- Research Article
- Open Access

# A Viscosity of Cesàro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

- Thanyarat Jitpeera
^{1}, - Phayap Katchang
^{1}and - Poom Kumam
^{1}Email author

**2011**:945051

https://doi.org/10.1155/2011/945051

© Thanyarat Jitpeera et al. 2011

**Received:**6 September 2010**Accepted:**15 October 2010**Published:**20 October 2010

## Abstract

We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesàro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.

## Keywords

- Variational Inequality
- Equilibrium Problem
- Monotone Mapping
- Nonexpansive Mapping
- Real Hilbert Space

## 1. Introduction

Throughout this paper, we assume that
is a real Hilbert space with inner product and norm are denoted by
and
, respectively and let
be a nonempty closed convex subset of
. A mapping
is called
if
, for all
. We use
to denote the set of *fixed points* of
, that is,
. It is assumed throughout the paper that
is a nonexpansive mapping such that
. Recall that a self-mapping
is a
on
if there exists a constant
and
such that

*x*is a solution of problem (1.1) implies that . If , then the mixed equilibrium problem (1.1) becomes the following

*equilibrium problem*is to find such that

The set of solutions of (1.2) is denoted by . The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).

*monotone*if

where and are all elements of and is an appropriate in They proved that converges strongly to an element of fixed point of which is the nearest to

where is a potential function for (i.e., for ).

for all
, where
is *W*-mapping. They proved the strong convergence theorems under some mind conditions.

In this paper, motivated by the above results and the iterative schemes considered in [9, 18–20], we introduce a new iterative process below based on viscosity and Cesàro mean approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for a -inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 21–24]. We extend and improve the corresponding results of Kumam and Katchang [9], Peng and Yao [20], Shimizu and Takahashi [18] and some authors.

## 2. Preliminaries

*metric projection*of onto It is well known that is a nonexpansive mapping of onto and satisfies

*C*into

*H*. In the context of the variational inequality problem the characterization of projection (2.5) implies the following:

*H*satisfies the Opial condition [25], that is, for any sequence with , the inequality

Then
is the *maximal monotone* and
if and only if
; see [26].

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and a proper extended real-valued function satisfies the following conditions:

(A2) is monotone, that is, for all

(A4)for each , is convex and lower semicontinuous;

(A5)for each , is weakly upper semicontinuous;

(B2)*C* is a bounded set.

We need the following lemmas for proving our main results.

Lemma 2.1 (Peng and Yao [20]).

for all . Then, the following hold.

(3) is firmly nonexpansive, that is, for any ,

Lemma 2.2 (Xu [27]).

where is a sequence in and is a sequence in such that

Lemma 2.3 (Osilike and Igbokwe [28]).

Lemma 2.4 (Suzuki [29]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.5 (Marino and Xu [17]).

Assume is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then .

Lemma 2.6 (Bruck [30]).

Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and a nonexpansive mapping. For each and the Cesàro means then

## 3. Main Results

In this section, we show a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of mixed equilibrium problem and the set of solutions of a variational inequality problem for a -inverse-strongly monotone mapping in a real Hilbert space by using the viscosity of Cesàro mean approximation method.

Theorem 3.1.

where , and , and satisfy the following conditions:

Proof.

It follows that , hence is nonexpansive.

Hence is bounded and also , and are bounded.

It is easy to see that is a contradiction of into itself. Hence is complete, there exists a unique fixed point , such that

Since is bounded, there exists a subsequence of which converge weakly to Without loss of generality, we can assume that From , we obtain

Dividing by we get From (A3) and the weakly lower semicontinuity of we have , for all and hence

which is a contradiction. Thus, we obtain

Noting that as and is -inverse-strongly monotone, hence from (3.46), we obtain as . Since is maximal monotone, we have , and hence . Therefore,

where By (3.47), (i) and (iii), we get Applying Lemma 2.2 to (3.48) we conclude that This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2.

where , for all satisfy the condition (i), (iii)–(v). Then, converges strongly to where

Proof.

Taking for , , and in Theorem 3.1, we can conclude the desired conclusion easily.

Corollary 3.3.

where , for all satisfy the condition (i), (iii)–(v). Then, converges strongly to where

Proof.

Taking and , for all in Corollary 3.2, we can conclude the desired conclusion easily.

## 4. Applications to Optimization Problem

*optimization problem:*

where is the set of fixed points of in and is a potential function for (i.e., for ). We have the following theorem.

