Best proximity point theorems in Hadamard spaces using relatively asymptotic center

In this article we survey the existence of best proximity points for a class of non-self mappings which‎ satisfy a particular nonexpansiveness condition. In this way, we improve and extend a main result of Abkar and Gabeleh [‎A‎. ‎Abkar‎, ‎M‎. ‎Gabeleh‎, Best proximity points of non-self mappings‎, ‎Top‎, ‎21, (2013)‎, ‎287-295]‎ which guarantees the existence of best proximity points for nonexpansive non-self mappings in the setting of uniformly convex Banach spaces. We also introduce a new notion, ‎ called relatively asymptotic center, on a nonempty, bounded,‎ closed ‎and ‎convex ‎pair ‎of ‎subsets ‎of a‎ ‎Hadamard‎ ‎metric space and ‎as a result of our main conclusions, we will show that the asymptotic center of any sequence in a nonempty, bounded, closed and convex subset of a Hadamard space is singleton. Moreover, we obtain the other existence results of best proximity points for generalized nonexpansive mappings using the appropriate geometric properties of Hadamard spaces. Finally, we provide some examples to illustrate our main results.