It is proved that if
q
i
j
(
t
,
s
)
ρ
j
(
s
)
[
ρ
i
(
t
)
]
−
1
{q_{ij}}(t,s){\rho _j}(s){[{\rho _i}(t)]^{ - 1}}
is bounded,
i
,
j
=
1
,
2
,
…
,
n
i,j = 1,2, \ldots ,n
, and
f
(
t
,
x
,
x
(
u
(
s
)
)
)
f(t,x,x(u(s)))
is “small",
\[
x
(
u
(
s
)
)
=
(
x
1
(
u
1
(
s
)
)
,
x
2
(
u
2
(
s
)
)
,
…
,
x
n
(
u
n
(
s
)
)
)
x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))
\]
with
u
i
(
t
)
⩽
t
{u_i}(t) \leqslant t
and
lim
t
→
∞
u
i
(
t
)
=
∞
{\lim _{t \to \infty }}{u_i}(t) = \infty
, the solutions of the integral equation
\[
x
(
t
)
=
h
(
t
)
+
∫
0
t
q
(
t
,
s
)
f
(
s
,
x
(
s
)
,
x
(
u
(
s
)
)
)
d
s
x\left ( t \right ) = h(t) + \int _0^t {q(t,s)f(s,x(s),x(u(s)))ds}
\]
satisfy the conditions
x
(
t
)
=
h
(
t
)
+
ρ
(
t
)
a
(
t
)
,
lim
t
→
∞
a
(
t
)
=
x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) =
constant where
ρ
(
t
)
\rho (t)
is a nonsingular diagonal matrix chosen in such a way that
ρ
−
1
(
t
)
h
(
t
)
{\rho ^{ - 1}}(t)h(t)
is bounded. The results contain, in particular, some results on the asymptotic behavior, stability and existence of nonoscillatory solutions of functional differential equations.