Explicit Scheme for Fixed Point Problem for Nonexpansive Semigroup and Split Equilibrium Problem in Hilbert Space

Pei Zhou,Gou-Jie Zhao

doi:10.1155/2014/858679

Abstract

We establish an iterative method for finding a common element of the set of fixed points of nonexpansive semigroup and the set of split equilibrium problems. Under suitable conditions, some strong convergence theorems are proved. Our works improve previous results for nonexpansive semigroup.

Highlights

Let H be a real Hilbert space whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively
Let C be a nonempty closed convex subset of H, and let F be a bifunction of C × C into R which is the set of real numbers
The equilibrium problem introduced by Blum and Oettli [1] for F : C × C → R is to find x ∈ C such that

Summary

Introduction

Let H be a real Hilbert space whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively. In 2010, Tian [10] introduced the following general iterative scheme for finding an element of set of solutions to the fixed point of nonexpansive mapping in a Hilbert space. In 2014, Kazmi and Rizvi [13] studied the following implicit iterative algorithm Under some assumptions, they obtain some strong convergence theorem for EP (1) and the fixed point problem: un = TrFn1 (xn + δA∗ (TrFn2 − I) Axn) , xn αnγf (xn). Motivated and inspired by [10,11,12,13], we introduce an explicit iterative scheme for finding a common element of the set of solutions SEP and fixed point for a nonexpansive semigroup in real Hilbert spaces. Some strong convergence theorems for approximating to these common elements are proved

Preliminaries

The General Explicit Iterative Method

Application