Let
{
X
s
:
α
⩽
s
⩽
β
}
\left \{ {{X_s}:\alpha \leqslant s \leqslant \beta } \right \}
be a scale of Banach spaces,
J
J
a real interval,
U
U
an open subset of
J
×
X
s
J \times {X_s}
for some
s
s
. In this paper we prove that the existence of solutions for
\[
x
′
=
A
(
t
)
x
+
f
(
t
,
x
)
,
x
(
t
0
)
=
x
0
,
x’ = A(t)x + f(t,x),\quad x({t_0}) = {x_0},
\]
is a generic property, when
A
(
t
)
A(t)
is an operator satisfying
\[
|
A
(
t
)
|
L
(
X
s
′
;
X
s
)
⩽
M
(
s
′
−
s
)
−
1
(
M
>
0
independent
of
s
,
s
′
,
t
)
{\left | {A(t)} \right |_{L({X_{s’}};{X_s})}} \leqslant M{(s’ - s)^{ - 1}}\quad (M > 0\;{\text {independent}}\;{\text {of}}\;s,s’,t)
\]
in the scale
{
X
s
}
\left \{ {{X_s}} \right \}
and
f
:
J
×
U
→
X
β
f:J \times U \to {X_\beta }
is continuous.