Author:
Ciesielski Maciej,Lewicki Grzegorz
Abstract
AbstractLet X be a real or complex Banach space and let $$ F \subset X $$
F
⊂
X
be a non-empty set. F is called an existence set of best coapproximation (existence set for brevity), if for any $$ x \in X $$
x
∈
X
, $$R_F(x) \ne \emptyset , $$
R
F
(
x
)
≠
∅
,
where $$\begin{aligned} R_F (x) = \{ d \in F : \Vert d-c\Vert \le \Vert x-c\Vert \hbox { for any } c \in F \} . \end{aligned}$$
R
F
(
x
)
=
{
d
∈
F
:
‖
d
-
c
‖
≤
‖
x
-
c
‖
for any
c
∈
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.
Publisher
Springer Science and Business Media LLC
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