Best proximity point theorems in metric spaces with property PC

Abstract

In the article [S. Sadiq Basha, Best approximation theorems for almost cyclic contractions, J. Fixed Point Theory Appl. 23, 32 (2021)], the author proved a best approximation theorem which ensures the existence of a best approximation for almost cyclic contractions in the richer framework of uniformly convex Banach space. Further, in the article [S. Sadiq Basha, Some best approximation theorems and best proximity point theorems, Acta Sci. Math. (Szeged) 89 (2023), 215-226], the author proved a best proximity point theorem which ascertains the existence of a best proximity point for almost cyclic contractions in the richer framework of uniformly convex Banach space. The purpose of this paper is to introduce a notion known as the property PC and extend the aforesaid best approximation theorem and best proximity point theorem to the less stringent framework of metric space with the property PC. As a consequence, the results obtained in this article generalize the well-renowned contraction principle due to Banach.