- Research Article
- Open Access

# Strong Convergence Theorems by Hybrid Methods for Strict Pseudocontractions and Equilibrium Problems

- Peichao Duan
^{1}Email author and - Jing Zhao
^{1}

**2010**:528307

https://doi.org/10.1155/2010/528307

© P. Duan and J. Zhao. 2010

**Received:**21 December 2009**Accepted:**9 May 2010**Published:**10 June 2010

## Abstract

Let
be *N* strict pseudocontractions defined on a closed convex subset
of a real Hilbert space
. Consider the problem of finding a common element of the set of fixed point of these mappings and the set of solutions of an equilibrium problem with the parallel and cyclic algorithms. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly by hybrid methods.

## Keywords

- Hilbert Space
- Hybrid Method
- Equilibrium Problem
- Parallel Algorithm
- Nonexpansive Mapping

## 1. Introduction

Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction from to , where is the set of real numbers.

for all . The set of such solutions is denoted by .

for all ; see [1]. We denote the set of fixed points of by (i.e., ).

for all . That is, is nonexpansive if and only if is a -strict pseudocontraction.

Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem (1.1); see for instance [2–5]. In particular, Combettes and Hirstoaga [6] proposed several methods for solving the equilibrium problem. On the other hand, Mann [7], Nakajo and Takahashi [8] considered iterative schemes for finding a fixed point of a nonexpansive mapping.

Recently, Acedo and Xu [9] considered the problem of finding a common fixed point of a finite family of strict pseudocontractive mappings by the parallel and cyclic algorithms. Very recently, Liu [3] considered a general iterative method for equilibrium problems and strict pseudocontractions. In this paper, motivated by [3, 5, 9–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the set of fixed points of a finite family of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) by the hybrid methods.

We will use the notation

(1) for weak convergence and for strong convergence,

(2) denotes the weak -limit set of .

## 2. Preliminaries

We need some facts and tools in a real Hilbert space which are listed as below.

Lemma 2.1.

Let be a real Hilbert space. There hold the following identities.

(i) , for all

(ii) , for all , for all

Lemma 2.2 (see [4]).

is convex (and closed).

for all . Such a is called the metric (or the nearest point) projection of onto .

Lemma 2.3 (see [4]).

Lemma 2.4 (see [13]).

Then .

Lemma 2.5 (see [9]).

Let be a nonempty closed convex subset of . Let is a sequence in and . Assume

(i)the weak -limit set ,

(ii)for each exists.

Then is weakly convergent to a point in .

Proposition 2.6 (see [9]).

Assume be a nonempty closed convex subset of a real Hilbert space .

(ii)If is a -strict pseudocontraction, then the mapping is demiclosed (at 0). That is, if is a sequence in such that and , then .

(iii)If is a -strict pseudocontraction, then the fixed point set of of is closed and convex so that the projection is well defined.

(iv)Given an integer , assume, for each be a -strict pseudocontraction for some . Assume is a positive sequence such that . Then is a -strict pseudocontraction, with

Lemma 2.7 (see [1]).

Let be a -strict pseudocontraction. Define by for each . Then, as is a nonexpansive mapping such that .

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

(A1) for all

(A2) is monotone, that is, for any

(A3)for each

(A4) is convex and lower semicontionuous for each

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.8 (see [14]).

Lemma 2.9 (see [6]).

for all . Then, the following statements hold:

(i) is single-valued;

(iii) ;

(iv) is closed and convex.

## 3. Parallel Algorithm

In this section, we apply the hybrid methods to the parallel algorithm for finding a common element of the set of fixed points of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) in Hilbert spaces.

Theorem 3.1.

for every , where for some for some , and satisfies . Then, converge strongly to .

Proof.

The proof is divided into several steps.

Step 1.

Show first that is well defined.

It is obvious that is closed and is closed convex for every . From Lemma 2.2, we also get is convex.

Step 2.

Show for all .

As by induction assumption, the inequality holds, in particular, for all . This together with the definition of implies that . Hence holds for all .

Step 3.

Then is bounded and (3.6) holds. From (3.3), (3.4), and Proposition 2.6(i), we also obtain and are bounded.

Step 4.

Then , that is, the sequence is nondecreasing. Since is bounded, exists. Then (3.8) holds.

Step 5.

It follows from that

Step 6.

We first show . To see this, we take and assume that as for some subsequence of .

So by the demiclosedness principle (Proposition 2.6(ii)), it follows that and hence holds.

Next we show take , and assume that as for some subsequence of . From (3.17), we obtain . Since and is closed convex, we get

for all and . Hence (3.21) holds.

Step 7.

From (3.6) and Lemma 2.4, we conclude that , where .

A very similar result obtained in a way completely different is Theorem of [11].

Theorem 3.2.

for every , where for some for some , and satisfies . Then, converge strongly to .

Proof.

The proof of this theorem is similar to that of Theorem 3.1.

Step 1.

is well defined for all

We show is closed convex for all by induction. For , we have is closed convex. Assume that for some is closed convex, from Lemma 2.2, we have is also closed convex. The assumption holds.

Step 2.

.

Step 3.

for all , where

Step 4.

Step 5.

Step 6.

Step 7.

.

The proof of Steps 2–7 is similar to that of Theorem 3.1.

A very similar result obtained in a way completely different is Theorem of [10].

## 4. Cyclic Algorithm

where , with .

Theorem 4.1.

for every , where for some for some , and satisfies . Then, converge strongly to .

Proof.

The proof of this theorem is similar to that of Theorem 3.1. The main points include the following.

Step 1.

is well defined for all

Step 2.

.

Step 3.

for all , where

Step 4.

Step 5.

To prove the above steps, one simply replaces with in the proof of Theorem 3.1.

Step 6.

Show that

Then the demiclosedness principle (Proposition 2.6(ii)) implies that for all . This ensures that .

The proof of is similar to that of Theorem 3.1.

Step 7.

Show that

The strong convergence to of is the consequence of Step 3, Step 5, and Lemma 2.4.

Theorem 4.2.

for every , where for some for some , and satisfies . Then, converge strongly to .

Proof.

The proof of this theorem can consult Step 1 of Theorem 3.2 and Steps 2–7 of Theorem 4.1.

## Declarations

### Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve the manuscript and the Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021) and the science research foundation program in Civil Aviation University of China (04-CAUC-15S) for their financial support.

## Authors’ Affiliations

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