Some remarks on contractive and existence sets

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 $$R_F(x)=\{d\in F:\Vert d-c\Vert \le \Vert x-c\Vert \; for\; anyc\in F\}.$$ 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.