Abstract
Let
X
0 ⊂
X
1 ⊂ ··· ⊂
X
p
be Banach spaces with continuous injection of
X
k
into
X
k + 1
for 0 ⩽
k ⩽
p − 1, and with
X
0 dense in
X
p
. We seek a function
u: [0, 1] →
X
0 such that its
kth derivative
u
(
k)
,
k = 0, 1,…,
p, is continuous from [0, 1] into
x
k
, and satisfies the initial condition
u
(
k)
(0) =
a
k
ϵ
X
k
. It is shown that such a function exists if and only if the initial values
a
0,
a
1, …,
a
p
satisfy a certain condition reminiscent of interpolation theory. This condition always holds when
p = 1; when
p ⩾ 2, the spaces
X
k
(
k = 0, 1, …,
p) may or may not be such that the desired function exists for
any given initial values
a
k
ϵ
X
k
.