[Paper Review] A contraction principle in semimetric spaces

Mihály Bessenyei, Zsolt Páles|arXiv (Cornell University)|Jan 8, 2014

Fixed Point Theorems Analysis10 references21 citations

TL;DR

This paper extends the Matkowski contraction principle to complete semimetric spaces endowed with a regular basic triangle function, proving the existence and uniqueness of fixed points for $φ$-contractions. It further establishes the stability of fixed points under pointwise convergence of mappings, generalizing the Banach Fixed Point Theorem to broader classes of spaces such as $c$-relaxed and $c$-inframetric spaces.

ABSTRACT

A branch of generalizations of the Banach Fixed Point Theorem replaces contractivity by a weaker but still effective property. The aim of the present note is to extend the contraction principle in this spirit for such complete semimetric spaces that fulfill an extra regularity property. The stability of fixed points is also investigated in this setting. As applications, fixed point results are presented for several important generalizations of metric spaces.

Motivation & Objective

• To generalize the Matkowski fixed point theorem beyond complete metric spaces to a broader class of semimetric spaces.
• To identify a regularity condition on semimetric spaces—via the continuity of the basic triangle function at the origin—that ensures topological well-behavedness, including Hausdorff separation and the Cauchy property of convergent sequences.
• To establish the stability of fixed points under pointwise convergence of a sequence of $φ$-contractions in such spaces.
• To demonstrate that the contraction principle holds in $c$-relaxed and $c$-inframetric spaces, which are special cases of regular semimetric spaces.
• To clarify the connection between the basic triangle function and topological properties such as idempotence of the closure operator, though this remains partially open.

Proposed method

• Define a semimetric space as a set $X$ with a symmetric, nonnegative function $d: X \times X \to \mathbb{R}_+$ that vanishes exactly on the diagonal.
• Introduce the basic triangle function $\Phi_d(u,v) = \sup\{d(x,y) \mid \exists p \in X: d(p,x) \leq u, d(p,y) \leq v\}$, which generalizes the triangle inequality.
• Define a regular semimetric space as one where $\Phi_d$ is continuous at the origin, ensuring the topology is Hausdorff and convergent sequences are Cauchy.
• Use the concept of a comparison function $\varphi: \mathbb{R}_+ \to \mathbb{R}_+$ such that $\varphi^n(t) \to 0$ pointwise for all $t > 0$, and define a $\varphi$-contraction by $d(Tx,Ty) \leq \varphi(d(x,y))$.
• Prove the existence and uniqueness of a fixed point for any $\varphi$-contraction on a complete regular semimetric space via contradiction and iterative estimates.
• Establish the stability of fixed points by showing that if a sequence of $\varphi$-contractions $T_n$ converges pointwise to $T_0$, then the fixed points of $T_n$ converge to the fixed point of $T_0$, under the assumption of self-continuity of $d$.

Experimental results

Research questions

• RQ1 Can the Matkowski fixed point theorem be extended to semimetric spaces that do not satisfy the standard triangle inequality?
• RQ2 What regularity condition on the basic triangle function ensures that the topology of a semimetric space is Hausdorff and that convergent sequences are Cauchy?
• RQ3 Under what conditions does the fixed point of a sequence of $\varphi$-contractions converge to the fixed point of the limit mapping?
• RQ4 How do $c$-relaxed and $c$-inframetric spaces fit into this generalized framework, and do they satisfy the required regularity and completeness conditions?
• RQ5 What is the relationship between the basic triangle function and the idempotence of the closure operator in Hausdorff semimetric spaces?

Key findings

• Every $\varphi$-contraction on a complete regular semimetric space has a unique fixed point, generalizing the Matkowski and Banach fixed point theorems.
• The basic triangle function $\Phi_d$ is optimal among all triangle functions for $d$, and its continuity at the origin characterizes regular semimetric spaces.
• In regular semimetric spaces, convergent sequences have unique limits and satisfy the Cauchy property.
• If a sequence of $\varphi$-contractions $T_n$ converges pointwise to $T_0$ and the semimetric $d$ is self-continuous, then the fixed points of $T_n$ converge to the fixed point of $T_0$.
• The result applies to complete $c$-relaxed metric spaces and $c$-inframetric spaces, where the basic triangle function is continuous at the origin.
• The self-continuity of $d$—which holds in metric and ultrametric spaces—ensures that the limit mapping $T_0$ inherits the $\varphi$-contractivity of the sequence $T_n$.