New iterations of the best proximity point with convergence rates

Abstract ‎Let (A, B) be a nonempty pair of disjoint subsets of a Banach space X‎. ‎It is well-known that if (A, B) be a bounded‎, ‎closed and convex pair in a uniformly convex‎ ‎Banach space X, ‎then every cyclic contraction mapping has a unique best proximity point‎. ‎The main purpose of this paper is to study the convergence of best proximity points for cyclic contraction mappings in the setting‎ ‎of uniformly convex Banach spaces‎. ‎Hence‎, ‎we consider the Mann and Ishikawa iteration schemes‎ ‎and obtain many of strong convergence results of the best proximity point for the cyclic contraction mapping in the‎ ‎uniformly convex Banach spaces‎. ‎To this end‎, ‎we consider a numerical example is presented to support the main result and we compare‎ ‎the convergence behavior of iterations‎. 2010 Mathematics Subject Classification: 90C48, 47H09, 46B20