Fixed points and best proximity points of a cyclic map

Abstract

This thesis contains two parts. In the first part, we provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of a best proximity point for a cyclic ℓ -contraction map in a reflexive Banach space. Moreover, we extend the Al-Thagafi and Shahzad theorem to metric spaces with property UC. In the second part, we prove that the Picard projection iteration sequence converges to a fixed point and the rate of convergence is given. We also prove the generalize Collage Theorem for a multi-valued weak contraction map. In addition, we extend the cyclic condition to multi-valued maps and study the existence of a fixed point and the existence of a best proximity point for the map having been extended. Moreover, we study data dependence problem for a special class of multi-valued maps.