- Research Article
- Open Access

# Weak and Strong Convergence Theorems for Equilibrium Problems and Countable Strict Pseudocontractions Mappings in Hilbert Space

- Rudong Chen
^{1}Email author, - Xilin Shen
^{2}and - Shujun Cui
^{1}

**2010**:474813

https://doi.org/10.1155/2010/474813

© Rudong Chen et al. 2010

**Received:**27 August 2009**Accepted:**10 January 2010**Published:**2 February 2010

## Abstract

We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of strict pseudocontractions in Hilbert Space. Then we study the weak and strong convergence of the sequences.

## Keywords

- Hilbert Space
- Banach Space
- Convergence Theorem
- Equilibrium Problem
- Weak Convergence

## 1. Introduction

Let be a nonempty closed convex subset of a Hilbert space and let be a self-mapping of . Then is said to be a strict pseudocontraction mappings if for all , there exists a constant such that

(if (1.1) holds, we also say that is a -strict pseudocontraction). We use to denote the set of fixed points of , to denote weak(strong) convergence, and to denote the set of .

Let be a bifunction where is the set of real numbers. Then, we consider the following equilibrium problem:

The set of such is denoted by . Numerous problems in physics, optimization, and economics can be reduced to find a solution of (1.2). Some methods have been proposed to solve the equilibrium problem (see [1–3]). Recently, S. Takahashi and W. Takahashi [4] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also studied the strong convergence of the sequences generated by their algorithm for a solution of the EP which is also a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space.

In this paper, thanks to the condition introduced by Aoyama et al. [5], We introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problems and the set of fixed points of a countable family of strict pseudocontractions mappings in Hilbert Space. Then we study the weak and strong convergence of the sequences. The additional condition is inspired by Marino and Xu [6] and Kim and Xu [7].

## 2. Preliminaries

For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions (see [3]):

(A3) is upper-hemicontinuous, that is, for each ,

(A4) is convex and lower semicontinuous for each .

Let be a real Hilbert space. Then there hold the following well-known results:

If is a sequence in weakly convergent to , then

Recall that the nearest point projection from onto assigns to each its nearest point denoted by in ; that is, is the unique point in with the property

Given and , then if and only if there holds the following relation:

Lemma (see [6]).

Let be a nonempty closed convex subset of a real Hilbert space . Let : be a -strict pseudocontraction such that .

()(Demi-closed principle) is demi-closed on , that is, if and , then .

()The fixed point set of is closed and convex so that the projection is well defined.

Lemma (see [5]).

Lemma (see [8]).

Lemma (see [9]).

## 3. Weak Convergence Theorems

Theorem 3.1.

Assume that for all , where is a small enough constant, and is a sequence in with Let for any bounded subset of and let be a mapping of into itself defined by and suppose that Then the sequences and converge weakly to an element of .

Proof.

Next, we claim that . since is bounded and is reflexive, is nonempty. Let be an arbitrary element. Then a subsequence of converges weakly to . Hence, from (3.11) we know that As , we obtain that . Let us show . Since , we have

and hence . Since is a strict pseudocontraction mapping, by Lemma 2.1( ) we know that the mapping is demiclosed at zero. Note that and . Thus, . Consequently, we deduce that . Since is an arbitrary element, we conclude that .

To see that and are actually weakly convergent, we take Since exist for every , by (2.2), we have

## 4. Strong Convergence Theorems

Theorem 4.1.

Assume that for all , where is a small enough constant, and is a sequence in with and . Let for any bounded subset of and let be a mapping of into itself defined by Suppose that Then, converges strongly to .

Proof.

So is closed and convex. Then, is closed and convex.

Next, we show by induction that for all . is obvious. Suppose that for some . Let . Putting for all , we know from (4.1) that

and hence . This implies that for all .

This implied that is well defined.

Lastly, we show that the sequence converges to . Since is bounded and is reflexive, is nonempty. Let be an arbitrary element. Then a subsequence of converges weakly to . From Lemma 2.1 and (4.16), we obtain that . Next, we show . Let be an arbitrary element of . From and , we have

and hence . Lemma 2.3 and (4.6) ensure the strong convergence of to . This completes the proof.

## Declarations

### Acknowledgment

This work is supported by the National Science Foundation of China, Grant 10771050.

## Authors’ Affiliations

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## Copyright

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