A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings

Abstract

In this paper, we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Yao and Yao [Y. Yao, J.-C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 186 (2) (2007) 1551–1558].

Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥·∥, respectively and let C be a closed convex subset of H. Let F be a bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for $F:C\times C\to \mathbf{R}$ is to find x ∈ C such that

$$F(x\text{,}y)⩾0\text{for all}y\in C\text{.}$$

The set of solutions of (1.1) is denoted by EP(F). Given a mapping $T:C\to H$, let $F(x\text{,}y)=〈\mathit{Tx}\text{,}y-x〉$ for all x, y ∈ C. Then z ∈ EP(F) if and only if 〈Tz, y − z〉 ⩾ 0 for all y ∈ C, i.e., z is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1). In 1997 Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to initial data when EP(F) is nonempty and proved a strong convergence theorem.

Let $A:C\to H$ be a mapping. The classical variational inequality, denoted by $\mathit{VI}(A\text{,}C)$, is to find x^∗ ∈ C such that

$$〈{\mathit{Ax}}^{\ast }\text{,}v-x^{\ast }〉⩾0$$

for all v ∈ C. The variational inequality has been extensively studied in the literature. See, e.g. [12], [14] and the references therein. A mapping A of C into H is called α-inverse-strongly monotone [2], [4] if there exists a positive real number α such that

$$〈\mathit{Au}-\mathit{Av}\text{,}u-v〉⩾\alpha \Vert \mathit{Au}-\mathit{Av}{\Vert }^2$$

for all u, v ∈ C. It is obvious that any α-inverse-strongly monotone mapping A is monotone and Lipschitz continuous. A mapping S of C into itself is called nonexpansive if

$$\Vert \mathit{Su}-\mathit{Sv}\Vert ⩽\Vert u-v\Vert$$

for all u, v ∈ C. We denote by F(S) the set of fixed points of S. For finding an element of $F(S)\cap \mathit{VI}(A\text{,}C)$, Takahashi and Toyoda [9] introduced the following iterative scheme:

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n){\mathit{SP}}_C(x_n-{\lambda }_n{\mathit{Ax}}_n)$$

for every n = 0, 1, 2, … , where x₀ = x ∈ C, α_n is a sequence in (0, 1), and λ_n is a sequence in (0, 2α). Recently, Nadezhkina and Takahashi [5] and Zeng and Yao [15] proposed some new iterative schemes for finding elements in $F(S)\cap \mathit{VI}(A\text{,}C)$. In 2006, Yao and Yao [13] introduced the following iterative scheme:

Let C be a closed convex subset of real Hilbert space H. Let A be an α-inverse-strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that $F(S)\cap \mathit{VI}(A\text{,}C)\ne \varnothing$. Suppose x₁ = u ∈ C and {x_n}, {y_n} are given by

$$\left\{\begin{array}{c}{y}_n=P_C(x_n-{\lambda }_n{\mathit{Ax}}_n)\text{,}\hfill \\ x_{n+1}={\alpha }_{n}u+{\beta }_n{x}_n+{\gamma }_n{\mathit{SP}}_C(y_n-{\lambda }_n{\mathit{Ay}}_n)\text{,}\hfill \end{array}\right.$$

where {α_n}, {β_n}, {γ_n} are three sequences in [0, 1] and {λ_n} is a sequence in $[0\text{,}2\alpha ]$. They proved that the sequence {x_n} defined by (1.3) converges strongly to common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings under some parameters controlling conditions.

Moreover, Takahashi and Takahashi [8] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result [3] and Wittmann’s result [10].

In this paper motivated by the iterative schemes considered in [8], [13] we will introduce a new iterative process (3.1) below for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem, and the solution set of the variational inequality problem for an α-inverse-strongly monotone mapping in a real Hilbert space. Then, we prove a strong convergence theorem which is connected with Yao and Yao’s result [13] and Takahashi and Takahashi’s result [8].