Fixed point results of pointwise contractions in modular metric spaces

Afrah An Abdou,Mohamed A Khamsi

doi:10.1186/1687-1812-2013-163

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 investigate the existence of fixed points of modular contractive mappings in modular metric spaces. These are related to the successive approximations of fixed points (via orbits) which converge to the fixed points in the modular sense, which is weaker than the metric convergence. MSC: 47H09, 46B20, 47H10, 47E10.

Highlights

The purpose of this paper is to give an outline of a fixed point theory for Lipschitzian mappings defined on some subsets of modular metric spaces which are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, CalderonLozanovskii spaces and many others
We look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [ ] on vector spaces and modular function spaces introduced by Musielack [ ] and Orlicz [ ]
The fixed point property in modular function spaces was initiated after the publication of the paper [ ] in

Summary

Introduction

The purpose of this paper is to give an outline of a fixed point theory for Lipschitzian mappings defined on some subsets of modular metric spaces which are natural generalizations of both function and sequence variants of many important, from applications perspective, spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, CalderonLozanovskii spaces and many others. The fixed point property in modular function spaces was initiated after the publication of the paper [ ] in . The authors presented a series of fixed point results for pointwise contractions and asymptotic pointwise contractions acting in modular functions spaces [ , ].

Objectives

Results

Conclusion