This is an announcement for the paper “Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings” by S. Rajeshhttps://arxiv.org/find/math/1/au:+Rajesh_S/0/1/0/all/0/1, P. Veeramanihttps://arxiv.org/find/math/1/au:+Veeramani_P/0/1/0/all/0/1.
Abstract: Let $(A,B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\cup B\rightarrow A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x\in A\cup B$ such that $|x - Tx| = \emph{dist}(A, B)$ whenever $T(A)\subset B, T(B)\subset A$. Also, we establish that if $T(A)\subset A, T(B)\subset B$, then there exist $x\in A$ and $y\in B$ such that $Tx=x, Ty=y$ and $|x - y| = \emph{dist}(A, B)$. 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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1611.02484