Best Proximity Point For The Proximal Non-Expansive Mapping

Let (A, B) be a pair of nonempty subsets of a metric space (X, d). A mapping T:A →B is said to be proximal nonexpansive if for all u1, u2, x1,x2 ∈A. d(u1, Tx1)= dist(A,B) and d(u2, Tx2)= dist(A,B), There have d(u1,u2)≤ d(x1,x2)-Our results are established on the star-shaped without the assumptions of continuity, affinity or P-property. Finally, as applications of the theorems, analogues for the nonexpansive mappings are also given. The paper improves and generalizes a number of the best proximity point theories have been concluded.