We are interested in the existence of solutions of the differential inclusion
{\dot x}\in F(t,x)
x
˙
∈
F
(
t
,
x
)
on the given time interval, say
[0,1]
[
0
,
1
]
. Here
F
F
is a set-valued mapping from
[0,1]\times \mathbf{R}^n
[
0
,
1
]
×
R
n
into
\mathbf{R}^n
R
n
(we shall write
F: [0,1] \times \mathbf{R}^n \rightrightarrows \mathbf{R}^n
F
:
[
0
,
1
]
×
R
n
⇉
R
n
in what follows) with closed values which will be assumed nonempty whenever necessary. The classical theorems of Filippov and Wazewski theorem uses, as the main assumption characterizing the dependence of
F
F
on
x
x
, the standard Lipschitz condition
h(F(t,x),F(t,x'))\le k(t)\| x-x'\|,
h
(
F
(
t
,
x
)
,
F
(
t
,
x
′
)
)
≤
k
(
t
)
∥
x
−
x
′
∥
,
where
h(P,Q)
h
(
P
,
Q
)
stands for the Hausdorff distance from
P
P
to
Q
Q
. This condition, quite reasonable when
F
F
is bounded-valued, becomes unacceptably strong if the values of
F
F
can be unbounded. Meanwhile unboundedness of the values of the right-hand side set-valued mapping is a fairly natural property of differential inclusions which appear in optimal control problems, e.g. when we deal with a Mayer problem obtained as a result of reformulation of a problem with integral functional. The main purpose of this note is to provide an existence theorem with a weaker version of the Lipschitz condition which is “more acceptable” when the values of
F
F
are unbounded. This condition which could be characterized as a “global” version of Aubin's pseudo-Lipschitz property is very close to that introduced by P. D. Loewen and R. T. Rockafellar [SIAM J. Control Optimization 32 (1994) 442–470].