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

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## 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 Open image in new window such that Open image in new window , then the point Open image in new window is called a fixed point of Open image in new window . The set of fixed points of Open image in new window is denoted by Open image in new window . It is well known that Open image in new window is closed convex and also nonempty if Open image in new window has a bounded trajectory (see [1]).

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

where Open image in new window is a potential function for Open image in new window (i.e., Open image in new window for all Open image in new window ).

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

In the case of Open image in new window , Open image in new window is deduced to Open image in new window . In the case of Open image in new window , Open image in new window is also denoted by Open image in new window .

and proved that, under certain appropriate conditions over Open image in new window and Open image in new window , the sequences Open image in new window and Open image in new window both converge strongly to Open image in new window .

where Open image in new window is a potential function for Open image in new window .

where Open image in new window .

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 Open image in new window , then Open image in new window is a nonexpansive mapping of Open image in new window into Open image in new window .

for all Open image in new window , then Open image in new window is called the solution of this variational inequality. The set of all solutions of the variational inequality is denoted by Open image in new window .

In this paper, we need the following lemmas.

Lemma 2.1 (see [21]).

Lemma 2.2 (see [22]).

where Open image in new window , Open image in new window , and Open image in new window satisfy the conditions:

(1) Open image in new window , Open image in new window or, equivalently, Open image in new window ;

(2) Open image in new window ;

(3) Open image in new window Open image in new window , Open image in new window .

Then Open image in new window .

Therefore, the following lemma naturally holds.

Lemma 2.3.

Lemma 2.4 (see [3]).

Assume that Open image in new window is a strongly positive linear bounded operator on a Hilbert space Open image in new window with coefficient Open image in new window and Open image in new window . Then Open image in new window .

Lemma 2.5 (see [2]).

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence in Open image in new window such that

(1) Open image in new window ;

(2) Open image in new window or Open image in new window .

Then Open image in new window .

Lemma 2.6 (see [23]).

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

(A1) Open image in new window for all Open image in new window ;

(A2) Open image in new window is monotone, that is, Open image in new window for all Open image in new window ;

(A3) For each Open image in new window ,

(A4) For each Open image in new window , Open image in new window is convex and lower semicontinuous.

Then Open image in new window is well defined and the following hold:

(1) Open image in new window is single-valued;

(2) Open image in new window is firmly nonexpansive, that is, for any Open image in new window ,

(3) Open image in new window ;

(4) Open image in new window is closed and convex.

*α*-inverse strongly monotone mapping, then Open image in new window . In fact, since Open image in new window , one has

Hence, Open image in new window .

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

Lemma 2.7 (see [26]).

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

Lemma 2.8 (see [5]).

## 3. Main Results

Now, we give our main results in this paper.

Theorem 3.1.

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

(E2) Open image in new window ;

(E3) Open image in new window ;

(E4) Open image in new window with Open image in new window ;

(E5) Open image in new window .

Proof.

Next, we proceed the proof with following steps.

Step 1.

Open image in new window is bounded.

which shows that Open image in new window is bounded, so is Open image in new window .

Step 2.

Open image in new window as Open image in new window .

where Open image in new window .

where Open image in new window .

By applying Lemma 2.2 to (3.23), we obtain Open image in new window as Open image in new window .

Step 3.

Open image in new window as Open image in new window .

Step 4.

Open image in new window as Open image in new window .

Step 5.

Without loss of generality, we may further assume that Open image in new window . Obviously, to prove Step 5, we only need to prove that Open image in new window .

which is a contradiction. Therefore, Open image in new window . Hence, Open image in new window .

Step 6.

The sequence Open image in new window strongly converges to some point Open image in new window .

Thus, applying Lemma 2.5 to (3.49), it follows that Open image in new window as Open image in new window . This completes the proof.

By Theorem 3.1, we have the following direct corollaries.

Corollary 3.2.

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

(E1) Open image in new window and Open image in new window ;

(E2) Open image in new window for each Open image in new window ;

(E3) Open image in new window for all Open image in new window ;

(E4) Open image in new window with Open image in new window ;

(E5) Open image in new window , Open image in new window , Open image in new window .

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 Open image in new window as Open image in new window (see [4, 5, 28]). The proof method of Open image in new window is simple and different with ones of others.

## 4. Applications for Multiobjective Optimization Problem

where Open image in new window and Open image in new window are both the convex and lower semicontinuous functions defined on a closed convex subset of Open image in new window of a Hilbert space Open image in new window .

Note that, if we find a solution Open image in new window , then one must have Open image in new window obviously.

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

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