1}Email author and - Sukanya Chornphrom
^{1}

**2010**:296759

https://doi.org/10.1155/2010/296759

© Somyot Plubtieng and Sukanya Chornphrom. 2010

**Received:**30 June 2010**Accepted:**13 December 2010**Published:**21 December 2010

## Abstract

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi (2007) and Iemoto and Takahashi (2009).

## Keywords

- Hilbert Space
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme
- Common Element

## 1. Introduction

The set of solution 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 reduce to find a solution of (1.1); see, for example, [1–9] and the references therein.

*nonexpansive*if for all , and a mapping is said to be

*firmly*

*nonexpansive*if for all . Let be a smooth, strictly convex and reflexive Banach space, and let be the duality mapping of and a nonempty closed convex subset of . A mapping is said to be

*nonspreading*if

for all
. Let
be the set of fixed points of
, and
nonempty; a mapping
is said to be *quasi-nonexpansive* if
for all
and
.

Remark 1.1.

In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [10, 11].

for all , where and is a sequence of . Strong convergence of this type iterative sequence has been widely studied: Wittmann [15] discussed such a sequence in a Hilbert space.

Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, [1, 2, 6, 7, 17–20] and the references therein. In 1997, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Recently, S. Takahashi and W. Takahashi [8] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let be a nonexpansive mapping. In 2008, Plubtieng and Punpaeng [7] introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem. Very recently, S. Takahashi and W. Takahashi [9] introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets.

In this paper, motivated by S. Takahashi and W. Takahashi [8] and Lemoto and Takahashi [16], we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces.

## 2. Preliminaries

holds for every with ; see [21, 22] for more details.

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

Lemma 2.1 (see [23]).

Lemma 2.2 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then the following are equivalent.

(1)There exists such that is bounded;

(2) is nonempty.

Lemma 2.3 (see [10]).

Let be a Hilbert space, a nonempty closed convex subset of . Let be a nonspreading mapping of into itself. Then is closed and convex.

Lemma 2.4.

Let be a real Hilbert space. Then for all ,

(1) ;

(2) .

Lemma 2.5 (see [24]).

Let , and let be sequences of real numbers such that

, for all ,

and .

Then, .

Lemma 2.6 (see [16]).

Let be a Hilbert space, a closed convex subset of , and a nonspreading mapping with . Then is demiclosed, that is, and imply .

Lemma 2.7 (see [16]).

Lemma 2.8 (see [25]).

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

(1) ;

(2) or .

Then .

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

(A1) ;

(A2) is monotone, that is, ;

(A3)for each , ;

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

The following lemma appears implicitly in [26].

Lemma 2.9 (see [26]).

The following lemma was also given in [4].

Lemma 2.10 (see [4]).

for all . Then, the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any , ;

(3) ;

(4) is closed and convex.

Lemma 2.11 (see [27]).

## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a nonspreading mapping and the set of solutions of the equilibrium problems.

Theorem 3.1.

for all , where and satisfy

, , ,

, ,

, and .

Then converges strongly to , where .

Proof.

for all . Put . We divide the proof into several steps.

Step 1.

By induction, we obtain that for all . So, is bound. Hence, , , and are also bounded.

Step 2.

Step 3.

Step 4.

Putting , we claim that the sequence converges strongly to . Indeed, we discuss two possible cases.

Case 1.

By (3.28) and , we immediately deduce by Lemma 2.8 that .

Case 2.

This completes the proof.

As direct consequences of Theorem 3.1, we obtain corollaries.

Corollary 3.2.

for all , where and satisfy

, , ,

, ,

, and .

Then converges strongly to , where .

## Declarations

### Acknowledgments

The authors would like to thank the referees for the insightful comments and suggestions. Moreover, the authors gratefully acknowledge the Thailand Research Fund Master Research Grants (TRF-MAG, MRG-WII515S029) for funding this paper.

## Authors’ Affiliations

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