One-local retract and common fixed point in modular function spaces

Saleh Abdullah Al-Mezel,Abdullah Al-Roqi,Mohamed Amine Khamsi

doi:10.1186/1687-1812-2012-109

Saleh Abdullah Al-Mezel, Abdullah Al-Roqi + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2012-109

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Abstract

In this paper, we introduce and study the concept of one-local retract in modular function spaces. In particular, we prove that any commutative family of ρ-nonexpansive mappings defined on a nonempty, ρ-closed and ρ-bounded subset of a modular function space has a common fixed point provided its convexity structure of admissible subsets is compact and normal.

Highlights

The purpose of this paper is to give an outline of a common fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces
The importance for applications of nonexpansive mappings in modular function spaces consists in the richness of structure of modular function spaces, that-besides being Banach spaces-are equipped with modular equivalents of norm or metric notions, and are equipped with almost everywhere convergence and convergence in submeasure
In order to prove that D is a one-local retract of C, let {Br(fi, ri)}iÎI be any family of r-closed balls such that fi Î D, for any i Î I, and i∈I

Summary

Introduction

The purpose of this paper is to give an outline of a common fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. These spaces are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces and many others. (5) Let fn ® f r - a.e. There exists a nondecreasing sequence of sets Hk ∈ P such that Hk ↑ Ω and {fn} converges uniformly to f on every Hk (Egoroff Theorem). The following definition plays an important role in the theory of modular function spaces

We say that r has the

Let h