[Paper Review] A fixed point theorem for contractions in modular metric spaces

V Chistyakov Vyacheslav|arXiv (Cornell University)|Dec 23, 2011

Fixed Point Theorems Analysis24 references21 citations

TL;DR

This paper establishes a fixed point theorem for contractive mappings in modular metric spaces, a generalization of metric spaces introduced by Chistyakov. Unlike classical fixed point theorems based on distance contraction, this work focuses on contracting generalized average velocities, with convergence occurring in a weaker modular sense rather than metric convergence.

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.

Motivation & Objective

• To extend fixed point theory to modular metric spaces, which generalize standard metric spaces.
• To address the lack of fixed point results for contractive maps in the context of modular metrics.
• To define convergence in terms of modular functionals rather than traditional metric distances.
• To establish conditions under which successive approximations converge to a fixed point in a modular topology.
• To generalize classical contraction mapping principles to the modular setting, where convergence is weaker than in metric spaces.

Proposed method

• Utilizes the concept of a modular metric on a set, defined via a family of convex functionals.
• Introduces a notion of contractive maps based on the modular metric rather than the standard metric.
• Applies the idea of 'generalized average velocities' to model contraction behavior in modular spaces.
• Establishes convergence of iterative sequences in the modular topology, defined by the modular functional.
• Employs techniques from modular space theory to prove existence and convergence of fixed points.
• Demonstrates that convergence is in the modular sense, meaning the modular of the difference tends to zero, not necessarily the metric.

Experimental results

Research questions

• RQ1 Can a fixed point theorem be established for contractive maps in modular metric spaces?
• RQ2 How does the convergence of iterative sequences behave in modular spaces compared to classical metric spaces?
• RQ3 What role do generalized average velocities play in defining contraction in modular settings?
• RQ4 In what sense does the fixed point emerge when convergence is defined via the modular functional?
• RQ5 How does the proposed framework generalize the classical Banach contraction principle?

Key findings

• A fixed point theorem is established for contractive maps in modular metric spaces, extending classical results to a broader framework.
• The convergence of successive approximations occurs in the modular topology, not in the stronger metric sense.
• The contraction condition is defined via the modular functional, reflecting behavior of generalized average velocities.
• The fixed point is guaranteed to exist under the proposed modular contraction condition.
• The result generalizes the Banach contraction principle to modular spaces, where convergence is weaker than in metric spaces.
• The framework allows for applications in spaces where standard metric tools are insufficient due to lack of uniform convexity or completeness.