On fixed point results in metric spaces for large triangle-perimeter contractions

TL;DR

This paper extends fixed point theory for large triangle-perimeter contractions in metric spaces, introducing a corrected approach that broadens previous theories and includes a crucial auxiliary assumption.

Contribution

It provides a corrected extension of existing contraction theories, introduces an auxiliary condition for fixed point results, and demonstrates the broader applicability of the new framework.

Findings

Counterexample to existing large triangle-perimeter contraction results

Fixed point theorem under an additional auxiliary condition

New framework is strictly broader than previous theories

Abstract

In this paper we introduce a corrected extension of Burton's theory of large contractions in the context of triangle-perimeter contractions introduced by Petrov. Combining these two lines of research, we prove a fixed point result for large triangle-perimeter contractions with an auxiliary assumption, which is of utmost importance. Firstly, we give a counterexample to the main result related to large triangle-perimeter contractions that currently exists in literature. Then, we prove that given an additional condition, a fixed point result for large triangle-perimeter contractions holds. Lastly, we illustrate with an example that this new framework is strictly broader than Burton's and Petrov's theory.