Research Article Results on the Existence and Convergence of Best Proximity Points | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 386037 10 pages doi 2010 386037 Research Article Results on the Existence and Convergence of Best Proximity Points Ali Abkar and Moosa Gabeleh Department of Mathematics Imam Khomeini International University . Box 288 Qazvin 34149 Iran Correspondence should be addressed to Moosa Gabeleh Received 24 February 2010 Accepted 10 June 2010 Academic Editor W. A. Kirk Copyright 2010 A. Abkar and M. 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. We first consider a cyclic -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 -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. 1. Introduction Let X X d be a metric space and let A B be two subsets of X. A mapping T A u B A u B is said to be cyclic provided that T A c B and T B c A. In 1 Kirk et al. proved the following interesting extension of the Banach contraction principle Theorem see 1 . Let A and B be two nonempty closed subsets of a complete metric space X. Suppose that T is a cyclic map such that d Tx Ty ad x y for some a e 0 1 and for all x e A y e B. Then T has a unique fixed point in A n B. Later on Eldred and Veeramani 2 considered the class of cyclic contractions. 2 Fixed Point Theory and Applications Definition see 2 . Let A and B be two nonempty subsets of a metric space X and let T A u B A u B T A c B and T B
