- Research Article
- Open Access

# Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems

- Hong Yu Li
^{1}Email author and - Hong Zhi Li
^{2}

**2009**:362191

https://doi.org/10.1155/2009/362191

© H. Li and H. Li. 2009

**Received:**26 August 2008**Accepted:**9 January 2009**Published:**28 January 2009

## Abstract

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed points of finite many nonexpansive mappings. We prove strong convergence of the iterative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for the minimization problem.

## Keywords

- Hilbert Space
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Strong Convergence

## 1. Introduction

The set of such solutions is denoted by EP( ). Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].

The set of solutions of variational inequality problem is denoted by . The variational inequality problem has been extensively studied in literatures, see, for example, [4, 5] and references therein.

where *T* is a nonexpansive mapping on
and
is a point on
.

where for all . Such a mapping is called -mapping generated by and . We know that is nonexpansive and , see [6].

They proved that converges strongly to , where is the metric projection from onto .

and proved that converges strongly to a point and also solves the variational inequality (1.8). For equilibrium problems, also see [10, 11].

and established a strong convergence theorem of to an element of .

where is defined by (1.5), is -cocoercive, and is a bounded linear operator. We should show that the sequences converge strongly to an element of . Our result extends the corresponding results of Qin et al. [16] and Colao et al. [9], and many others.

## 2. Preliminaries

Let be a monotone mapping of into , then if and only if . The following result is useful in the rest of this paper.

Lemma 2.1 (see [17]).

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

Lemma 2.2 (see [18]).

Let be bounded sequences in Banach space satisfying and . Let be a sequence in with . Then, .

Lemma 2.3.

Lemma 2.4 (see [7]).

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

For solving the equilibrium problem for a bifunction , we assume that satisfies the following conditions:

(A4)for all is convex and lower semicontinuous.

The following result is in Blum and Oettli [1].

Lemma 2.5 (see [1]).

We also know the following lemmas.

Lemma 2.6 (see [19]).

for all . Then, the following holds:

It is known that in this case is maximal monotone, and if and only if .

## 3. Strong Convergence Theorem

Theorem 3.1.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , be in and be in , is a constant in . Assume and

Proof.

We divide the proof into several steps.

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

, where is the unique solution of variational inequality

Since is bounded, without loss of generality, we assume itself converges weakly to a point . We should prove .

with the same argument as used in [14], we can derive , since is maximal monotone, we know .

Divide by in both side yields , let , by (A3) we conclude . Therefore, .

Step 6.

The sequence converges strongly to .

so, by Lemma 2.1, we conclude . This completes the proof.

Putting and for all in Theorem 3.1, we obtain the following corollary.

Corollary 3.2.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself. Let be relaxed -cocoercive and -Lipschitzian. Let be an -contraction with and a strong positive linear bounded operator with coefficient , be a constant with . Let sequences , in and be a constant in . Assume and

Putting and , , in Theorem 3.1, we obtain the following corollary.

Corollary 3.3.

Let be a real Hilbert space and be a nonempty closed convex subset of . a finite family of nonexpansive mappings from into itself and a bifunction satisfying (A1)–(A4). Let be an -contraction with and a strong positive linear bounded operator with coefficient , is a constant with . Let sequences , in and in . Assume and

## 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 - Combettes PL, Hirstoaga SA:
**Equilibrium programming in Hilbert spaces.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):117–136.MathSciNetMATHGoogle Scholar - Moudafi A:
**Second-order differential proximal methods for equilibrium problems.***Journal of Inequalities in Pure and Applied Mathematics*2003,**4**(1, article 18):1–7.MathSciNetMATHGoogle Scholar - Yao J-C, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications*.*Volume 76*. Kluwer Academic Publishers, Dordrecht, The Nederlands; 2005:501–558.View ArticleGoogle Scholar - Zeng LC, Schaible S, Yao J-C:
**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 - Bauschke HH:
**The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.***Journal of Mathematical Analysis and Applications*1996,**202**(1):150–159. 10.1006/jmaa.1996.0308MathSciNetView 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 - Plutbieng S, Punpaeng R:
**A general iterative method for equlibrium 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 - 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 - 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, Article ID 64363,**2007:**-12.Google Scholar - Shang MJ, Su YF, Qin XL:
**A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces.***Fixed Point Theorey and Applications*2007, Article ID 95412,**2007:**-9.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 - Yao Y, Yao J-C:
**On modified iterative method for nonexpansive mappings and monotone mappings.***Applied Mathematics and Computation*2007,**186**(2):1551–1558. 10.1016/j.amc.2006.08.062MathSciNetView ArticleMATHGoogle Scholar - Chen JM, Zhang LJ, Fan TG:
**Viscosity approximation methods for nonexpansive mappings and monotone mappings.***Journal of Mathematical Analysis and Applications*2007,**334**(2):1450–1461. 10.1016/j.jmaa.2006.12.088MathSciNetView ArticleMATHGoogle Scholar - Qin XL, Shang MJ, Zhou HY:
**Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces.***Applied Mathematics and Computation*2008,**200**(1):242–253. 10.1016/j.amc.2007.11.004MathSciNetView 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 - 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 - Takahashi W, Zembayashi K:
**Strong convergence theorem for equilibrium problems and relatively nonexpansive mappings in Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2008,**70**(1):45–57.MathSciNetView 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.