Fixed Points of Asymptotic Pointwise Contractions under Local Uniform Convergence

TL;DR

This paper introduces a new weak asymptotic contraction concept in metric spaces, proving fixed point convergence under local uniform convergence conditions, extending Kirk's theorem.

Contribution

It extends Kirk's asymptotic contraction theorem by replacing global uniform convergence with local uniform convergence.

Findings

Iterates of continuous mappings with bounded orbits converge to a unique fixed point.

The new notion generalizes previous contraction conditions by using local uniform convergence.

The limit function satisfies a Boyd--Wong type control condition.

Abstract

We introduce a weak asymptotic version of nonlinear contraction, termed \emph{asymptotic pointwise contraction}. For a mapping on a metric space, this notion requires the existence of a sequence of functions that dominate the distances between the $n$-th iterates of any two points. The sequence is assumed to converge pointwise to a limit function, and the convergence is required to be uniform on every bounded set (i.e., locally uniform). The limit function is then controlled by a Boyd--Wong type condition: there exists a nondecreasing, right upper semicontinuous function strictly below the identity on positive numbers, and the limit function is bounded above by this function evaluated at a maximum term that involves not only the distance between the two points but also distances from each point to its image and mutual distances between each point and the image of the other. By standard analytic arguments we prove that if the mapping is continuous on a complete metric space and possesses a bounded orbit, then its iterates converge to a unique fixed point. This result extends Kirk's asymptotic contraction theorem by replacing global uniform convergence on $[0,\infty)$ with the weaker condition of local uniform convergence.