Some remarks on contractive and existence sets

TL;DR

This paper investigates conditions under which existence sets of best coapproximation in Banach spaces are equivalent to contractive sets, providing insights into their structural relationship.

Contribution

It establishes specific conditions on sets and spaces that make existence sets and contractive sets equivalent in Banach spaces.

Findings

Identifies conditions for equivalence of existence and contractive sets.

Provides criteria linking set properties to Banach space structure.

Enhances understanding of best coapproximation in functional analysis.

Abstract

Let X be a real or complex Banach space and let F in X be a non-empty set. F is called an existence set of best coapproximation (existence set for brevity), if for any x in X, $R_F(x)$ is not the empty set, where $$ R_F (x) = \{ d \in F : \|d-c\| \leq \|x-c\| \hbox{ for any } c \in F \}.$$ It is clear that any existence set is a contractive subset of X. The aim of this paper is to present some conditions on F and X under which the notions of exsistence set and contractive set are equivalent.