# An Iterative Method for Generalized Equilibrium Problems, Fixed Point Problems and Variational Inequality Problems

- Qing-you Liu
^{1}, - Wei-you Zeng
^{2}and - Nan-jing Huang
^{2}Email author

**2009**:531308

**DOI: **10.1155/2009/531308

© Qing-you Liu et al. 2009

**Received: **11 January 2009

**Accepted: **28 May 2009

**Published: **29 June 2009

## Abstract

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for -inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. (2007).

## 1. Introduction

It is well known that GEP(1.1) contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalities (see, e.g., [1–6] and the reference therein).

for all . We denote by the set of fixed points of , that is, . If is bounded, closed and convex and is a nonexpansive mappings of into itself, then is nonempty (see [8]).

In 1997, Flåm and Antipin [9] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. In 2003, Iusem and Sosa [10] presented some iterative algorithms for solving equilibrium problems in finite-dimensional spaces. They have also established the convergence of the algorithms. Recently, Huang et al. [11] studied the approximate method for solving the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem.

where , and proved the strong convergence theorems for iterative scheme (1.8) under some conditions on parameters. In 2007, S. Takahashi and W. Takahashi [14] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga's result [3] and Wittmann's result [15]. Tada and W. Takahashi [16] introduced the Mann type iterative algorithm for finding a common element of the set of solutions of the and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm. Yao et al. [17] introduced an iteration process for finding a common element of the set of solutions of the and the set of common fixed points of infinitely many nonexpansive mappings in Hilbert spaces. They proved a strong-convergence theorem under mild assumptions on parameters. Very recently, Moudafi [18] proposed an iterative algorithm for finding a common element of , where is an -inverse-strongly monotone mapping, and obtained a weak convergence theorem. There are some related works, we refer to [19–22] and the references therein.

Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. [17].

## 2. Preliminaries

where is a parameter in .

Then is the maximal monotone and if and only if ; see [23].

Such a mapping is called the -mapping generated by and see [24]. It is obvious that is nonexpansive and if then .

Lemma 2.1 (see [24]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that and let be a sequence in for some . Then, for every and , the limit exists.

Remark 2.2 (see [17]).

for some constant .

Lemma 2.3 (see [24]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive mappings of into itself such that and let be a sequence in for some . Then, .

For solving the generalized equilibrium problem, we assume that the bifunction 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.

The following lemma appears implicitly in [1].

Lemma 2.4 (see [1]).

The following lemma was also given in [3].

Lemma 2.5 (see [3]).

for all . Then, the following hold:

(b1) is single-valued;

(b2) is firmly nonexpansive, that is, for any ,

(b3) ;

(b4) is closed and convex.

Remark 2.6.

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.7 (see [25]).

then .

Lemma 2.8 (see [26]).

where is a sequence in (0,1) and is a sequence in such that

(1) ;

(2) or .

Then .

## 3. Main Results

In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP(1.1), and the solution set of the variational inequality problem for an -inverse-strongly monotone mapping in real Hilbert spaces.

Theorem 3.1.

for all , where , , and are three sequences in , is a sequence in for some and for some satisfying

(i) ;

(ii) and ;

(iii) ;

(iv) and ;

(v) and .

Then , , and converge strongly to the point , where .

Proof.

This implies that is bounded. Therefore, , , , and are also bounded.

It is easy to see that and hence .

we get .

This implies that .

This is a contradiction. Hence, .

Applying Lemma 2.8 to (3.56), we conclude that converges strongly to . Consequently, and converge strongly to . This completes the proof.

As direct consequences of Theorem 3.1, we have the following two corollaries.

Corollary 3.2.

for all , where , , are three sequences in , is a sequence in for some satisfying conditions (i)–(iv). Then converges strongly to the point , where .

Proof.

Let and for all and in Theorem 3.1. Then for Letting (the identity mapping) for all , then for It is easy to see that all conditions of Theorem 3.1 hold. Therefore, we know that the sequence generated by (3.59) converges strongly to . This completes the proof.

