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The aim of this article is to study the existence of best proximity pairs for asymptotic relatively nonexpansive mappings using a geometric notion of the Pythagorean property. In this way, we establish an existence theorem under some different conditions with respect to the same result of Rajesh and Veeramani (Numer Funct Anal Optim 37:80â91, 2016). We also prove a best proximity point theorem for cyclic relatively nonexpansive mappings which are compact in the setting of reflexive Busemann convex spaces. An illustrative example will be presented to show the usability of our conclusions.
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