A remark on the paper "A note on the paper Best proximity point results for $p$-proximal contractions"

TL;DR

This paper clarifies that for $p$-proximal contractions with a constant less than one-third, the existence of best proximity points can be derived directly from the Banach contraction principle, simplifying previous results.

Contribution

It demonstrates that under certain conditions, the best proximity point results for $p$-proximal contractions are a consequence of the Banach contraction principle, providing a simpler proof.

Findings

Best proximity points exist for $p$-proximal contractions with $k<1/3$.

Such points can be obtained via the Banach contraction principle.

The result simplifies previous proofs in the literature.

Abstract

Recently, In the year 2020, Altun et al. \cite{AL} introduced the notion of $p$-proximal contractions and discussed about best proximity point results for this class of mappings. Then in the year 2021, Gabeleh and Markin \cite{GB} showed that the best proximity point theorem proved by Altun et al. in \cite{AL} follows from the fixed point theory. In this short note, we show that if the $p$-proximal contraction constant $k<\frac{1}{3}$ then the existence of best proximity point for $p$-proximal contractions follows from the celebrated Banach contraction principle.