Pointwise orbital proximal contractions and best proximity results in hyperconvex metric spaces

Abstract

We introduce new classes of pointwise orbital contractions for mappings defined on pairs of subsets of metric spaces and investigate their best proximity properties in hyperconvex metric spaces. First, we define the notion of a pointwise orbital cyclic contraction and establish the existence of best proximity points for such mappings acting on admissible pairs of subsets of a hyperconvex metric space. As a consequence, a novel fixed point theorem is obtained for pointwise orbital contractions defined on a single admissible subset. We then study the noncyclic case by introducing pointwise orbital noncyclic contractions. Using minimal admissible invariant pairs and the concept of semi-sharp proximinality, we prove the existence of best proximity pairs for distance-preserving noncyclic mappings in hyperconvex setting. Furthermore, we show that in Banach spaces the semi-sharp proximinality of every convex pair characterizes strict convexity. Finally, motivated by asymptotic contraction techniques, we introduce the class of asymptotic proximal pointwise cyclic contractions and prove a best proximity point theorem under a suitable geometric notion, called property UC, for admissible pairs.