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.