A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization
Introduction
Throughout this paper, we always assume that is a real Hilbert space with inner product and norm , respectively, is a nonempty closed convex subset of and is the metric projection of onto . In the following, we denote by “” strong convergence and by “” weak convergence. Recall that a mapping is called nonexpansive, if We denote by the set of fixed points of the mapping . Recall that a mapping is called -inverse-strongly monotone [2], if there exists a positive real number such that
Remark It is easy to see that if is -inverse-strongly monotone, then it is a -Lipschitzian mapping.
Let be a mapping. The classical variational inequality problem is to find a such that The set of solutions of variational inequality (1.2) is denoted by .
Let be a bifunction. The “so-called” equilibrium problem for the function is to find a point such that We denote the set of solutions of the equilibrium problem (1.3) by .
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 is nonempty and proved some strong convergence theorems in Hilbert spaces.
For finding a common element of , Takahashi and Toyoda [14] introduced the following iterative scheme: and and obtained a weak convergence theorem in a Hilbert space, where is a sequence in (0, 1) and is a sequence in .
For finding a common element of , Takahashi and Takahashi [13] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: and
For finding a common element of , recently Su, Shang and Qin [11] introduced the following iterative scheme: 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 , very recently Plubtieng and Punpaeng [8] also introduced the following iterative scheme: and 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 -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 be a real Hilbert space. It is well known that for any Let be a nonempty closed convex subset of . For each , there exists a unique nearest point in , denoted by , such that is called a metric projection of onto . It is known that is a nonexpansive mapping and satisfies: Moreover, is characterized by the following property: 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 -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 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: where is a nonempty closed convex subset of a Hilbert space, and is a convex and lower semi-continuous functional. We denote by the set of solutions of (4.1). Let be a bifunction defined by . We consider the following equilibrium problem, that is to find such that 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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