1. Introduction
In this paper, we study the oscillatory behavior of the differential equations with damping terms
where
During this study, we will assume the following:
- (H1)
and is a ratio of odd natural numbers;
- (H2)
, and
- (H3)
, is not eventually zero on for
- (H4)
, , and .
Let
. We say that a real-valued function
is a solution of (
1) if
,
satisfies (
1) on
, and
for every
. A solution of (
1) is called oscillatory if it has arbitrarily large zeros on
; otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.
There are many applications described by neutral differential equations. Such equations arise naturally in the modeling of physical and biological phenomena such as oscillations of neuromuscular systems, deformation of structures, or problems of elasticity (see [
1,
2] for more details).
We know that there is great interest in studying the qualitative behaviors of differential equations, such as the asymptotic behavior, the property of stability, the property of boundary, and the oscillatory properties (see [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15]).
We will now present some of the previous results that were mentioned in the literature.
Grace and Akin [
16] discuss the oscillations of nonlinear delay differential equations
where
for
They stipulated in their studies that the differential equation
is nonoscillatiory or oscillatory.
Elabbasy et al. [
17] studied the fourth-order delay differential equation with the middle term
where
They established new oscillation results by using generalized Riccati transformations and comparative principles.
In [
18], Dassios and Bazighifan studied the oscillatory behaviour of nonlinear differential equations
where
and by using the Riccati technique, they proved that the solutions to (
4) are oscillatory or converge to zero as
Yang and Bai [
19] investigated the oscillation behavior of solutions of the fourth-order
p-Laplacian differential equations with the middle term
where
. Furthermore, they investigated (
5) under the condition
and by using the Riccati transformations and comparison method with first-order and second-order differential equations, they proved that the solutions to (
5) are oscillatory.
We know that there are many works that deal with oscillating solutions of fourth-order differential equations with damping terms, but most of them—as far as we know—are concerned only with the canonical case. We also know that most of the results obtained in the noncanonical case guarantee that the fourth-order differential equations with damping terms are oscillatory or converge to zero. In light of this, this paper is a continuation of the above recent work on the noncanonical case in which we will introduce some new conditions that guarantee the oscillation of differential equations with damping terms that use Riccati transformations. We will also show that our criteria take into account the influence of the delay argument , which has been neglected in previous studies. The criteria we obtained improved and completed some of the criteria in previous studies. We will also mention some lemmas that will help us prove the main results of this paper. Finally, we present examples that show the applicability of our results.
Lemma 1.
([
20]).
Assume that is of one sign, eventually. Then, there exists a and is integer, with , such thatfor all .
Lemma 2.
([
21], Lemma 2.2.3).
Let and be of fixed sign and not identically zero on a subray of . Suppose that there exists a such that for . If then there exists a such thatfor every and
2. Main Results
Before presenting the main results of our paper, we will start by mentioning some notations. Let
, where
We define the functions
and
Now, we define class
as the category of all eventually positive solutions of (
1).
Lemma 3.
Assume that . Then , , and one of the following cases hold, for :
are positive and is negative; and
are positive for all .
Proof. Assume that
, then, there exists
such that
and
for all
. Hence, we see that
for
. Multiplying (
1) by
, we get
and so
Now, by using Lemma 1 with
, we readily get the cases (
A)–(
C). □
Lemma 4.
Let . Assume that and and Then Proof. Assume that
, and then there exists a
, such that
and
for
. Considering the fact that
, and from (
7) we have
from (
9) we have that
is increasing and thus
Integrating (
10) from
s to
, we get
and so
From (
11), we obtain
which leads to
From (
2) and (
14), we have
From (
9) and (
15), we find (
8) holds. This completes the proof. □
Lemma 5.
Let . Assume that and and Then
is bounded;
where and .
Proof. Assume that , and then there exists a , such that and for .
From Lemma 4, we have
is increasing. By using (
11) and (
16), we obtain
If
from
and (
17), we have that
is bounded.
If
, using (
11), we have
and by using (
16), we obtain
Because is decreasing, then is bounded. Thus, is bounded, where .
