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 $T:A\cup B\to A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x\in A\cup B$ such that $\Vert x-Tx\Vert =\text{dist}(A,B)$ whenever $T(A)\subseteq B$, $T(B)\subseteq A$. Also, we establish that if $T(A)\subseteq A$ and $T(B)\subseteq B$, then there exist $x\in A$ and $y\in B$ such that $Tx=x$, $Ty=y$ and $\Vert x-y\Vert =\text{dist}(A,B)$. 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.

References in corpus (1)

• Best proximity points for proximal contractions