Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

Abstract

ABSTRACTLet (A, B) be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and T: A ∪ B → A ∪ B be a continuous and asymptotically relatively nonexpansive map. We prove that there exists x ∈ A ∪ B such that ‖x − Tx‖ = dist(A, B) whenever T(A) ⊆ B, T(B) ⊆ A. Also, we establish that if T(A) ⊆ A and T(B) ⊆ B, then there exist x ∈ A and y ∈ B such that Tx = x, Ty = y and ‖x − y‖ = dist(A, B). We prove the aforementioned results when the pair (A, B) has the rectangle property and property UC. In the case of A = B, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.