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

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Part of the following topical collections:
  1. Selected Papers from the 10th International Conference 2009 on Nonlinear Functional Analysis and Applications

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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]):

(A1) Open image in new window

(A2) Open image in new window

(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 .

() Open image in new window satisfies the Lipschitz condition

()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]).

Let Open image in new window be a nonempty closed convex subset of a Banach space and let Open image in new window be a sequence of mapping of Open image in new window into itself. Suppose Open image in new window Then, for each Open image in new window , Open image in new window converges strongly to some point of Open image in new window . Moreover, let Open image in new window be a mapping of Open image in new window into itself defined by

Then Open image in new window

Lemma (see [8]).

then Open image in new window .

Lemma (see [9]).

Let 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 bifunction from Open image in new window satisfying (A1), (A2), (A3), and(A4). Then, for any Open image in new window and Open image in new window , there exists Open image in new window such that

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 firmly nonexpansive, that is,

() Open image in new window

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

3. Weak Convergence Theorems

Theorem 3.1.

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 sequence of Open image in new window -strict pseudocontractions mappings on Open image in new window into itself with Open image in new window . Assume that Open image in new window Let Open image in new window be a bifunction satisfying (A1), (A2), (A3), (A4), and Open image in new window . Let Open image in new window and Open image in new window be sequence generated by Open image in new window and

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.

Pick Open image in new window . Then from the definition of Open image in new window in Lemma 2.4, we have Open image in new window , and therefore Open image in new window . It follows from (3.1) that
Since Open image in new window for all Open image in new window , we get Open image in new window that is, the sequence Open image in new window is decreasing. Hence Open image in new window exists. In particular, Open image in new window is bounded. Since Open image in new window is firmly nonexpensive, Open image in new window is also bounded. Also (3.2) implies that
Taking the limit as Open image in new window yields that
Since Open image in new window is bounded, it follows that
We apply Lemma 2.2 to get
Next, we claim that Open image in new window . Indeed, let Open image in new window be an arbitrary element of Open image in new window . Then as above
and hence
Therefore, from (3.2), we have
and hence
So, from the existence of Open image in new window , we have

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

By (A2), we have
and hence
From (A4), we have
Then, for Open image in new window and Open image in new window , from (A1), and (A4), we also have
Taking Open image in new window and using (A3), we get

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.

Let Open image in new window be a closed convex subset of a real Hilbert space Open image in new window . Let Open image in new window be a sequence of Open image in new window -strict pseudocontractions mappings on Open image in new window into itself with Open image in new window . Assume that Open image in new window Let Open image in new window be a bifunction satisfying (A1), (A2), (A3), (A4) and Open image in new window . For Open image in new window and Open image in new window , let Open image in new window and Open image in new window be sequence generated by Open image in new window and

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.

First, we show that Open image in new window is closed and convex. It is obvious that Open image in new window is closed and convex. Suppose that Open image in new window is closed and convex for some Open image in new window . For Open image in new window , we know that Open image in new window is equivalent to

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

In fact, from (4.8), we have
Since Open image in new window exists, we have that Open image in new window . On the other hand Open image in new window implies that
Further, we have
From (4.3), we have
On the other hand, we have
Then, we have
Therefore, we have
We apply Lemma 2.2 to get

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
Therefore, we have
As in the proof of Theorem 3.1, we have
By (A2), we have
and hence
From (A4), we have
Then, for Open image in new window and Open image in new window , from (A1) and (A4), we also have
Taking Open image in new window and using (A3), we get

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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Copyright information

© Rudong Chen et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsTianjin Polytechnic UniversityTianjinChina
  2. 2.Department of MathematicsXinxiang CollegeXinxiangChina

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