Bregman weak relatively nonexpansive mappings in Banach spaces

Eskandar Naraghirad,Jen-Chih Yao

doi:10.1186/1687-1812-2013-141

Abstract

In this paper, we introduce a new class of mappings called Bregman weak relatively nonexpansive mappings and propose new hybrid iterative algorithms for finding common fixed points of an infinite family of such mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ-inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein-type is presented. Our results improve and generalize many known results in the current literature. MSC:47H10, 37C25.

Highlights

The hybrid projection method was first introduced by Hangazeau in [ ]
In this paper, using Bregman functions, we introduce new hybrid iterative algorithms for finding common fixed points of an infinite family of Bregman weak relatively nonexpansive mappings in Banach spaces
Theorem . improves Theorem . in the following aspects. ( ) For the structure of Banach spaces, we extend the duality mapping to a more general case, that is, a convex, continuous and strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets. ( ) For the mappings, we extend the mapping from a relatively nonexpansive mapping to a countable family of Bregman weak relatively nonexpansive mappings

Summary

Introduction

The hybrid projection method was first introduced by Hangazeau in [ ]. If C is a nonempty, closed and convex subset of a reflexive Banach space E and g : E → R is a strongly coercive Bregman function, for each x ∈ E, there exists a unique x ∈ C such that 1.6 Some facts about Bregman quasi-nonexpansive mappings Let C be a nonempty, closed and convex subset of a reflexive Banach space E.

Results

Conclusion