Best proximity results in regular cone metric spaces
In 1980, \textit{B. Rzepecki} introduced a generalized metric \(d_{E}\) on a set \(X\) in a way that \(d_{E} : X \times{X} \longrightarrow K\), replacing the set of real numbers with a Banach space \(E\) in the metric function where \(K\) is a normal cone in \(E\) with a partial order \(\leq\) [Publ. Inst. Math., Nouv. Sér. 28(42), 179--186 (1980; Zbl 0482.47029)]. Seven years later, \textit{S.-D. Lin} considered the notion of \(K\)-metric spaces by replacing the set of non-negative real numbers with a cone \(K\) in the metric function [Indian J. Pure Appl. Math. 18, 685--690 (1987; Zbl 0622.47057)]. In 1997, \textit{P. P. Zabrejko} gave a survey of the theory of \(K\)-metric spaces [Collect. Math. 48, No.4-6, 825--859 (1997; Zbl 0892.46002)]. Ten years after Zabrejko's work, \textit{L.-G. Huang} and \textit{X. Zhang} [J. Math. Anal. Appl. 332, No. 2, 1468--1476 (2007; Zbl 1118.54022)] announced the notion of a cone metric space by replacing real numbers with an ordered Banach space, which is the same as the definition of Rzepecki, or of Lin, or of Zabreiko. \textit{A. Sönmez} [Appl. Math. Lett. 23, No. 4, 494--497 (2010; Zbl 1187.54022)] defined the notion of distance between two subsets \(A\) and \(B\) of a cone metric space \(X\). In the paper under review, the authors establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.
scientific article; zbMATH DE number 6004387