On Berinde's paper "Comments on some fixed point theorems in metric spaces"

Constantin Zalinescu

Published 2020-03-23Version 1

In Theorem 1 of the paper [V. Pata, A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), 299-305] it is proved that Picard's iterates for a function converge to a fixed point if a certain condition (C) is verified for all parameters in the interval [0,1]. In the recent paper [V. Berinde, Comments on some fixed point theorems in metric spaces, Creat. Math. Inform. 27 (2018), 15-20], the author claims that Pata's result does not hold at least in the two extremal cases for the parameter involved in (C). In this note we point out that Berinde's Theorem 1.1 has only a visual similarity with Pata's Theorem 1, and the conclusion of Theorem 1.1 is verified only by constant functions.