Theorem 4.1.

and the sequence converges strongly to some point , which solves the following optimization problem (4.1).

Proof.

Since is the unique solution of (4.3), therefor, . This complete the proof of Theorem 4.1.

## Declarations

### Acknowledgments

The authors would like to thank the Faculty of science KMUTT. Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5380044.

## Authors’ Affiliations

## References

- 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.2007.02.022MathSciNetView ArticleMATHGoogle Scholar - Burachik RS, Lopes JO, Da Silva GJP:
**An inexact interior point proximal method for the variational inequality problem.***Computational & Applied Mathematics*2009,**28**(1):15–36.MathSciNetMATHGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Kumam P:
**Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space.***Turkish Journal of Mathematics*2009,**33**(1):85–98.MathSciNetMATHGoogle Scholar - Kumam P:
**A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping.***Nonlinear Analysis: Hybrid Systems*2008,**2**(4):1245–1255. 10.1016/j.nahs.2008.09.017MathSciNetMATHGoogle 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/s12190-008-0129-1MathSciNetView ArticleMATHGoogle Scholar - Kumam P, Jaiboon C:
**A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):510–530. 10.1016/j.nahs.2009.04.001MathSciNetMATHGoogle Scholar - Kumam P, Katchang P:
**A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):475–486. 10.1016/j.nahs.2009.03.006MathSciNetMATHGoogle Scholar - Moudafi A, Théra M:
**Proximal and dynamical approaches to equilibrium problems.**In*Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Econom. and Math. Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar - Wang Z, Su Y:
**Strong convergence theorems of common elements for equilibrium problems and fixed point problems in Banach paces.***Journal of Applied Mathematics & Informatics*2010,**28**(3–4):783–796.MATHGoogle Scholar - Wangkeeree R, Wangkeeree R:
**Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings.***Nonlinear Analysis: Hybrid Systems*2009,**3**(4):719–733. 10.1016/j.nahs.2009.06.009MathSciNetMATHGoogle 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.2008.12.055MathSciNetView ArticleMATHGoogle Scholar - Yao Y, Cho YJ, Chen R:
**An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(7–8):3363–3373. 10.1016/j.na.2009.01.236MathSciNetView ArticleMATHGoogle Scholar - Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optim. Appl.*.*Volume 76*. Springer, New York, NY, USA; 2005:501–558. 10.1007/0-387-23393-8_12View ArticleGoogle Scholar - Zeng LC, Schaible S, Yao JC:
**Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities.***Journal of Optimization Theory and Applications*2005,**124**(3):725–738. 10.1007/s10957-004-1182-zMathSciNetView ArticleMATHGoogle Scholar - Marino G, Xu H-K:
**A general iterative method for nonexpansive mappings in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2006,**318**(1):43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView ArticleMATHGoogle Scholar - Shimizu T, Takahashi W:
**Strong convergence to common fixed points of families of nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1997,**211**(1):71–83. 10.1006/jmaa.1997.5398MathSciNetView ArticleMATHGoogle Scholar - Plubtieng S, Punpaeng R:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Journal of Mathematical Analysis and Applications*2007,**336**(1):455–469. 10.1016/j.jmaa.2007.02.044MathSciNetView ArticleMATHGoogle 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.2008.11.014MathSciNetView ArticleMATHGoogle 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.2007.08.045MathSciNetView ArticleMATHGoogle Scholar - Wangkeeree R:
**An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-17.Google Scholar - Liou Y-C, Yao Y:
**Iterative algorithms for nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-10.Google Scholar - Yao Y, Liou Y-C, Yao J-C:
**Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings.***Fixed Point Theory and Applications*2007,**2007:**-12.Google Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Rockafellar RT:
**On the maximality of sums of nonlinear monotone operators.***Transactions of the American Mathematical Society*1970,**149:**75–88. 10.1090/S0002-9947-1970-0282272-5MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Osilike MO, Igbokwe DI:
**Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations.***Computers & Mathematics with Applications*2000,**40**(4–5):559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals.***Journal of Mathematical Analysis and Applications*2005,**305**(1):227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView ArticleMATHGoogle Scholar - Bruck RE:
**On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces.***Israel Journal of Mathematics*1981,**38**(4):304–314. 10.1007/BF02762776MathSciNetView ArticleMATHGoogle Scholar - Xu HK:
**An iterative approach to quadratic optimization.***Journal of Optimization Theory and Applications*2003,**116**(3):659–678. 10.1023/A:1023073621589MathSciNetView ArticleMATHGoogle Scholar

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