On proximal contractions via implicit relations and best proximity points

TL;DR

This paper introduces new types of proximal contractions using implicit relations and establishes conditions for the existence of best proximity points, extending and generalizing previous results in the field.

Contribution

It defines novel proximal contractions via implicit relations and proves new existence theorems for best proximity points, generalizing earlier work by Sadiq Basha.

Findings

Established existence of best proximity points for new proximal contractions.

Generalized previous results by Sadiq Basha.

Provided examples validating the theoretical results.

Abstract

In this paper, we employ two types of implicit relations to define some new kind of proximal contractions and study about their best proximity points. More precisely, we use two class of functions $\mathcal{A}$ and $\mathcal{A}'$ to explore proximal $\mathcal{A}$, $\mathcal{A}'$-contractions of first and second type and strong proximal $\mathcal{A}$, $\mathcal{A}'$-contractions. We investigate the existence of best proximity point results of the same. It is worth mentioning that the well-known results of Sadiq Basha [J. Approx. Theory, $2011$] on proximal contractions are the special cases of our obtained results. We authenticate our results by suitable examples.