Best Proximity Point Results for Perimetric Contractions

TL;DR

This paper introduces new types of proximal contractions called perimetric proximal contractions, establishes conditions for the existence of at most two best proximity points, and provides illustrative examples demonstrating these properties.

Contribution

It defines two novel classes of proximal contractions and derives new best proximity point results, expanding the understanding of such mappings.

Findings

Best proximity points are not necessarily unique.

At most two best proximity points can exist for these mappings.

Examples demonstrate mappings with either one or two best proximity points.

Abstract

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.