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

Let A and B be two nonempty subsets of a normed space X, and let T:A boolean OR B -> A boolean OR B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T: A \cup B \to A \cup B$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: C \to C$\end{document} with a cyclic (resp., noncyclic) relatively nonexpansive mapping to obtain the best proximity pair. Additionally, we provide an application to a functional equation.