On coupled best proximity points and Ulam-Hyers stability

TL;DR

This paper investigates the existence and stability of coupled best proximity points for specific mappings in uniformly convex Banach spaces, extending the theory to p-cyclic contractions and nonexpansive mappings.

Contribution

It establishes the existence of coupled best proximity points for p-cyclic contraction and nonexpansive mappings, and studies their Ulam-Hyers stability.

Findings

Existence of coupled best proximity points for p-cyclic contractions.

Existence of coupled best proximity points for p-cyclic nonexpansive mappings.

Ulam-Hyers stability of the best proximity point problem.

Abstract

For two nonempty, closed, bounded and convex subsets $A$ and $B$ of a uniformly convex Banach space $X$ consider a mapping $T:(A \times B) \cup (B \times A) \rightarrow A \cup B$ satisfying $T(A,B) \subset B$ and $T(B, A) \subset A$. In this paper the existence of a coupled best proximity point is established when $T$ is considered to be a p-cyclic contraction mapping and a p-cyclic nonexpansive mapping. The Ulam-Hyers stability of the best proximity point problem is also studied.