Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

Moosa Gabeleh,Hans-Peter A Künzi

doi:10.1515/dema-2020-0005

Abstract

AbstractIn this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

Highlights

Throughout this study, (A, B) is a pair of nonempty and disjoint subsets of a normed linear space X
[6] A convex pair (K1, K2) in a Banach space X is said to have a proximal normal structure (PNS) if for any bounded, closed, convex and proximinal pair (H1, H2) ⊆ (K1,K2) for which dist(H1, H2) = dist(K1, K2) and δ(H1, H2) > dist(H1, H2), there exists (x1, x2) ∈ H1 × H2 such that max{δx1 (H2), δx2 (H1)} < δ (H1, H2). It was announced in [6] that every nonempty, bounded, closed and convex pair in a uniformly convex Banach space X has PNS
Theorem 1.9. [8, Theorem 3.2] Let (A, B) be a nonempty, closed and convex pair in a uniformly convex Banach space X and T a noncyclic contraction mapping defined on A ∪ B

Summary

Introduction

Throughout this study, (A, B) is a pair of nonempty and disjoint subsets of a normed linear space X. A point (p, q) ∈ A × B is said to be a best proximity pair for the noncyclic mapping T: A ∪ B → A ∪ B provided that p = Tp, q = Tq and d (p, q) = dist(A, B) := inf{d (x, y): (x, y) ∈ A × B}. Some existence results of best proximity points (pairs) can be found in [1,2,3,4,5].

Objectives

Results

Conclusion