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Corrigendum to the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings”

Gabeleh Moosa

Demonstratio Mathematica

2021

Vol. 54, nr 1

9--10

The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings, ”Demonstr. Math. 53 (2020), 38-43.

Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

Gabeleh Moosa, Künzi Hans-Peter A.

Demonstratio Mathematica

2020

Vol. 53, nr 1

38--43

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and 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 [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.