# Convergence Theorems Concerning Hybrid Methods for Strict Pseudocontractions and Systems of Equilibrium Problems

- Peichao Duan
^{1}Email author

**2010**:396080

https://doi.org/10.1155/2010/396080

© Peichao Duan. 2010

**Received: **23 May 2010

**Accepted: **26 August 2010

**Published: **30 August 2010

## Abstract

Let be strict pseudo-contractions defined on a closed and convex subset of a real Hilbert space . We consider the problem of finding a common element of fixed point set of these mappings and the solution set of a system of equilibrium problems by parallel and cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem. Furthermore, we prove that these schemes converge strongly by hybrid methods. The results presented in this paper improve and extend some well-known results in the literature.

## 1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty, closed, and convex subset of .

The solution set of (1.2) is denoted by .

for all ; see [2]. We denote the fixed point set of by , that is, .

for all . That is, is nonexpansive if and only if is a 0-strict pseudocontraction.

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 3, 4] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.1); related work can also be found in [5–8].

Recently, Acedo and Xu [9] considered the problem of finding a common fixed point of a finite family of strict pseudo-contractive mappings by the parallel and cyclic algorithms. Very recently, Duan and Zhao [10] considered new hybrid methods for equilibrium problems and strict pseudocontractions. In this paper, motivated by [5, 8–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the fixed point set of a finite family of strict pseudocontractions and the solution set of the system of equilibrium problems (1.1) by the hybrid methods.

We will use the following notations:

(1) for the weak convergence and for the strong convergence,

## 2. Preliminaries

We will use the facts and tools in a real Hilbert space which are listed below.

Lemma 2.1.

Let be a real Hilbert space. Then the following identities hold:

Lemma 2.2 (see [6]).

is convex (and closed).

Lemma 2.3 (see [13]).

Proposition 2.4 (see [9]).

Let be a nonempty, closed, and 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 is closed and convex. Therefore the projection is well defined.

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

(v)Let and be given as in item (iv). Suppose that has a common fixed point. Then

Lemma 2.5 (see [2]).

Let be a -strict pseudocontraction. Define by for any . Then, for any , is a nonexpansive mapping with .

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

(A2) is monotone, that is, for any

(A4) is convex and lower semicontinuous for each

Lemma 2.6 (see [3]).

Lemma 2.7 (see [1]).

## 3. Parallel Algorithm

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

Theorem 3.1.

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

Proof.

Denote for every and for all . Therefore . The proof is divided into six steps.

Step 1.

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

As by induction assumption, the inequality holds, in particular, for all . This together with the definition of implies that . Hence holds for all Thus , and therefore the sequence is well defined.

Step 2.

Then is bounded and (3.6) holds. From (3.3), (3.4), and Proposition 2.4 (i), we also obtain that , ,and are bounded.

Step 3.

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

Step 4.

Now, it follows from that as .

Step 5.

So by the demiclosedness principle (Proposition 2.4 (ii)), it follows that , and hence .

for all and for each that is, Hence (3.23) holds.

Step 6.

From (3.6) and Lemma 2.3, we conclude that , where .

Remark 3.2.

In this paper, we first turn the strict pseudocontraction into nonexpansive mapping then replace with a more simple form in the iterative algorithm.

Remark 3.3.

If , , and , we can obtain [14, Theorem ].

Remark 3.4.

If , , , and and we use to replace , we can get the result that has been studied by Tada and Takahashi in [8] for nonexpansive mappings. If , , , and , we can get [7, Theorem ].

Theorem 3.5.

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

Proof.

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

Step 1.

The sequence is well defined. We will show by induction that is closed and convex for all . For , we have which is closed and convex. Assume that for some is closed and convex; from Lemma 2.2, we have that is also closed and convex; The proof of is similar to the one in Step 1 of Theorem 3.1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

The proof of Step 2–Step 6 is similar to that of Theorem 3.1.

Remark 3.6.

If , we can obtain the two corresponding theorems in [10].

## 4. Cyclic Algorithm

Theorem 4.1.

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

Proof.

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

Step 1.

Step 2.

Step 3.

Step 4.

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

Step 5.

Then the demiclosedness principle implies that for all . This ensures that .The Proof of is similar to that of Theorem 3.1.

Step 6.

The sequence converges strongly to

The strong convergence to of is a consequence of Step 2, Step 5, and Lemma 2.3.

Theorem 4.2.

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

Proof.

The proof of this theorem is similar to that of Step 1 in Theorem 3.5 and Step 2–Step 6 in Theorem 4.1.

Remark 4.3.

If , we can obtain the two corresponding theorems in [10].

## Declarations

### Acknowledgment

This research is supported by Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).

## Authors’ Affiliations

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