On Best Proximity Points in Metric and Banach Spaces

Abstract

Abstract. In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.

Existence and uniqueness of best proximity points in geodesic metric spaces

“…In [6], it was proved that property UC implies property WUC and also a result on existence, uniqueness and convergence of iterates of a cyclic contraction to a best proximity point for a pair of sets with property WUC was obtained.…”

“…Remark 2.5. Any uniformly convex complete metric space with either a monotone or lower semicontinuous from the right modulus of convexity has the nonempty intersection property (see Definition 2.1 in [6] and Proposition 2.2 in [13]). …”

“…Existence, uniqueness and convergence of iterates to a best proximity point for cyclic contractions have recently been studied in [2,3,6,7].…”

“…We begin this section by recalling the main properties on pairs of sets defined in [6,7]. Firstly, Property UC was defined in [7] as follows.…”

“…In this regard, we define properties HW and HWT, and compare them with properties UC and WUC. In Section 4 we make use of properties HW and HWT to state the main result of the paper and we provide its main consequences in relation to results given in [2,3,6].…”