- Research Article
- Open Access

# A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems

- Phayap Katchang
^{1}and - Poom Kumam
^{1, 2}Email author

**2010**:370197

https://doi.org/10.1155/2010/370197

© P. Katchang and P. Kumam 2010

**Received:**27 October 2009**Accepted:**16 March 2010**Published:**31 May 2010

## Abstract

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.

## Keywords

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

## 1. Introduction

*nonexpansive*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

*contraction*on if there exists a constant and such that . Let be a strongly positive bounded linear operator on : that is, there is a constant with property

*variational inclusion problem*, which is to find a point such that

where is the zero vector in . The set of solutions of problem (1.2) is denoted by .

which is called the *mixed quasivariational inequality* (see [1]).

This problem is called *Hartman-Stampacchia variational problem* (see [2–4]).

A set-valued mapping is called monotone if, for all , and imply that . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if, for , for every implies that .

*resolvent operator*that is associate with and as follows:

where is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone (see [5, 6]).

*mixed equilibrium problem*for finding such that

*x*is a solution of problem (1.7) implying that . If , then the mixed equilibrium problem (1.7) becomes the following

*equilibrium problem*that is to find such that

The set of solutions of (1.8) is denoted by EP*(F)*. Given a mapping
, let
for all
. Then
if and only if
for all
that is,
is a solution of the variational inequality. 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 (1.8). Some methods have been proposed to solve the equilibrium problem (see [8–21]).

They proved that if and satisfy appropriate conditions, then converges strongly to , where .

where is a potential function for , that is, , for all .

where
. Such a mapping
is called the *W-mapping* generated by
and
. Nonexpansivity of each
ensures the nonexpansivity of
. Moreover, in [24, Lemma
], it is shown that
.

The concept of -mappings was introduced in [25, 26]. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in [24, 27, 28] and the references cited therein.

where is monotone and Lipschitz continuous mapping and is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences and of parameters satisfy appropriate conditions.

In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in [6], Peng et al. in [22], Peng and Yao in [29], and Plubtieng and Sriprad in [23], we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al. [6], Peng et al. [22], Peng and Yao [29], Plubtieng and Sriprad [23], and some authors.

## 2. Preliminaries

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

for all .

*inverse-strongly monotone*if there exists a positive real number such that

So, if then is a nonexpansive mapping from into itself.

*H*satisfies the Opial condition [30], that is, for any sequence with , the inequality

holds for every with .

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction *F*,
, and the set *C*.

(A1) for all

(A2) is monotone, that is, for all

(A3)For each

(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 result.

Lemma 2.1 (Peng and Yao [31]).

for all . Then, the following hold.

(1)For each .

(2) is single valued.

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

(4)

(5) is closed and convex.

Lemma 2.2 (Xu [32]).

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

(1)

(2) or

Then

Lemma 2.3 (Osilike and Igbokwe [33]).

Lemma 2.4 (Colao et al. [28]).

Lemma 2.5 (Suzuki [34]).

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

Lemma 2.6 (Marino and Xu [35]).

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

Lemma 2.7 (Brézis [5]).

Let be a maximal monotone mapping and be a Lipschitz continuous mapping. Then the mapping is a maximal monotone mapping.

Remark 2.8.

Lemma 2.7 implies that is closed and convex if is a maximal monotone mapping and is a Lipschitz continuous mapping.

Lemma 2.9 (Zhang et al. [6]).

## 3. Main Result

In this section, we prove 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 a mixed equilibrium problem, and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping in a real Hilbert space.

Theorem 3.1.

for every , where and satisfy the following conditions:

(i) and ,

(ii) ,

(iii) ,

(iv) for all .

Proof.

Therefore, is bounded. We also obtain that and , are all bounded.

where .

where is an approximate constant such that , .

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

which is a contradiction. Thus, we obtain .

It follows from the maximal monotonicity of that , that is, . This implies that .

where . By (3.65), we get . Hence by Lemma 2.2 to (3.66), 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 conditions (i)–(iii) in Theorem 3.1. Then converges strongly to

Proof.

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

Corollary 3.3.

Equivalently, one has

Proof.

From Theorem 3.1 put ; then . So we have and . The conclusion of Corollary 3.3 can be obtained from Theorem 3.1 immediately.

## 4. Application

where is a convex and lower semicontinuous functional defined convex subset of a Hilbert space . We denote by the set of solutions of (4.1). Let be a bifunction defined by . We consider the equilibrium problem (1.8); it is obvious that . Therefore, from Theorem 3.1, we give the following corollary.

