Fixed Point Theorems for Relaxed Asymptotic Contractions via Two Quasi-Metrics

TL;DR

This paper introduces a new class of asymptotic contractions using two quasi-metrics, relaxing existing fixed point theorems and ensuring convergence to a unique fixed point under weaker conditions.

Contribution

It extends Kirk's asymptotic fixed point theorem by relaxing hypotheses through a novel quasi-metric framework and convergence analysis.

Findings

Established existence and uniqueness of fixed points under weaker assumptions

Proved convergence of iterates to the fixed point in complete metric spaces

Generalized previous fixed point theorems with a new contraction condition

Abstract

We introduce a new class of asymptotic contractions that employs two quasi-metrics defined directly in terms of the underlying mapping. The contraction condition compares these two quantities via a sequence of bounding functions that converge locally uniformly to a Boyd-Wong function. This framework relaxes the hypotheses of Kirk's asymptotic fixed point theorem and strictly contains it as a special case. Assuming only the continuity of the map and the boundedness of some orbit in a complete metric space, we prove both the existence and uniqueness of a fixed point, along with the convergence of all iterates to that point.