On the conditions of fixed-point theorems concerning $F$-contractions

TL;DR

This paper introduces a generalized fixed-point theorem for $F$-contractions that relaxes previous conditions, extends to $b$-metric spaces and $ heta$-contractions, and simplifies the theoretical framework.

Contribution

It provides a more general and natural fixed-point theorem for $F$-contractions, removing or relaxing earlier constraints and broadening the applicable context.

Findings

Unified fixed-point theorem for $F$-contractions.

Applicable in $b$-metric spaces and $ heta$-contractions.

Simplified and more natural formulation of the theory.

Abstract

We prove a fixed-point theorem that generalises and simplifies a number of results in the theory of $F$-contractions. We show that all of the previously imposed conditions on the operator can be either omitted or relaxed. Furthermore, our result is formulated in the more general context of $b$-metric spaces and $\varphi$-contractions. We also point out that the framework of $F$-contractions can be reformulated in an equivalent way that is both closer in spirit to the classical syntax of Banach-type fixed point theorems, and also more natural and easier to deal with in the proofs.