A note on the paper "Best proximity point of generalized $F$-proximal non-self contractions

TL;DR

This note connects the existence of best proximity points for generalized $F$-proximal contractions to fixed point theory, simplifying the understanding of their existence and uniqueness.

Contribution

It demonstrates that the existence of best proximity points for these contractions can be derived directly from fixed point theory results.

Findings

Best proximity point existence follows from fixed point theory.

Simplifies previous proofs for generalized $F$-proximal contractions.

Establishes a direct link between best proximity points and fixed points.

Abstract

In the year 2021, Beg et al. \cite{beg} [J. Fixed Point Theory Appl.(2021)] introduced two classes of non-self mappings namely, generalized $F$-proximal contraction of the first kind and generalized $F$-proximal contraction of the second kind. Then authors studied the existence and uniqueness of best proximity points for this two classes of mappings. In this short note, we show that the existence of best proximity point for generalized $F$-proximal contraction of the first kind follows from the same conclusion in fixed point theory.