A fixed point theorem for contractions in modular metric spaces

TL;DR

This paper extends fixed point theorems to modular metric spaces, which generalize traditional metric spaces, by establishing convergence of contractive maps based on generalized average velocities.

Contribution

It introduces a fixed point theorem for contractions in modular spaces, broadening the scope beyond classical metric space fixed point results.

Findings

Fixed points exist for contractive maps in modular spaces.

Convergence of approximations is weaker than in metric spaces.

The theorem relates to generalized average velocities rather than distances.

Abstract

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V. V. Chistyakov, Metric modulars and their application, Dokl. Math. 73(1) (2006) 32-35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72(1) (2010) 1-14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather ``generalized average velocities'' than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.