Remark 3.3.

From Corollary 3.2, we can get an iterative scheme for finding the solution of the variational inequality involving the -inverse-strongly monotone mapping .

Corollary 3.4 (see [17, Theorem 3.5]).

for all , where , , are three sequences in , and is a sequence in satisfying conditions (i)–(iii) and (v). Then, the sequences and converge strongly to the point , where .

Proof.

Let for and and for all in Theorem 3.1. Since , we get that . It follows from Theorem 3.1 that the sequences and converge strongly to the point . This completes the proof.

Remark 3.5.

The main result of Yao et al. [17, Corollary 3.2] improved and extended the corresponding theorems in Combettes and Hirstoaga [3] and S. Takahashi and W. Takahashi [14]. Our Theorem 3.1 improves and extends Theorem 3.5 of Yao et al. [17] in the following aspects:

(1)the equilibrium problem is extended to the generalized equilibrium problem;

(2)our iterative process (3.1) is different from Yao et al. iterative process (3.60) because there are a project operator and an -inverse-strongly monotone mapping;

(3)our iterative process (3.1) is more general than Yao et al. iterative process (3.60) because it can be applied to solving the problem of finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality for -inverse-strongly monotone mapping.

## Declarations

### Acknowledgments

This work was supported by the National Natural Science Foundation of China (50674078, 50874096, 10671135, 70831005) and the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University).

## Authors’ Affiliations

## References

- Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MathSciNetMATHGoogle Scholar - Li J, Huang N-J, Kim JK:
**On implicit vector equilibrium problems.***Journal of Mathematical Analysis and Applications*2003,**283**(2):501–512. 10.1016/S0022-247X(03)00277-4MathSciNetView ArticleMATHGoogle Scholar - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Wu K-Q, Huang N-J, Yao J-C:
**Existence theorems of solutions for a system of nonlinear inclusions with an application.***Journal of Inequalities and Applications*2007,**2007:**-12.Google 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 - 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 - Takahashi W:
*Nonlinear Functional Analysis, Fixed Point Theory and Its Applications*. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Flåm SD, Antipin AS:
**Equilibrium programming using proximal-like algorithms.***Mathematical Programming*1997,**78**(1):29–41.MathSciNetView ArticleMATHGoogle Scholar - Iusem AN, Sosa W:
**Iterative algorithms for equilibrium problems.***Optimization*2003,**52**(3):301–316. 10.1080/0233193031000120039MathSciNetView ArticleMATHGoogle Scholar - Huang N-J, Lan H-Y, Teo KL:
**On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces.***Journal of Inequalities and Applications*2007,**2007:**-14.Google 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 - 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 - 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 - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Tada A, Takahashi W:
**Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem.***Journal of Optimization Theory and Applications*2007,**133**(3):359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleMATHGoogle 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 - Moudafi A:
**Weak convergence theorems for nonexpansive mappings and equilibrium problems.***Journal of Nonlinear and Convex Analysis*2008,**9**(1):37–43.MathSciNetMATHGoogle Scholar - Peng J-W, Yao J-C:
**A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems.***Taiwanese Journal of Mathematics*2008,**12**(6):1401–1432.MathSciNetMATHGoogle 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 - 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.2008.02.042MathSciNetView ArticleMATHGoogle Scholar - Zeng W-Y, Huang N-J, Zhao C-W:
**Viscosity approximation methods for generalized mixed equilibrium problems and fixed points of a sequence of nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-15.Google 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 - Shimoji K, Takahashi W:
**Strong convergence to common fixed points of infinite nonexpansive mappings and applications.***Taiwanese Journal of Mathematics*2001,**5**(2):387–404.MathSciNetMATHGoogle Scholar - Suzuki T:
**Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(1):103–123. 10.1155/FPTA.2005.103View ArticleMathSciNetMATHGoogle 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 - 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

## 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.