Best approximations in metric spaces with property strongly UC

In this article, we introduce a geometrical notion, called property strongly UC which is stronger than property UC and prove the existence of best approximations for a new class of almost cyclic ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-contraction maps defined on a pair of subsets of a metric space. As a particular case of this existence theorem, we obtain the main results of [Sadiq Basha, S., Best approximation theorems for almost cyclic contractions. J. Fixed Point Theory Appl. 23 (2021)] and [Eldred, A. Anthony; Veeramani, P., Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2006)]. Moreover, we study the existence of a best approximation and continuity properties of almost cyclic contractions in the context of a reflexive Banach space and a metric space.