Suppose that X is a real or complex Banach space and that A is a continuous function from
[
0
,
∞
)
×
X
[0,\infty ) \times X
into X. Suppose also that there is a continuous real valued function
ρ
\rho
defined on
[
0
,
∞
)
[0,\infty )
such that
A
(
t
,
⋅
)
−
ρ
(
t
)
I
A(t, \cdot ) - \rho (t)I
is dissipative for each t in
[
0
,
∞
)
[0,\infty )
. In this note we show that, for each z in X, there is a unique differentiable function u from
[
0
,
∞
)
[0,\infty )
into X such that
u
(
0
)
=
z
u(0) = z
and
u
′
(
t
)
=
A
(
t
,
u
(
t
)
)
u’(t) = A(t,u(t))
for all t in
[
0
,
∞
)
[0,\infty )
. This is an improvement of previous results on this problem which require additional conditions on A.