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

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

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

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

The set of such Open image in new window is denoted by Open image in new window . 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, 2, 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 Open image in new window satisfies the following conditions (see [3]):

(A3) Open image in new window is upper-hemicontinuous, that is, for each Open image in new window , Open image in new window

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

Let Open image in new window be a real Hilbert space. Then there hold the following well-known results:

If Open image in new window is a sequence in Open image in new window weakly convergent to Open image in new window , then

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

Given Open image in new window and Open image in new window , then Open image in new window if and only if there holds the following relation:

Lemma (see [6]).

Let Open image in new window be a nonempty closed convex subset of a real Hilbert space Open image in new window . Let Open image in new window : Open image in new window be a Open image in new window -strict pseudocontraction such that Open image in new window .

()(Demi-closed principle) Open image in new window is demi-closed on Open image in new window , that is, if Open image in new window and Open image in new window , then Open image in new window .

()The fixed point set Open image in new window of Open image in new window is closed and convex so that the projection Open image in new window is well defined.

Lemma (see [5]).

Lemma (see [8]).

then Open image in new window .

Lemma (see [9]).

Further, if Open image in new window , then the following holds:

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

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

## 3. Weak Convergence Theorems

Theorem 3.1.

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

Proof.

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

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

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

Hence Open image in new window and proof is completed.

## 4. Strong Convergence Theorems

Theorem 4.1.

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

Proof.

So Open image in new window is closed and convex. Then, Open image in new window is closed and convex.

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

and hence Open image in new window . This implies that Open image in new window for all Open image in new window .

This implied that Open image in new window is well defined.

From Open image in new window , we have

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

and hence Open image in new window . Lemma 2.3 and (4.6) ensure the strong convergence of Open image in new window to Open image in new window . This completes the proof.

## Notes

### Acknowledgment

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

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