- Research Article
- Open Access

# Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

- Yekini Shehu
^{1}Email author

**2011**:852789

https://doi.org/10.1155/2011/852789

© Yekini Shehu. 2011

**Received:**6 September 2010**Accepted:**25 November 2010**Published:**14 December 2010

## Abstract

We introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of -strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature.

## Keywords

- Convex Subset
- Equilibrium Problem
- Nonexpansive Mapping
- Iterative Scheme
- Maximal Monotone

## 1. Introduction

*monotone*if

*inverse-strongly monotone*(see, e.g., [1, 2]) if there exists a positive real number such that , for all . For such a case, is called -inverse-strongly monotone. A -inverse-strongly monotone is sometime called

*-cocoercive*. A mapping is said to be

*relaxed*

*-cocoercive*if there exists such that

(See, e.g., [3, 4].) We will denote the set of solutions of the variational inequality problem (1.5) by .

*maximal*if the graph is not properly contained in the graph of any other monotone map, where for a multivalued mapping . It is also known that is maximal if and only if for for every implies . Let be a monotone mapping defined from into and a normal cone to at , that is, . Define a mapping by

Then, is maximal monotone and (see, e.g., [5]).

for all
. If
, then the mapping
is *nonexpansive.* A point
is called *a fixed point* of
if
. The fixed points set of
is the set
. Iterative approximation of fixed points of
-strictly pseudocontractive mappings have been studied extensively by many authors (see, e.g., [1, 6–9] and the references contained therein).

for all . The set of solutions of (1.10) is denoted by .

for all . The set of solutions of (1.11) is denoted by EP.

for all . The set of solutions of (1.12) is denoted by MEP.

The generalized mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [24]). Numerous problems in Physics, optimization, and economics reduce to find a solution of problem (1.8). Several methods have been proposed to solve the fixed-point problems, variational inequality problems and equilibrium problems in the literature (see, e.g., [5, 11, 12, 20, 25–30]).

Recently, Ceng and Yao [25] introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space . In their results, they imposed the following condition on a nonempty closed and convex subset of :

(E) is -strongly convex and its derivative is sequentially continuous from weak topology to the strong topology.

We remark here that this condition (E) has been used by many authors for approximation of solution to mixed equilibrium problem in a real Hilbert space (see, e.g., [31, 32]). However, it is observed that the condition (E) does not include the case and . Furthermore, Peng and Yao [21], R. Wangkeeree and R. Wangkeeree [30], and many other authors replaced condition (E) with the following conditions:

or

(B2) is a bounded set.

Consequently, conditions (B1) and (B2) have been used by many authors in approximating solution to generalized mixed equilibrium (mixed equilibrium) problems in a real Hilbert space (see, e.g., [21, 30]).

Recently, Takahashi et al. [33] proved the following convergence theorem using hybrid method.

Theorem 1.1 (Takahashi et al. [33]).

Assume that satisfies . Then, converges strongly to .

Motivated by the results of Takahashi et al. [33], Kumam [28] studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space. In particular, he proved the following theorem.

Theorem 1.2 (Kumam, [28]).

Then, converges strongly to .

Motivated by the ongoing research and the above-mentioned results, we introduce *a new iterative scheme for finding a common element of the set of fixed points of an infinite family of*
*-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space*. Furthermore, we show that our new iterative scheme converges strongly to a common element of the three afore mentioned sets. In our results, we use conditions (B1) and (B2) mentioned above. Our result extends many important recent results. Finally, we give some applications of our results.

## 2. Preliminaries

Let be a real Hilbert space with inner product and norm and let be a nonempty closed and convex subset of . The strong convergence of to is denoted by as .

*metric projection*of onto . We know that is a nonexpansive mapping of onto . It is also known that satisfies

So, if , then is a nonexpansive mapping of into .

For solving the generalized mixed equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

(A1) for all ,

(A2) is monotone, that is, for all ,

(A3) for each ,

(A4)for each is convex and lower semicontinuous.

We need the following technical result.

Lemma 2.1 (R. Wangkeeree and R. Wangkeeree [30]).

for all . Then, the following hold:

(1)for each ,

(2) is single-valued,

(4) ,

(5) is closed and convex.

## 3. Main Results

Theorem 3.1.

*μ*-Lipschitzian, relaxed -cocoercive mapping of into . Suppose . Let , , , and be generated by , , ,

Assume that and satisfy

(i) ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

Proof.

This implies that . By following the same arguments, we can show that .

which implies that . We have and hence . Therefore, .

for all . By (2.3) again, we conclude that . This completes the proof.

Corollary 3.2.

Assume that , and satisfy

(i) ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

*polar*of in to be the set

The set of solutions of the complementarity problem is denoted by . We shall assume that satisfies the following conditions:

(E1) is -inverse strongly monotone,

(E2) .

Also, we replace conditions (B1) and (B2) with

(D2) is a bounded set.

Theorem 3.3.

Assume that , and satisfy

(i) ,

(ii) ,

(iii) ,

(iv) .

Then, converges strongly to .

Proof.

Using Lemma 7.1.1 of [34], we have that . Hence, by Theorem 3.1, we obtain the desired conclusion.

Remark 3.4.

Our Corollary 3.2 extends Theorems 1.1 and 1.2.

Remark 3.5.

Our iterative scheme (3.1) is simpler than the iterative schemes (5.1) and (5.11) of Acedo and Xu [6]. Furthermore, in our results, we use iterative scheme (3.1) to approximate a common fixed point of an *infinite family of*
*-strictly pseudocontractive mappings* while the iterative schemes (5.1) and (5.11) of Acedo and Xu [6] are used to approximate a common fixed point of a *finite family of*
*-strictly pseudocontractive mappings.*

Remark 3.6.

Our results also hold for infinite family of uniformly continuous quasistrict pseudocontractions. Hence, we can adapt our results for an infinite family of uniformly continuous quasi-nonexpansive mappings in a real Hilbert space.

## Declarations

### Acknowledgments

The author is extremely grateful to Professor S. Al-Homidan and the anonymous referees for their valuable comments and useful suggestions which improve the presentation of this paper. This research work is dedicated to Professor C. E. Chidume with admiration and respect.

## Authors’ Affiliations

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