Fixed point results of pointwise contractions in modular metric spaces
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RESEARCH
Open Access
Fixed point results of pointwise contractions in modular metric spaces Afrah A N Abdou1* and Mohamed A Khamsi2,3 *
Correspondence: [email protected] Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract
1
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: Primary 47H09; secondary 46B20; 47H10; 47E10 Keywords: 2 -condition; fixed point; modular function space; modular metric spaces; Orlicz spaces; pointwise contraction mapping
1 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. Modular metric spaces were introduced in [, ]. The main idea behind this new concept is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) ‘field of (generalized) velocities’: to each ‘time’ λ > (the absolute value of ) an average velocity wλ (x, y) is associated in such a way that in order to cover the - ‘distance’ between points x, y ∈ X, it takes time λ to move from x to y with velocity wλ (x, y). But the way we approached the concept of modular metric spaces is different. Indeed, 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 []. In recent years, there was an uptake of interest in the study of electrorheological fluids, sometimes referred to as ‘smart fluids’ (for instance, lithium polymetachrylate). For these fluids, modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces, Lp and W ,p , where p is a fixed constant, is not adequate, but rather the exponent p should be able to vary [, ]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [, ]. The classical technique used so far in studying this problem is to convert the energy functional, naturally defined by a modular, to a convoluted © 2013 Abdou and Khamsi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http:
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