Corollary 4.1.

for every , where and satisfy the following conditions:

(i) and .

(ii) .

(iii) .

(iv) for all .

Proof.

From Theorem 3.1 put and . The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.

## Declarations

### Acknowledgments

The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.

## Authors’ Affiliations

## References

- Noor MA: Generalized set-valued variational inclusions and resolvent equations.
*Journal of Mathematical Analysis and Applications*1998, 228(1):206–220. 10.1006/jmaa.1998.6127MathSciNetView ArticleMATHGoogle Scholar - Browder FE: Nonlinear monotone operators and convex sets in Banach spaces.
*Bulletin of the American Mathematical Society*1965, 71(5):780–785. 10.1090/S0002-9904-1965-11391-XMathSciNetView ArticleMATHGoogle Scholar - Hartman P, Stampacchia G: On some non-linear elliptic differential-functional equations.
*Acta Mathematica*1966, 115(1):271–310. 10.1007/BF02392210MathSciNetView ArticleMATHGoogle Scholar - Lions J-L, Stampacchia G: Variational inequalities.
*Communications on Pure and Applied Mathematics*1967, 20: 493–519. 10.1002/cpa.3160200302MathSciNetView ArticleMATHGoogle Scholar - Brézis H:
*Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans Les Espaces de Hilbert, North-Holland Mathematics Studies, no. 5. Notas de Matemática (50)*. North-Holland, Amsterdam, The Netherlands; 1973:vi+183.Google Scholar - Zhang S-S, Lee JHW, Chan CK: Algorithms of common solutions to quasi variational inclusion and fixed point problems.
*Applied Mathematics and Mechanics*2008, 29(5):571–581. 10.1007/s10483-008-0502-yMathSciNetView ArticleMATHGoogle 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 - Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.
*The Mathematics Student*1994, 63(1–4):123–145.MathSciNetMATHGoogle 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.2009.02.106MathSciNetView ArticleMATHGoogle Scholar - Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms.
*Mathematical Programming*1997, 78(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Jaiboon C, Kumam P: A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings.
*Fixed Point Theory and Applications*2009, 2009:-32.Google Scholar - Jaiboon C, Kumam P, Humphries UW: Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems.
*Bulletin of the Malaysian Mathematical Sciences Society*2009, 32(2):173–185.MathSciNetMATHGoogle 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, 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 - Katchang P, Kumam P: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space.
*Journal of Applied Mathematics and Computing*2010, 32(1):19–38. 10.1007/s12190-009-0230-0MathSciNetView ArticleMATHGoogle Scholar - Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In
*Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems*.*Volume 477*. Springer, Berlin, Germany; 1999:187–201.View 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.2009.06.042MathSciNetView ArticleMATHGoogle 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.2008.06.011MathSciNetView 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 - Peng J-W, Wang Y, Shyu DS, Yao J-C: Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems.
*Journal of Inequalities and Applications*2008, 2008:-15.Google Scholar - Plubtieng S, Sriprad W: A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces.
*Fixed Point Theory and Applications*2009, 2009:-20.Google Scholar - Atsushiba S, Takahashi W: Strong convergence theorems for a finite family of nonexpansive mappings and applications.
*Indian Journal of Mathematics*1999, 41(3):435–453.MathSciNetMATHGoogle Scholar - Takahashi W, Shimoji K: Convergence theorems for nonexpansive mappings and feasibility problems.
*Mathematical and Computer Modelling*2000, 32(11–13):1463–1471.MathSciNetView ArticleMATHGoogle Scholar - Takahashi W: Weak and strong convergence theorems for families of nonexpansive mappings and their applications.
*Annales Universitatis Mariae Curie-Skłodowska. Sectio A*1997, 51(2):277–292.MATHMathSciNetGoogle Scholar - Ceng LC, Cubiotti P, Yao JC: Strong convergence theorems for finitely many nonexpansive mappings and applications.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(5):1464–1473. 10.1016/j.na.2006.06.055MathSciNetView ArticleMATHGoogle Scholar - Colao V, Marino G, Xu H-K: An iterative method for finding common solutions of equilibrium and fixed point problems.
*Journal of Mathematical Analysis and Applications*2008, 344(1):340–352. 10.1016/j.jmaa.2008.02.041MathSciNetView ArticleMATHGoogle Scholar - Peng J-W, Yao J-C: Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings.
*Computers & Mathematics with Applications*2009, 58(7):1287–1301. 10.1016/j.camwa.2009.07.040MathSciNetView ArticleMATHGoogle 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 - 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 - 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 - 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

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