# Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications

- Thanyarat Jitpeera
^{1}, - Uamporn Witthayarat
^{1}and - Poom Kumam
^{1}Email author

**2011**:971479

https://doi.org/10.1155/2011/971479

© Thanyarat Jitpeera et al. 2011

**Received: **5 January 2011

**Accepted: **20 February 2011

**Published: **13 March 2011

## Abstract

We introduce new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others.

## 1. Introduction

*nonexpansive*if , for all . The set of

*fixed points*of denoted by ; that is, . If is bounded, closed, and convex and is a nonexpansive mapping of into itself, then ; see, for instance, [1]. Let be a bifunction of into , where is the set of real numbers, a mapping, and a real-valued function. The

*generalized mixed equilibrium problem*is for finding such that

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problems as special cases [2–7].

*mixed equilibrium problem*[8] for finding such that

The set of solutions of (1.3) is denoted by .

The set of solutions of (1.4) is denoted by , which this problem was studied by S. Takahashi and W. Takahashi [10].

*equilibrium problem*which is to find such that

*variational inequality problem,*denoted by , is to find such that

*monotone*if

where is the -mapping and , are , -inverse-strongly monotone mappings of into , respectively. He proved that if the sequences , , , , , , and of parameters satisfies appropriate conditions, then generated by (1.11) converges strongly to .

Recently, Shehu [34] motivated Chantarangsi et al. [35] who studied the problem of approximating a common element of the set of fixed points of an infinite family of nonexpansive mapping, the set of common solutions of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert spaces.

In this paper, motivated by the above results, we present a new hybrid iterative scheme for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces. Then, we prove strong convergence theorems under some mild conditions. Finally, we give some applications of our results. The results presented in this paper generalize, extend, and improve the results of Takahashi et al. [32], Kumam [20], Kangtunyakarn [33], and many authors.

## 2. Preliminaries

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

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

For solving the generalized mixed equilibrium problem, let us assume that the bifunction , the nonlinear mapping is continuous monotone, is convex, and lower semicontinuous satisfies the following conditions:

(A2) is monotone; that is, for any ,

(A4) is convex and lower semicontinuous for each ,

The following lemma appears implicitly in [2]. We need the following lemmas for proving our main result.

Lemma 2.1 (see [2]).

The following lemma was also given in [36].

Lemma 2.2 (see [36]).

for all . Then, the following hold:

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

Lemma 2.3 (see [37]).

for all . Then, the following hold:

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

Lemma 2.4 (see [38]).

## 3. Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces.

Theorem 3.1.

for every , where , and satisfying the following conditions:

Proof.

Next, we will divide the proof into four steps.

Step 1.

which shows that , for all , for all . So, , for all , for all . Therefore, it follows that , for all . This implies that is well defined.

Step 2.

Step 3.

We claim that the following statements hold:

By (3.46) and (3.48), we have that , for all .

Step 4.

which is a contradiction. Thus, we obtain .

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

Lastly, by the same proof of [39, Theorem 3.1, pages 346-347], we can show that and . Therefore, ; that is, .

By (2.4), again, we conclude that . This completes the proof.

Using Theorem 3.1, we obtain the following corollaries.

Corollary 3.2.

for every , where , , and satisfy the following conditions:

Proof.

Taking for , to be nonexpansive mappings, in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.

where denotes the identity operator on .

Lemma 3.3 (see [41]).

Let be a nonempty closed convex subset of a real Hilbert space , and let be a -strict pseudocontraction. Define by for each . Then, as ) is nonexpansive such that .

Using Theorem 3.1, we obtain the following result.

Theorem 3.4.

for every , where , , and satisfy the following conditions:

Proof.

From Theorem 3.1, is a finite family of -strict pseudocontraction. By Lemma 3.3, we have that is nonexpansive mappings. The conclusion of Theorem 3.4 can be obtained from Theorem 3.1 immediately.

## 4. Some Applications

### 4.1. Complementarity Problem

*polar*of in to be the set

The set of solution of the complementarity problem is denoted by , . We will assume that , satisfies the following conditions:

(E1) , is a , -inverse-strongly monotone mapping, respectively,

Corollary 4.1.

for every , where , , and satisfy the following conditions:

Proof.

Using Lemma 7.1.1 of [42], we have that and . Hence, by Corollary 4.1, we can conclude the desired conclusion easily. This completes the proof.

### 4.2. Optimization Problem

*optimization problem*with nonempty set of solutions:

where is a convex and lower semicontinuous functional, and define as a closed convex subset of a real Hilbert space . We denote the set of solutions of (4.5) by and . Let be a bifunction defined by . We consider the equilibrium problem, it is obvious that , . Therefore, from Theorem 3.1, we obtained the following corollary.

Corollary 4.2.

for every , where , , and satisfy the following conditions:

Proof.

From Theorem 3.1, put , , and . The conclusion of Corollary 4.2 can be obtained from Theorem 3.1 immediately.

### 4.3. Minimization Problem

In this section, we study the problem for finding a minimizer of a continuously Frèchet differentiable convex functional in a Hilbert space.

First, we use the following lemma in our result.

Lemma 4.3 (see [43]).

Let be a Banach space, let be a continuously Frèchet differentiable convex functional on , and let be the gradient of . If is -Lipschitz continuous, then is an -inverse-strongly monotone.

Let , be functionals on which satisfies the following conditions:

(C1)let, , be a continuously Frèchet differentiable convex functional on , and let, , be , -Lipschitz continuous, respectively,

Corollary 4.4.

for every , where , , and satisfying the following conditions:

Proof.

We know form condition (C1) and Lemma 4.3 that , is a , -inverse-strongly monotone operator from in to itself, respectively. The conclusion of Corollary 4.4 can be obtained from Theorem 3.1 immediately.

