On possible generalisations of quasi-contractions

TL;DR

This paper examines the limits of fixed point theorems for quasi-contractions in metric spaces, showing that further generalizations do not guarantee fixed points, supported by counterexamples.

Contribution

It demonstrates the impossibility of further straightforward relaxations of quasi-contractive conditions ensuring fixed points, through counterexamples and theoretical analysis.

Findings

Counterexamples of operators without fixed points

Further relaxations do not guarantee fixed points

Fixed point theorems for quasi-contractions are essentially optimal

Abstract

This paper investigates whether some fixed point theorems for quasi-contractions on metric spaces introduced by \`Cir\`ic in [1] and generalised by Kumam et al. in [2] can be improved further. It turns out that the answer is negative. We provide two examples of complete metric spaces and two operators without fixed points. We prove that for any possible straightforward relaxation of generalised quasi-contractive conditions, one of these operators satisfies the condition.