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In this article, we introduce a geometrical notion, property strongly UC which is stronger than property UC and prove the existence of best approximations for a new class of almost cyclic $\psi$-contraction maps in a metric space. As a particular case, we obtain the main results of [Sadiq Basha, S., Best approximation theorems for almost cyclic contractions. J. Fixed Point Theory Appl. 23 (2021), MR4254305] and [Eldred, A. Anthony; Veeramani, P., Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2006), MR2260159]. Moreover, we study the existence of best approximations for almost cyclic contractions in a reflexive Banach space.

We provide a few characterizations of a strictly convex Banach space. Using this we improve the main theorem of [Digar, Abhik; Kosuru, G. Sankara Raju; Cyclic uniform Lipschitzian mappings and proximal uniform normal structure. Ann. Funct. Anal. 13 (2022)].

The main objective of this article is to provide an alternative approach to the central result of [Eldred, A. Anthony, Kirk, W. A., Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., vol 171(3), (2005) 283-293] using proximal uniform normal structure. Also we provide characterizations of a strictly convex space. Finally, sufficient conditions for the existence of a non-empty proximal sub-pair for a pair in a Banach space are discussed.

The aim of this paper is to address an open problem given in [Kirk, W. A., Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. {\bf{vol 463(2)}}, (2018) 461--476]. We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction wrt orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces.

In this article we introduce a new type of cyclic contraction mapping on a pair of subsets of a metric space with a graph and prove best proximity points results for the same. Also, we demonstrate that the number of such points is same with the number of connected subgraphs. Hereafter, we introduce a fixed point mapping obtained from the aforesaid cyclic contraction and prove some fixed point theorems which will be used to find a common solution for a system of periodic boundary value problems. Our results unify and subsume many existing results in the literature.