Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces

Nawab Hussain,Abdullah Alotaibi,Eskandar Naraghirad

doi:10.1186/1687-1812-2013-113

Nawab Hussain, Abdullah Alotaibi + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2013-113

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Abstract

In this paper, we first introduce the concepts of Bregman nonexpansive retract and Bregman one-local retract and then use these concepts to establish the existence of common fixed points for Banach operator pairs in the framework of reflexive Banach spaces. No compactness assumption is imposed either on C or on T, where C is a closed and convex subset of a reflexive Banach space E and is a Bregman nonexpansive mapping. We also establish the well-known De Marr theorem for a Banach operator family of Bregman nonexpansive mappings. MSC: Primary 06F30; 46B20, 47E10.

Highlights

This paper is motivated by the recent papers [ – ]
In [ ] the authors introduced the concept of NR-maps and they used this concept to establish the existence of common fixed points for Banach operator pairs in the context of uniformly convex geodesic metric spaces
We first introduce the concepts of Bregman NR-map and Bregman one-local retract and use these concepts to establish the existence of common fixed points for Banach operator pairs in reflexive Banach spaces

Summary

Introduction

This paper is motivated by the recent papers [ – ]. ]. In this paper we establish some common fixed point results for the Banach operator and symmetric Banach operator pairs in reflexive Banach spaces for Bregman nonexpansive mappings that generalize the concept of nonexpansivity. The ordered pair (S, T) of two self-maps of a closed and convex subset C of a Banach space E is called a Banach operator pair if the set Fix(T) is S-invariant, namely S(Fix(T)) ⊆ Fix(T). It is well known that if a continuous convex function g : E → R is Gâteaux differentiable, ∇g is norm-to-weak∗ continuous 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. The function g is said to be uniformly convex if the function δg : [ , +∞) → [ , +∞], defined by x+y δg(t) := sup g(x) + g(y) – g

Let g

This means that

Our assumption implies