Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

TL;DR

This paper establishes new proximity point theorems for asymptotically relatively nonexpansive mappings in nearly uniformly convex Banach spaces, extending fixed point results to pairs of sets with specific properties.

Contribution

It introduces proximity point theorems for asymptotically relatively nonexpansive maps on convex pairs with rectangle and UC properties, generalizing existing fixed point theorems.

Findings

Existence of points satisfying proximity conditions under specified mappings.

Extension of fixed point theorems to pairs of convex sets with additional properties.

Results include classical fixed point theorems as special cases.

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 $T(A) \subseteq B$, $T(B) \subseteq A$. 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 $\|x - y\| = \emph{dist}(A, B)$. 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.