Abstract
In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation
[
r
(
t
)
(
x
(
t
)
+
P
(
t
)
x
(
t
−
τ
)
)
′
]
′
+
∑
i
=
1
m
Q
i
(
t
)
f
i
(
x
(
t
−
σ
i
)
)
=
0
,
t
⩾
t
0
,
where
m
⩾
1
is an integer,
τ
>
0
,
σ
i
⩾
0
,
r
,
P
,
Q
i
∈
C
(
[
t
0
,
∞
)
,
R
)
,
f
i
∈
C
(
R
,
R
)
(
i
=
1
,
2
,
…
,
m
), are studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general
P
(
t
)
and
Q
i
(
t
)
(
i
=
1
,
2
,
…
,
m
) which means that we allow oscillatory
P
(
t
)
and
Q
i
(
t
)
(
i
=
1
,
2
,
…
,
m
). In particular, our results improve essentially and extend some known results in the recent references.