Summary: We first consider a cyclic
-contraction map on a reflexive Banach space
and provide a positive answer to a question raised by M. A. Al-Thagafi
and N. Shahzad
on the existence of best proximity points for cyclic
-contraction maps in reflexive Banach spaces in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 10 (A), 3665–3671 (2009; Zbl 1197.47067
)]. In the second part of the paper, we discuss the existence of best proximity points in the framework of more general metric spaces. We obtain some new results on the existence of best proximity points in hyperconvex metric spaces as well as in ultrametric spaces.