One-Local Retract and Common Fixed Point in Modular Metric Spaces

Afrah A N Abdou

doi:10.1155/2013/672069

Abstract

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we introduced and study the concept of one-local retract in modular metric space. In particular, we investigate the existence of common fixed points of modular nonexpansive mappings defined on nonempty -closed -bounded subset of modular metric space.

Highlights

The purpose of this paper is to give an outline of a common fixed-point theory for nonexpansive mappings on some subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, CalderonLozanovskii, and many other spaces
We study the concept of one-local retract in more general setting in modular metric space; we prove the existence of common fixed points for a family of modular nonexpansive mappings defined on nonempty ωclosed ω-bounded subsets in modular metric space
Note that Aω(Xω) is stable by intersection. At this point we will need to define the concept of Chebyshev center and radius in modular metric spaces

Summary

Introduction

The purpose of this paper is to give an outline of a common fixed-point theory for nonexpansive mappings (i.e., mappings with the modular Lipschitz constant 1) on some subsets of modular metric spaces which are natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, CalderonLozanovskii, and many other spaces. In [26], Khamsi introduced the concept of one-local retract in metric spaces and proved that any commutative family of nonexpansive mappings defined on a metric space with a compact and normal convexity structure has a common fixed point. In [27], the authors introduced the concept of one-local retract in modular function spaces and proved the existence of common fixed points for commutative mappings. We study the concept of one-local retract in more general setting in modular metric space; we prove the existence of common fixed points for a family of modular nonexpansive mappings defined on nonempty ωclosed ω-bounded subsets in modular metric space. For more on metric fixed point theory, the reader may consult the book [28] and for modular function spaces the book [29]

Basic Definitions and Properties

One-Local Retract Subsets in Modular Metric Spaces

Common Fixed Point Result