Given that and are two nonempty subsets of the convex metric space , a mapping is noncyclic relatively nonexpansive, provided that , , and for all . A point is called a best proximity pair for the mapping if , , and . In this work, we study the existence of best proximity pairs for noncyclic relatively nonexpansive mappings by using the notion of nonconvex proximal normal structure. In this way, we generalize a main result of Eldred, Kirk, and Veeramani. We also establish a common best proximity pair theorem for a commuting family of noncyclic relatively nonexpansive mappings in the setting of convex metric spaces, and as an application we conclude a common fixed-point theorem.
"Nonconvex proximal normal structure in convex metric spaces." Banach J. Math. Anal. 10 (2) 400 - 414, April 2016. https://doi.org/10.1215/17358787-3495759