- Research Article
- Open Access

# Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

- Shenghua Wang
^{1, 2}Email author and - Baohua Guo
^{1, 2}

**2011**:392741

https://doi.org/10.1155/2011/392741

© S.Wang and B. Guo. 2011

**Received:**15 October 2010**Accepted:**18 November 2010**Published:**2 December 2010

## Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al. (2008) and some others.

## Keywords

- Hilbert Space
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme

## 1. Introduction

If there exists a point such that , then the point is called a fixed point of . The set of fixed points of is denoted by . It is well known that is closed convex and also nonempty if has a bounded trajectory (see [1]).

Let be a fixed point, be a strongly positive linear bounded operator on and be a finite family of nonexpansive mappings of into itself such that .

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

where is a potential function for (However, Colao et al. pointed out in [5] that there is a gap in Yao's proof).

In the case of , is deduced to . In the case of , is also denoted by .

and proved that, under certain appropriate conditions over and , the sequences and both converge strongly to .

where is a potential function for .

where .

The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al. [5] and some others.

## 2. Preliminaries

*α*-inverse-strongly monotone mapping if there exists a positive real number

*α*such that

Hence, if , then is a nonexpansive mapping of into .

for all , then is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Lemma 2.2 (see [22]).

where , , and satisfy the conditions:

(1) , or, equivalently, ;

(2) ;

(3) , .

Then .

Therefore, the following lemma naturally holds.

Lemma 2.3.

Lemma 2.4 (see [3]).

Assume that is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.5 (see [2]).

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

(1) ;

(2) or .

Then .

Lemma 2.6 (see [23]).

Let C be a nonempty closed convex subset of a Hilbert space H and let be a bifunction which satisfies the following:

(A1) for all ;

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

(A3) For each ,

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

Then is well defined and the following hold:

(1) is single-valued;

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

(3) ;

(4) is closed and convex.

*α*-inverse strongly monotone mapping, then . In fact, since , one has

Hence, .

Such a mapping is called the - generated by and (see [5, 24, 25]).

Lemma 2.7 (see [26]).

Let be a nonempty closed convex subset of a Banach space. Let be nonexpansive mappings of into itself such that and let be real numbers such that for each and . Let be the -mapping of generated by and . Then .

Lemma 2.8 (see [5]).

## 3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

Let be a Hilbert space and be a nonempty closed convex subset of . Let be a contraction with coefficient , , be strongly positive linear bounded self-adjoint operators with coefficients and , respectively, be a finite family of nonexpansive mappings, be an infinite family of bifunctions satisfying be an infinite family of inverse-strongly monotone mappings with constants such that . Let and be two sequences in , be asequence in with , be a sequence in with and be a strictly decreasing sequence Set . Take a fixed number with . Assume that

(E1)

(E2) ;

(E3) ;

(E4) with ;

(E5) .

Proof.

Next, we proceed the proof with following steps.

Step 1.

is bounded.

which shows that is bounded, so is .

Step 2.

as .

where .

where .

By applying Lemma 2.2 to (3.23), we obtain as .

Step 3.

as .

Step 4.

as .

Step 5.

.

Without loss of generality, we may further assume that . Obviously, to prove Step 5, we only need to prove that .

which is a contradiction. Therefore, . Hence, .

Step 6.

The sequence strongly converges to some point .

Thus, applying Lemma 2.5 to (3.49), it follows that as . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

Let
be a Hilbert space and
be a nonempty closed convex subset of
. Let
be a contraction with coefficient
,
be strongly positive linear bounded self-adjoint operator with coefficient
,
be a finite family of nonexpansive mappings,
be a bifunction satisfying (A1)–(A4), and
be an *α*-inverse strongly monotone mapping such that
. Let
and
be two sequences in
,
be a sequence in
with
,
be a number in
, and
be a sequence
. Take a fixed number
with
. Assume that

(E1) and ;

(E2) for each ;

(E3) for all ;

(E4) with ;

(E5) , , .

Remark 3.3.

In the proof process of Theorem 3.1, we do not use Suzuki's Lemma (see [27]), which was used by many others for obtaining as (see [4, 5, 28]). The proof method of is simple and different with ones of others.

## 4. Applications for Multiobjective Optimization Problem

where and are both the convex and lower semicontinuous functions defined on a closed convex subset of of a Hilbert space .

Note that, if we find a solution , then one must have obviously.

By Theorem 3.1 with , , and for all , the sequence converges strongly to a solution , which is a solution of the multiobjective optimization problem (4.1).

## Authors’ Affiliations

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