Markov-Kakutani's theorem for best proximity pairs in Hadamard spaces

In the current paper, we consider two classes of noncyclic mappings, called quasi-noncyclic relatively nonexpansive and noncyclic relatively u-continuous, and survey the existence of best proximity pairs as well as the structure of best proximity pair sets for these classes of mappings in Busemann convex spaces. We also study the existence of a common best proximity pair for families if noncyclic mappings in Hadamard spaces. In this way, we obtain a generalization of Markov-Kakutani's fixed point theorem in Hadamard spaces.