Copyright © 2010 Ali Abkar and Moosa Gabeleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first consider a cyclic φ-contraction map on a reflexive Banach space X and provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of best proximity points for cyclic φ-contraction maps in reflexive Banach spaces in one of their works (2009). In the second part of the paper, we will 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.