- 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

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]).

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:

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

(A4)for each is convex and lower semicontinuous.

We need the following technical result.

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

## 3. Main Results

Theorem 3.1.

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

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.

*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,

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

Theorem 3.3.

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

## References

- Browder FE, Petryshyn WV:
**Construction of fixed points of nonlinear mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1967,**20:**197–228. 10.1016/0022-247X(67)90085-6MATHMathSciNetView ArticleGoogle Scholar - Iiduka H, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.***Nonlinear Analysis: Theory, Methods & Applications*2005,**61**(3):341–350. 10.1016/j.na.2003.07.023MATHMathSciNetView ArticleGoogle Scholar - Bruck RE Jr.:
**On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space.***Journal of Mathematical Analysis and Applications*1977,**61**(1):159–164. 10.1016/0022-247X(77)90152-4MATHMathSciNetView ArticleGoogle Scholar - Ceng L-C, Khan AR, Ansari QH, Yao J-C:
**Viscosity approximation methods for strongly positive and monotone operators.***Fixed Point Theory*2009,**10**(1):35–72.MATHMathSciNetGoogle Scholar - Rockafellar RT:
**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877–898. 10.1137/0314056MATHMathSciNetView ArticleGoogle Scholar - Acedo GL, Xu H-K:
**Iterative methods for strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(7):2258–2271. 10.1016/j.na.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Qin X, Kang JI:
**Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(9):4203–4214. 10.1016/j.na.2009.02.106MATHMathSciNetView ArticleGoogle Scholar - Liu Y:
**A general iterative method for equilibrium problems and strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4852–4861. 10.1016/j.na.2009.03.060MATHMathSciNetView ArticleGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MATHMathSciNetView ArticleGoogle Scholar - Liu M, Chang S, Zuo P:
**On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi--asymptotically nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2010,**2010:**-18.Google Scholar - Petrot N, Wattanawitoon K, Kumam P:
**A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces.***Nonlinear Analysis: Hybrid Systems*2010,**4**(4):631–643. 10.1016/j.nahs.2010.03.008MATHMathSciNetGoogle Scholar - Zhang S:
**Generalized mixed equilibrium problem in Banach spaces.***Applied Mathematics and Mechanics. English Edition*2009,**30**(9):1105–1112. 10.1007/s10483-009-0904-6MATHMathSciNetView ArticleGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MATHMathSciNetGoogle Scholar - Plubtieng S, Punpaeng R:
**A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings.***Applied Mathematics and Computation*2008,**197**(2):548–558. 10.1016/j.amc.2007.07.075MATHMathSciNetView ArticleGoogle Scholar - Qin X, Shang M, Su Y:
**Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems.***Mathematical and Computer Modelling*2008,**48**(7–8):1033–1046. 10.1016/j.mcm.2007.12.008MATHMathSciNetView ArticleGoogle Scholar - Su Y, Shang M, Qin X:
**An iterative method of solution for equilibrium and optimization problems.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(8):2709–2719. 10.1016/j.na.2007.08.045MATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W:
**Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**331**(1):506–518. 10.1016/j.jmaa.2006.08.036MATHMathSciNetView ArticleGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(1):99–112. 10.1016/j.na.2009.06.042MATHMathSciNetView ArticleGoogle Scholar - Shehu Y:
**Fixed point solutions of generalized equilibrium problems for nonexpansive mappings.***Journal of Computational and Applied Mathematics*2010,**234**(3):892–898. 10.1016/j.cam.2010.01.055MATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W:
**Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(3):1025–1033. 10.1016/j.na.2008.02.042MATHMathSciNetView ArticleGoogle Scholar - Peng J-W, Yao J-C:
**Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems.***Mathematical and Computer Modelling*2009,**49**(9–10):1816–1828. 10.1016/j.mcm.2008.11.014MATHMathSciNetView ArticleGoogle Scholar - Plubtieng S, Sombut K:
**Weak convergence theorems for a system of mixed equilibrium problems and nonspreading mappings in a Hilbert space.***Journal of Inequalities and Applications*2010,**2010:**-12.Google Scholar - Yao Y, Liou Y-C, Yao J-C:
**A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MATHMathSciNetGoogle Scholar - Ceng L-C, Yao J-C:
**A hybrid iterative scheme for mixed equilibrium problems and fixed point problems.***Journal of Computational and Applied Mathematics*2008,**214**(1):186–201. 10.1016/j.cam.2007.02.022MATHMathSciNetView ArticleGoogle Scholar - Nadezhkina N, Takahashi W:
**Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2006,**128**(1):191–201. 10.1007/s10957-005-7564-zMATHMathSciNetView ArticleGoogle Scholar - Noor MA:
**General variational inequalities and nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2007,**331**(2):810–822. 10.1016/j.jmaa.2006.09.039MATHMathSciNetView ArticleGoogle Scholar - Kumam P:
**A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping.***Journal of Applied Mathematics and Computing*2009,**29**(1–2):263–280. 10.1007/s12190-008-0129-1MATHMathSciNetView ArticleGoogle Scholar - Qin X, Cho YJ, Kang SM:
**Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces.***Journal of Computational and Applied Mathematics*2009,**225**(1):20–30. 10.1016/j.cam.2008.06.011MATHMathSciNetView ArticleGoogle Scholar - Wangkeeree R, Wangkeeree R:
**A general iterative method for variational inequality problems, mixed equilibrium problems, and fixed point problems of strictly pseudocontractive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2009,**2009:**-32.Google Scholar - Liu M, Chang SS, Zuo P:
**An algorithm for finding a common solution for a system of mixed equilibrium problem, quasi-variational inclusion problem, and fixed point problem of nonexpansive semigroup.***Journal of Inequalities and Applications*2010,**2010:**-23.Google Scholar - Yao Y, Noor MA, Zainab S, Liou Y-C:
**Mixed equilibrium problems and optimization problems.***Journal of Mathematical Analysis and Applications*2009,**354**(1):319–329. 10.1016/j.jmaa.2008.12.055MATHMathSciNetView ArticleGoogle Scholar - Takahashi W, Takeuchi Y, Kubota R:
**Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):276–286. 10.1016/j.jmaa.2007.09.062MATHMathSciNetView ArticleGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar

## Copyright

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.