In this paper the problem of existence of solutions to the initial value problem
u
′
(
t
)
=
A
(
t
,
u
(
t
)
)
,
u
(
a
)
=
z
u’(t) = A(t,u(t)),u(a) = z
, is considered where
A
:
[
a
,
b
)
×
D
→
E
A:[a,b) \times D \to E
is continuous, D is a closed subset of a Banach space E, and
z
∈
D
z \in D
. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution. Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D.