# Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings

- Atid Kangtunyakarn
^{1}Email author

**2010**:836714

https://doi.org/10.1155/2010/836714

© Atid Kangtunyakarn. 2010

**Received: **7 October 2010

**Accepted: **2 November 2010

**Published: **21 November 2010

## Abstract

We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.

## 1. Introduction

Let
be a real Hilbert space, and let
be a nonempty closed convex subset of
. Let
be a nonlinear mapping, and let
be a bifunction. A mapping
of
into itself is called *nonexpansive* if
for all
. We denote by
the set of fixed points of
(i.e.,
). Goebel and Kirk [1] showed that
is always closed convex, and also nonempty provided
has a bounded trajectory.

*strongly positive*with coefficient if there is a constant with the property

The set of solutions of (1.2) is denoted by . Many problems in physics, optimization, and economics are seeking some elements of , see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, 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.

In the case of , . Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of (1.5)

*inverse-strongly monotone*, see [5], if there exists a positive real number such that

The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping (see [6, 7]).

The ploblem of finding a common element of and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed point problem of nonexpansive mappings, see [8–10].

where is an -inverse strongly monotone mapping of into with positive real number , and , , , and proved strong convergence of the scheme (1.7) to , where in the framework of a Hilbert space, under some suitable conditions on , , and bifunction .

where is a contraction mapping and is -mapping generated by infinite family of nonexpansive mappings and infinite real number. Under suitable conditions of these parameters they proved strong convergence of the scheme (1.8) to , where .

## 2. Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

The following characterizes the projection .

Lemma 2.1 (see [13]).

Lemma 2.2 (see [14]).

where , satisfy the conditions

Lemma 2.3 (see [15]).

for is well defined, nonexpansive, and hold.

Lemma 2.4 (see [16]).

Let be a uniformly convex Banach space, a nonempty closed convex subset of , and a nonexpansive mapping. Then is demiclosed at zero.

Lemma 2.5 (see [17]).

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

(A4) for all is convex and lower semicontinuous.

The following lemma appears implicitly in [2].

Lemma 2.6 (see [2]).

Lemma 2.7 (see [3]).

for all . Then, the following hold:

In 2009, Kangtunyakarn and Suantai [18] defined a new mapping and proved their lemma as follows.

Definition 2.8.

This mapping is called
*-mapping* generated by
and
.

Lemma 2.9.

Let be a nonempty closed convex subset of strictly convex. Let be a finite family of nonexpanxive mappings of into itself with , and let , , where , , for all , for all . Let be the mapping generated by and . Then .

Lemma 2.10.

Let be a nonempty closed convex subset of Banach space. Let be a finite family of nonexpansive mappings of into itself and , , where , and such that as for and Moreover, for every , let and be the -mappings generated by and and and , respectively. Then for every .

Lemma 2.11 (see [19]).

## 3. Main Result

Theorem 3.1.

where such that , , , , . Assume that

Proof.

Thus is nonexpansive. By using the same proof, we obtain that and are nonexpansive.

We will divide our proof into 6 steps.

Step 1.

By Lemma 2.7, we have . By the same argument as above, we obtaine that

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

By Step 5, (3.87), and Lemma 2.2, we have , where . It easy to see that sequences , , and converge strongly to .

## 4. Application

Using our main theorem (Theorem 3.1), we obtain the following strong convergence theorems involving finite family of -strict pseudocontractions.

To prove strong convergence theorem in this section, we need definition and lemma as follows.

Definition 4.1.

Lemma 4.2 (see [20]).

Let be a nonempty closed convex subset of a real Hilbert space and a -strict pseudo contraction. Define by for each . Then, as is nonexpansive such that .

Theorem 4.3.

where , , such that , , , , . Assume that

Proof.

For every , by Lemma 4.2, we have is nonexpansive mappings. From Theorem 3.1, we can concluded the desired conclusion.

Theorem 4.4.

where , , such that , , . Assume that

Proof.

For every , by Lemma 4.2, we have that is nonexpansive mappings, putting , , , , and . From Theorem 3.1, we can conclude the desired conclusion.

## Declarations

### Acknowledgment

The authors would like to thank Professor Dr. Suthep Suantai for his suggestion in doing and improving this paper.

## Authors’ Affiliations

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