# Some remarks on "Convergence of Picard's iteration using projection algorithm for noncyclic contractions" [Indag. Math. 30 (2019) 227--239]

## TL;DR

This paper establishes the equivalence of best proximity points and pairs for certain mappings in Banach spaces and analyzes the convergence of iterative sequences, linking recent methods to Picard's iteration.

## Contribution

It proves the equivalence between best proximity points and pairs in Banach spaces and connects recent convergence methods to classical Picard's iteration.

## Key findings

- Best proximity points for cyclic and noncyclic mappings are equivalent in strictly convex Banach spaces.
- Convergence of best proximity pairs for noncyclic contractions is achieved via Picard's iteration.
- Recent convergence methods are shown to be special cases of Picard's iteration.

## Abstract

In this note, at first we prove that the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that a main result of the paper "Proximal normal structure and relatively nonexpansive mappings", Studia Math., (171~(2005) 283--293) immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper published in Indag. Math., 30(1) (2019) 227--239 is obtained exactly from Picard's iteration sequence.