From (
16) and (
8), we find
and from (
12) and (
13), we get
If
, from
, then
is increasing. Letting
(if
, then
), the Inequality (
19) becomes
Because
, then from (
21) we have
where
(if
then
).
From (
20) and (
22), we obtain
where
and
This completes the proof. □
Lemma 6.
Let . Assume that , and Proof. Assume that , and then there exists a , such that and for .
Because
for
. Recall that
. Hence,
By using (
7) and (
25) we get
From the definition of the function
, we have
and
from (
26) and (
27), we find (
24) holds. This completes the proof. □
Lemma 7.
Let . Assume that , and Proof. Assume that , and then there exists a , such that and for .
Because
is nonincreasing, then
Letting
, we get
from (
30), we have
for
Therefore, we have
which, from (
7), it follows that
By using (
25) and (
28), we have
which can be written as
From Lemma 2, we get
and hence (
32), becomes
Thus, we find that (
29) holds. This completes the proof. □
Theorem 1.
Assume that hold, and If there exists a positive nondecreasing function , such that hold, where and Proof. Assume that
, and then there exists a
, such that
and
for
. From
for
and (
7), we have
and hence
is nonincreasing. Therefore, there exists a
, such that the cases
hold, for all
.
Assume that case
holds. From Lemma 4, we get
From (
16), we have that
for all
. From Lemma 5, we have
By taking the integral from
to
s after multiplying the Inequality (
37) by
we get
By using the inequality
we get
and so
where
because
is bounded. Letting
in (
40), we obtain a contradiction with (
34).
Assume that case
holds. From Lemma 6, we have that (
24) holds.
If
. Note that
is a positive nonincreasing, and then
By using Lemma 2 with
we obtain
by using (
41) and (
42) in (
24), we get
Because
, thus there exists a constant
and
, such that
If
, then
; thus, (
43) still holds.
If
, because
, we have
. Because
, hence
, therefore
is nondecreasing. Thus, there exist constant
,
, such that
By combining (
24) and (
44), we then obtain
which, together with (
43), implies that
where
and
By using (
39) in (
45), we have
Integrating this inequality from
to
s, we get
Letting
in (
46), we get a contradiction with (
35).
Assume that case
holds. From Lemma 7, we see that (
29) holds.
If
. Note that
is a nonincreasing, then
which is
integrating from
s to
∞, we obtain
By using (
47) in (
24), we obtain
If
, then
; thus, (
48) still holds.
If
, because
is nonincreasing, there exists a
, such that
and so
By combining (
29) and (
49), we then have
which, together with (
48), implies that
where
By using the inequality
we find that
which is
By using (
39) and (
52), we obtain
By integrating (
53) from
to
s, we get
which is
which contradicts (
36). This completes the proof. □
Theorem 2.
Theorem 1 still holds if conditions (34)–(36) are replaced by respectively.
Proof. Suppose that (
54) holds. Then there exists a sufficiently large
, for any
, such that
By integrating (
57) from
S to
s, we have
Letting
in (
58), we have that (
34) holds.
Moreover, condition (
55) can be obtained directly by
into (
35).
Now, condition (
56) can be obtained directly by substituting
into (
36).
This completes the proof. □
Now, we present some examples to illustrate the possibility of applying the results that we obtained. First, we present a special case, which is when .
Example 1.
Consider the differential equation where and It is easy to verify that and Moreover, we see that and then condition (55) holds. Now, the condition (54) becomes and condition (56) becomes By using Theorem 2, we have that (59) is oscillatory if the conditions (60) and (61) verified. Example 2.
Consider the differential equation where Let , then we have and
Now, the condition (54) becomes The condition (55) becomes The condition (56) becomes Hence (54)–(56) are satisfied, and by using Theorem 2, we see that (62) is oscillatory. Remark 1.
Note that our criterion (60) essentially takes into account the influence of the delay argument , which has been neglected in [18]. Moreover, note that our criteria guarantee the oscillation of (59), whereas the criteria that were deduced in [18] guarantee that (59) is oscillatory or converges to zero. Therefore, our criteria are an improvement and complement the criteria in [18].