A Study of Kirk's Asymptotic Contractions via Leader Contractions

TL;DR

This paper broadens fixed point theory for nonlinear contractions by removing the bounded orbit requirement, showing that asymptotic Kirk contractions are Leader contractions, and weakening continuity conditions, thus unifying and generalizing key results.

Contribution

It proves that asymptotic Kirk contractions are Leader contractions, eliminating the boundedness assumption and weakening the upper semicontinuity condition, thereby unifying and extending fixed point theorems.

Findings

Every asymptotic Kirk contraction is a Leader contraction.

Bounded orbit condition can be removed from fixed point results.

Weakened upper semicontinuity suffices for fixed point existence.

Abstract

This paper investigates asymptotic fixed point results for nonlinear contractions, with emphasis on Kirk-type theorems and their generalizations. A central difficulty in the literature has been the requirement that the mapping possesses a bounded orbit, a condition that is often hard to verify and traditionally viewed as essential for guaranteeing the existence of fixed points. We eliminate this boundedness assumption by proving that every asymptotic Kirk contraction is a Leader contraction, which inherently guarantees orbit boundedness. This observation simplifies fixed point arguments and broadens the scope of applicable mappings. We also resolve an open question by showing that the standard upper semicontinuity condition on the control function phi can be weakened to right-upper semicontinuity, addressing a conjecture posed by Jachymski et al. These contributions unify and generalize several foundational results, including those of Boyd-Wong, Kirk, Chen, Arav et al., and Reich and Zaslavski, under the more flexible framework of Leader contractions. The results offer streamlined and more practical conditions for convergence and fixed point existence in generalized metric spaces.