Hybrid shrinking iterative solutions to convex feasibility problems for countable families of relatively nonexpansive mappings and a system of generalized mixed equilibrium problems

Wei-Qi Deng,Shanguang Qian

doi:10.1186/1687-1812-2014-148

Abstract

We propose a new hybrid shrinking iterative scheme for approximating common elements of the set of solutions to convex feasibility problems for countable families of relatively nonexpansive mappings of a set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces. The results extend those of other authors, in which the involved mappings consist of just finitely many ones.

Highlights

1 Introduction Throughout this paper we assume that E is a real Banach space with its dual E∗, C is a nonempty, closed, convex subset of E, and J : E → E∗ is the normalized duality mapping defined by
A mapping T : C → C is said to be relatively nonexpansive if F(T) = F (T) = ∅ and φ(p, Tx) ≤ φ(p, x), ∀x ∈ C, p ∈ F(T), ( . )
A strong convergence theorem is established in the framework of Banach spaces

Summary

Introduction

Throughout this paper we assume that E is a real Banach space with its dual E∗, C is a nonempty, closed, convex subset of E, and J : E → E∗ is the normalized duality mapping defined by. In , Plubtieng and Ungchittrakool [ ] established strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the following hybrid method in mathematical programming:. Inspired and motivated by the studies mentioned above, in this paper, we use a modified hybrid iteration scheme for approximating common elements of the set of solutions to convex feasibility problem for a countable families of relatively nonexpansive mappings, of set of solutions to a system of generalized mixed equilibrium problems. A strong convergence theorem is established in the framework of Banach spaces The results extend those of the authors, in which the involved mappings consist of just finitely many ones. (iii) If E is a smooth, strictly convex and reflexive Banach space, the normalized duality mapping J : E → E∗ is single valued, one-to-one, and onto.

Main results

Applications