Some Properties of Distances and Best Proximity Points of Cyclic Proximal Contractions in Metric Spaces

Abstract

This paper presents some results concerning the properties of distances and existence and uniqueness of best proximity points of p-cyclic proximal, weak proximal contractions, and some of their generalizations for the non-self-mapping $T:{\bigcup }_{i\in \stackrel{-}{p}}{A}_i\to {\bigcup }_{i\in \stackrel{-}{p}}{B}_i(p\ge 2)$, where $A_i$ and $B_i$, $\forall i\in \stackrel{-}{p}=\{\mathrm{1,2},\dots ,p\}$, are nonempty subsets of $X$ which satisfy $T\left(A_i\right)\subseteq B_i,\forall i\in \stackrel{-}{p}$, such that $(X,d)$ is a metric space. The boundedness and the convergence of the sequences of distances in the domains and in their respective image sets of the cyclic proximal and weak cyclic proximal non-self-mapping, and of some of their generalizations are investigated. The existence and uniqueness of the best proximity points and the properties of convergence of the iterates to such points are also addressed.