Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

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Let (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 aforesaid results when the pair (A, B) has the rectangle property and property UC. In 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.