Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings. (2nd January 2016) - Rajesh, S.; Veeramani, P.
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 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.