Some best approximation theorems and best proximity point theorems

Abstract

A best approximation theorem for almost cyclic contractions has been proved in the recent article (Sadiq Basha in J. Fixed Point Theory Appl 23:32, 2021). The purpose of this note is to show that, with the same hypotheses as in the preceding best approximation theorem, the conclusion of the theorem can be strengthened to produce a best proximity point rather than a best approximation and hence a best proximity point theorem for almost cyclic contractions in the framework of a uniformly convex Banach space. Further, it is interesting to observe that such a best proximity point theorem for almost cyclic contractions generalizes/subsumes the well known best proximity point theorem, due to Eldred and Veeramani (J Math Anal Appl 323:1001–1006, 2006), for cyclic contractions in the framework of a uniformly convex Banach space. On the other hand, these best approximation theorems and best proximity point theorems for some types of contractions do not generalize the most elegant Banach’s contraction principle because of the underlying richer framework of a uniformly convex Banach space rather than a simpler framework like a complete metric space. Therefore, the purpose of this note is to bring forth the framework of utmost complete space and establish a best proximity point theorem for almost cyclic contractions in such a simpler framework, thereby generalizing the contraction principle.