A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization

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Abstract

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].

Introduction

Throughout this paper, we always assume that H is a real Hilbert space with inner product , and norm , respectively, C is a nonempty closed convex subset of H and PC is the metric projection of H onto C. In the following, we denote by “” strong convergence and by “” weak convergence. Recall that a mapping S:CC is called nonexpansive, if SxSyxy,x,yC. We denote by F(S) the set of fixed points of the mapping S. Recall that a mapping A:CH is called α-inverse-strongly monotone [2], if there exists a positive real number α such that AxAy,xyαAxAy2,x,yC.

Remark It is easy to see that if A:CH is α-inverse-strongly monotone, then it is a 1α-Lipschitzian mapping.

Let A:CH be a mapping. The classical variational inequality problem is to find a uC such that Au,vu0,vC. The set of solutions of variational inequality (1.2) is denoted by V I(C,A).

Let ϕ:C×CR be a bifunction. The “so-called” equilibrium problem for the function ϕ is to find a point xC such that ϕ(x,y)0,yC. We denote the set of solutions of the equilibrium problem (1.3) by EP(ϕ).

This equilibrium problem contains the fixed point problem, optimization problem, variational inequality problem and Nash equilibrium problem as its special cases (see, for example, Blum and Oetti [1]).

In 1977 Combettes and Hirstoaga [5] introduced an iterative scheme of finding the best approximation to initial data when EP(ϕ) is nonempty and proved some strong convergence theorems in Hilbert spaces.

For finding a common element of F(S)VI(C,A), Takahashi and Toyoda [14] introduced the following iterative scheme: x1C and xn+1=αnxn+(1αn)SPC(xnλnAxn),n1, and obtained a weak convergence theorem in a Hilbert space, where {αn} is a sequence in (0, 1) and {λn} is a sequence in (0,2α).

For finding a common element of F(S)EP(ϕ), Takahashi and Takahashi [13] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: x1H and {ϕ(un,y)+1rnyun,unxn0,yC,xn+1=αnf(xn)+(1αn)Sun,n1n1.

For finding a common element of F(S)VI(C,A)EP(ϕ), recently Su, Shang and Qin [11] introduced the following iterative scheme: x1H{ϕ(un,y)+1rnyun,unxn0,yC,xn+1=αnf(xn)+(1αn)SPC(unλnAun),n1n1. Under suitable conditions some strong convergence theorems are proved which extend and improve the results of Iiduka et al. [7] and Takahashi et al. [13].

On the other hand, in order to find a common element of F(S)VI(C,A)EP(ϕ), very recently Plubtieng and Punpaeng [8] also introduced the following iterative scheme: x1=uC and {ϕ(un,y)+1rnyun,unxn0,yC,yn=PC(unλnAun),xn+1=αnu+βnxn+γnSPC(ynλnAyn),n1, under suitable conditions, some strong convergence theorems are proved which extend some recent results of Yao and Yao [15].

In this paper, motivated and inspired by the above results, we will introduce a new iterative scheme (3.1) below for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality problem for an α-inverse-strongly monotone mapping in a Hilbert space. Then we prove some strong convergence theorems which extend and improve the corresponding results of Yao and Yao [15], Iiduka et al. [7], Takahashi et al. [14], Takahashi and Takahashi [13], Su, Shang and Qin [11], and Chang, Cho and Kim [3].

Section snippets

Preliminaries

Let H be a real Hilbert space. It is well known that for any λ[0,1]λx+(1λ)y2=λx2+(1λ)y2λ(1λ)xy2. Let C be a nonempty closed convex subset of H. For each xH, there exists a unique nearest point in C, denoted by PCx, such that xPCxxy,xC.P is called a metric projection of H onto C. It is known that PC is a nonexpansive mapping and satisfies: PCxPC2PCxPC,xy,x,yC. Moreover, PCx is characterized by the following property: PcxCandxPCx,PCxy0,yC. In the context

Main results

In this section, we shall introduce an iterative scheme by using the viscosity approximation method for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the variational inequality for an α-inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets.

Theorem 3.1

Let H be a real Hilbert space,

Applications

(I) Application to Optimization problem.

In this section we shall utilize the results presented in the paper to study the following optimization problem: mimxCh(x), where C is a nonempty closed convex subset of a Hilbert space, and h:CR is a convex and lower semi-continuous functional. We denote by S the set of solutions of (4.1). Let ϕ:C×CR be a bifunction defined by ϕ(x,y)=h(y)h(x). We consider the following equilibrium problem, that is to find xC such that ϕ(x,y)0,yC. It is easy to

Acknowledgment

The first author was supported by the Natural Science foundation of Yibin University (No. 2007-Z003).

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