We consider the system of differential inclusions
$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$
, where
F,G:
D →
Kυ (
\(R^{m_1 } \)),
Kυ (
\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces
\(R^{m_1 } \) and
\(R^{m_2 } \), respectively,
D =
R + ×
\(R^{m_1 } \) ×
\(R^{m_2 } \) × 0,
a],
a > 0, and µ is a small parameter. The functions
F and
G and the right-hand side of the averaged problem
\(\dot u\) ∈ µ
F 0(
u),
u(0) =
x 0,
F 0(
u) ∈
Kυ (
\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each
? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ
0] and any solution
x µ(·),
y µ(·) of the original problem, there exists a solution
u µ(·) of the averaged problem such that ∥
x µ(
t) ?
y µ(
t) ∥ ≤
? for
t ∈ 0, 1/µ]. Furthermore, for each solution
u µ(·)of the averaged problem, there exists a solution
x µ(·),
y µ(·) of the original problem with the same estimate.
