Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

Abstract

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 existsAlso, we establish that if T (A) ⊆ A and T (B) ⊆ B, then there exist x ∈ A and y ∈ B such that T x = x, T y = y and x − y = dist(A, B). We prove the aforesaid results when the pair (A, B) has the rectangle property and property U C. 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.

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