## Declarations

### Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this work. The authors would like to thank The National Research University Project of Thailand's Office of the Higher Education Commission for financial support (under the NRU-CSEC Project no. 54000267). Furthermore, P. Kumam's research supported by the Commission on Higher Education and the Thailand Research Fund under Grant no. MRG5380044.

## Authors’ Affiliations

## References

- Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Chadli O, Schaible S, Yao JC:
**Regularized equilibrium problems with application to noncoercive hemivariational inequalities.***Journal of Optimization Theory and Applications*2004,**121**(3):571–596.MathSciNetView ArticleMATHGoogle Scholar - Chadli O, Wong NC, Yao JC:
**Equilibrium problems with applications to eigenvalue problems.***Journal of Optimization Theory and Applications*2003,**117**(2):245–266. 10.1023/A:1023627606067MathSciNetView ArticleMATHGoogle Scholar - Konnov IV, Schaible S, Yao JC:
**Combined relaxation method for mixed equilibrium problems.***Journal of Optimization Theory and Applications*2005,**126**(2):309–322. 10.1007/s10957-005-4716-0MathSciNetView ArticleMATHGoogle Scholar - Moudafi A, Théra M:
**Proximal and dynamical approaches to equilibrium problems.**In*Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View ArticleGoogle Scholar - Zeng L-C, Wu S-Y, Yao J-C:
**Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems.***Taiwanese Journal of Mathematics*2006,**10**(6):1497–1514.MathSciNetMATHGoogle 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.2007.02.022MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Toyoda M:
**Weak convergence theorems for nonexpansive mappings and monotone mappings.***Journal of Optimization Theory and Applications*2003,**118**(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle 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):506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView 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 - Cho YJ, Petrot N, Suantai S:
**Fixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems.***Journal of Nonlinear Analysis and Optimization*2010,**1**(1):45–53.MathSciNetGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Jitpeera T, Kumam P:
**A composite iterative method for generalized mixed equilibrium problems and variational inequality problems.***Journal of Computational Analysis and Applications*2011,**13**(2):345–361.MathSciNetMATHGoogle Scholar - Jitpeera T, Kumam P:
**An extragradient type method for a system of equilibrium problems, variational inequality problems and fixed points of finitely many nonexpansive mappings.***Journal of Nonlinear Analysis and Optimization*2010,**1:**71–91.MathSciNetGoogle Scholar - Jaiboon C, Kumam P:
**Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities.***Journal of Inequalities and Applications*2010,**2010:**-43.Google Scholar - Jaiboon C, Kumam P:
**A general iterative method for addressing mixed equilibrium problems and optimization problems.***Nonlinear Analysis. Theory, Methods & Applications*2010,**73**(5):1180–1202. 10.1016/j.na.2010.04.041MathSciNetView ArticleMATHGoogle 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.2008.09.017MathSciNetView 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 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.2009.04.001MathSciNetView ArticleMATHGoogle Scholar - Kumam P, Jaiboon C:
**A system of generalized mixed equilibrium problems and fixed point problems for pseudocontractive mappings in Hilbert spaces.***Fixed Point Theory and Applications*2010,**2010:**-33.Google 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.2009.03.006MathSciNetView ArticleMATHGoogle Scholar - Kumam W, Jaiboon C, Kumam P, Singta A:
**A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings.***Journal of Inequalities and Applications*2010,**2010:**-25.Google Scholar - Wang Z, Su Y:
**Strong convergence theorems of common elements for equilibrium problems and fixed point problems in Banach paces.***Journal of Application Mathematics and 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.2009.06.009MathSciNetView ArticleMATHGoogle Scholar - Wangkeeree R, Petrot N, Kumam P, Jaiboon C:
**Convergence theorem for mixed equilibrium and variational inequality problems for relaxed cocoercive mappings.***Journal of Computational Analysis and Applications*2011,**13**(3):425–449.MathSciNetMATHGoogle 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*.*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 - 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.2007.09.062MathSciNetView ArticleMATHGoogle Scholar - Kangtunyakarn A:
**Iterative methods for finding common solution of generalized equilibrium problems and variational inequality problems and fixed point problems of a finite family of nonexpansive mappings.***Fixed Point Theory and Applications*2010,**2010:**-29.Google Scholar - Shehu Y: Strong convergence theorems for family of nonexpansive mappings and sys- tem of generalized mixed equilibrium problems and variational inequality problems. International Journal of Mathematics and Mathematical Sciences. In pressGoogle Scholar
- Chantarangsi W, Jaiboon C, Kumam P:
**A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces.***Abstract and Applied Analysis*2010,**2010:**-39.Google Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Peng J-W, Liou Y-C, Yao J-C:
**An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions.***Fixed Point Theory and Applications*2009,**2009:**-21.Google 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 - 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.2003.07.023MathSciNetView ArticleMATHGoogle Scholar - 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)90085-6MathSciNetView ArticleMATHGoogle Scholar - Zhou H:
**Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces.***Nonlinear Analysis. Theory, Methods & Applications*2008,**69**(2):456–462. 10.1016/j.na.2007.05.032MathSciNetView ArticleMATHGoogle Scholar - Takahashi W:
**Weak and strong convergence theorems for families of nonexpansive mappings and their applications.***Annales Universitatis Mariae Curie-Skłodowska*1997,**51**(2):277–292.MathSciNetMATHGoogle Scholar - Baillon J-B, Haddad G:
**Quelques propriétés des opérateurs angle-bornés et -cycliquement monotones.***Israel Journal of Mathematics*1977,**26**(2):137–150. 10.1007/BF03007664MathSciNetView ArticleMATHGoogle 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.