# A Method for a Solution of Equilibrium Problem and Fixed Point Problem of a Nonexpansive Semigroup in Hilbert's Spaces

- Nguyen Buong
^{1}Email author and - NguyenDinh Duong
^{2}

**2011**:208434

https://doi.org/10.1155/2011/208434

© Nguyen Buong and Nguyen Dinh Duong. 2011

**Received: **3 October 2010

**Accepted: **13 January 2011

**Published: **15 February 2011

## Abstract

We introduce a viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert's spaces.

## 1. Introduction

Let be a nonempty, closed, and convex subset of a real Hilbert space . Denote the metric projection from onto by . Let be a nonexpansive mapping on , that is, and , for all . We use to denote the set of fixed points of , that is, .

Let be a nonexpansive semigroup on a closed convex subset , that is,

(1)for each , is a nonexpansive mapping on ,

(4)for each , the mapping from into is continuous.

Denote by . We know [1, 2] that is a closed, convex subset in and if is bounded.

Assume that the bifunction satisfies the following set of standard properties:

(A3)for every , is weakly lower semicontinuous and convex,

Denote the set of solutions of (1.1) by . We also know [3] that is a closed convex subset in .

where and denote the set of solutions of an equilibrium problem involving by a bifunction on and the fixed point set of a nonexpansive semigroup on a closed convex subset , respectively.

In the case that , , , and , a nonexpansive mapping on , for all , (1.2) is the fixed point problem of a nonexpansive mapping. In 2000, Moudafi [4] proved the following strong convergence theorem.

Theorem 1.1.

Then, converges strongly to , where .

where , is a contraction, and are sequences of positive real numbers satisfying the conditions: , , and as .

where and are in satisfying the following conditions: , , , and is a positive divergent real sequence.

There were some methods proposed to solve equilibrium problem (1.1); see for instance [8–12]. In particular, Combettes and Histoaga [3] proposed several methods for solving the equilibrium problem.

In 2007, S. Takahashi and W. Takahashi [13] combinated the Moudafi's method with the Combettes and Histoaga's result in [3] to find an element . They proved the following strong convergence theorem.

Theorem 1.2.

Then, and converge strongly to , where .

for finding an element in the case that under the uniformly asymptotic regularity condition on the nonexpansive semigroup on .

for all , , , and be the sequences in (0,1), and , are the sequences in satisfy the following conditions:

The strong convergence of (1.12)-(1.13) and its corollaries are showed in the next section.

## 2. Main Results

We formulate the following facts needed in the proof of our results.

Lemma 2.1.

Lemma 2.2 (see [15]).

Let be a nonempty, closed, convex subset of a real Hilbert space . For any , there exists a unique such that , for all , and if and only if for all .

Lemma 2.3 (see [16]).

where and are sequences of real numbers such that , , and . Then, .

Lemma 2.4 (see [9]).

Lemma 2.5 (see [9]).

Then, the following statements hold:

Lemma 2.6 (see [17]).

Lemma 2.7 (Demiclosedness Principle [18]).

If is a closed convex subset of , is a nonexpansive mapping on , is a sequence in such that and , then .

Lemma 2.8 (see [19]).

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

Now, we are in a position to prove the following result.

Theorem 2.9.

Let and be two nonempty, closed, convex subsets in a real Hilbert space . Let be a bifunction from to satisfying conditions (A1)–(A4) with replaced by , let be a nonexpansive semigroup on such that and let be a contraction of into itself. Then, and generated by (1.12)-(1.13) converge strongly to , where .

Proof.

Hence, is a contraction of into itself. Since is complete, there exists a unique element such that . Such a is an element of , because .

So, for all and hence is bounded. Therefore, , , and are also bounded.

As is bounded, there exists a subsequence of the sequence which converges weakly to . From (2.37), we also have that converges weakly to . Since and and , are two closed convex subsets in , we have that .

Therefore, also converges weakly to , as .

From (2.33) it follows that as . This completes the proof.

Corollary 2.10.

where , , , and satisfy conditions (i)–(v). Then, and converge strongly to , where .

Proof.

From the proof of the theorem, in (2.12).

Corollary 2.11.

where is defined by (1.13) for all and , , , and satisfy conditions (i)–(v). Then, the sequences and converge strongly to , where .

Proof.

where is defined by (1.13) and , , , and satisfy conditions (i)–(v). This algorithm is different from Yao and Noor's algorithm (1.6), in which for all . It likes completely the Plubtieng and Punpaeng's algorithm (1.8), but converges under a new condition on .

## Declarations

### Acknowledgment

This work was supported by the Vietnamese National Foundation of Science and Technology Development.

## Authors’ Affiliations

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