In this paper we obtain sufficient conditions under which every solution of the retarded differential equation
\[
(
1
)
x
′
(
t
)
+
p
(
t
)
x
(
t
−
τ
)
=
0
,
t
⩾
t
0
,
(1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},
\]
, where
τ
\tau
is a nonnegative constant, and
p
(
t
)
>
0
p(t) > 0
, is a continuous function, tends to zero as
t
→
∞
t \to \infty
. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as
t
→
∞
t \to \infty
. More precisely the following theorems have been established. Theorem 1. Assume that
∫
t
0
∞
p
(
t
)
d
t
=
+
∞
\int _{{t_0}}^\infty {p(t)dt = + \infty }
and
lim
t
→
∞
∫
t
−
τ
t
p
(
s
)
d
s
>
π
/
2
{\lim _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > \pi /2}
or
lim
sup
t
→
∞
∫
t
−
τ
t
p
(
s
)
d
s
>
1
\lim {\sup _{t \to \infty }}\int _{t - \tau }^t {p(s)ds > 1}
. Then every solution of (1) tends to zero as
t
→
∞
t \to \infty
. Theorem 2. Assume that
lim
sup
t
→
∞
∫
t
→
τ
t
p
(
s
)
d
s
>
1
\lim {\sup _{t \to \infty }}\int _{t \to \tau }^t {p(s)ds > 1}
. Then every oscillatory solution of (1) tends to zero as
t
→
∞
t \to \infty
.