On a generalization of a relatively nonexpansive mapping and best proximity pair

Abstract

Abstract Let A and B be two nonempty subsets of a normed space X, and let T : A ∪ B → A ∪ B $T: A \cup B \to A \cup B$ be a cyclic (resp., noncyclic) mapping. The objective of this paper is to establish weak conditions on T that ensure its relative nonexpansiveness. The idea is to recover the results mentioned in two papers by Matkowski (Banach J. Math. Anal. 2:237–244, 2007; J. Fixed Point Theory Appl. 24:70, 2022), by replacing the nonexpansive mapping f : C → C $f: C \to C$ with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation.

Keywords

• Fixed point
• Cyclic and noncyclic
• Relatively nonexpansive mapping
• Uniformly convex Banach space