1. Introduction
Consider the 2nd-order delay differential equation (DDE) of the neutral type:
(1)
where
and
. In this entry, we obtain new sufficient criteria for the oscillation of solutions of (
1) under the following hypotheses:
(A1) is a ratio of odd integers;
(A2) for a constant (this constant plays an important role in the results), and does not vanish identically on any half-line with
(A3) and for
By a proper solution of (
1), we mean a
with
and
for
and u satisfies (
1) on
. A solution u of (
1) is called nonoscillatory if it is eventually positive or eventually negative; otherwise, it is called oscillatory.
The oscillatory properties of solutions of second-order neutral DDE (
1) in the noncanonical case, that is:
(2)
where
2. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order
We begin with the following notations: is the set of all eventually positive solutions of (1), ,
Lemma 1. Assume that and there exists a such that:
![](/media/common/202112/blobid2-61c2e65a6a17d.png)
(3)
Then, v eventually satisfies:
![](/media/common/202112/blobid3-61c2e6b8e1124.png)
and:
![](/media/common/202112/blobid2-61c2f0e49f8bb.png)
Proof. Let . Then, we have that , and are positive for , for some . Therefore, it follows from (1) that:
Using (1) and Lemma 1 in [1], we see that:
![](/media/common/202112/blobid0-61c2f0873f0c7.png)
and so:
![](/media/common/202112/blobid1-61c2f0d62c4e9.png)
(4)
Integrating this inequality from to t and using the fact , we find:
![](/media/common/202112/blobid3-61c2f17acbc23.png)
(5)
Assume the contrary, that for . Thus, from (5), we have:
![](/media/common/202112/blobid4-61c2f1dbc609e.png)
This, from (3), implies:
![](/media/common/202112/blobid5-61c2f26feb7dc.png)
Letting and taking the fact that as , we obtain , which contradicts the positivity of .
Next, since v is positive decreasing, we have that . Assume the contrary, that . Then, for all , for some . Thus, from (3) and (5), we have:
![](/media/common/202112/blobid0-61c2f33397c16.png)
or
![](/media/common/202112/blobid1-61c2f34673ceb.png)
and so,
![](/media/common/202112/blobid2-61c2f361da024.png)
(6)
Using the fact that , we obtain that for all . Hence, by integrating (6) from to t, we obtain:
![](/media/common/202112/blobid3-61c2f3d141dd5.png)
Letting and taking the fact that as , we obtain , which contradicts the positivity of . Therefore, .
Since is decreasing, we obtain:
![](/media/common/202112/blobid4-61c2f47892f21.png)
and:
![](/media/common/202112/blobid5-61c2f4f06892b.png)
(7)
Then, .
From (7), we obtain:
![](/media/common/202112/blobid6-61c2f539bd262.png)
Thus, from (4) and the fact , we obtain:
![](/media/common/202112/blobid7-61c2f573434df.png)
and then:
![](/media/common/202112/blobid8-61c2f58d51913.png)
The proof is complete.
Lemma 2. Assume that and there exists a such that (3) holds. Then:
![](/media/common/202112/blobid9-61c2f5ee23320.png)
Proof. Let . From Lemma 1, we have that – hold for .
Integrating from to t, we arrive at:
![](/media/common/202112/blobid10-61c2f72e03edb.png)
From (3), we obtain:
![](/media/common/202112/blobid11-61c2f753d3d30.png)
and:
![](/media/common/202112/blobid12-61c2f780be5c4.png)
(8)
Using , we eventually have:
![](/media/common/202112/blobid13-61c2f7fdec355.png)
Hence, (8) becomes:
This implies that is a decreasing function.
The proof is complete.
2.2. Oscillation Theorems
In the next theorem, by using the principle of comparison with an equation of the first-order, we obtain a new criterion for the oscillation of (
1).
Theorem 6. Assume that
and there exists a
such that (
3) holds. If the delay differential equation:
(9)
is oscillatory, then every solution of (
1) is oscillatory.
Proof. Assume the contrary, that (
1) has a solution
. Then, we have that
and
are positive for
, for some
. From Lemmas 1 and 2, we have that
–
hold for
.
Next, we define:
![](/media/common/202112/blobid1-61c31d91d6708.png)
From
,
for
. Thus,
![](/media/common/202112/blobid2-61c31e2103408.png)
Thus, it follows from
that:
(10)
Using
, we obtain that:
![](/media/common/202112/blobid4-61c31ea04444a.png)
which with (
10) gives:
(11)
Now, we set:
![](/media/common/202112/blobid6-61c31f1b78108.png)
Then,
, and so, (
11) becomes:
which has a positive solution. In view of
[2] (Theorem 1), (
9) also has a positive solution, which is a contradiction.
The proof is complete.
Corollary 1. Assume that
and there exists a
such that (
3) holds. If:
(12)
then every solution of (
1) is oscillatory.
Proof. It follows from Theorem 2 in
[3] that the condition (
12) implies the oscillation of (
9).
Next, by introducing two Riccati substitution, we obtain a new oscillation criterion for (
1).
Theorem 7. Assume that
and there exists a
such that (
3) holds. If:
(13)
then every solution of (
1) is oscillatory.
Proof. Assume the contrary, that (
1) has a solution
. Then, we have that
and
are positive for
, for some
. From Lemmas 1 and 2, we have that
–
hold for
.
Now, we define the functions:
and:
Then,
and
are negative for
. From
, we obtain:
![](/media/common/202112/blobid10-61c323b685ce6.png)
Hence,
![](/media/common/202112/blobid11-61c323d8bb2c6.png)
and:
![](/media/common/202112/blobid12-61c324044f4a1.png)
Then:
(14)
and:
(15)
Combining (
14) and (
15), we obtain:
![](/media/common/202112/blobid15-61c3247fdaeae.png)
Integrating this inequality from
to
t, we have:
![](/media/common/202112/blobid16-61c3252f4e5d1.png)
From
we obtain
. Therefore,
where:
![](/media/common/202112/blobid17-61c32597482b4.png)
Since
and
, we find:
Taking
and using (
13), we arrive at a contradiction.
The proof is complete.
2.3. Applications and Discussion
Remark 1. It is easy to see that the previous works that dealt with the noncanonical case required either or . Since η is decreasing and , we have that . Then, the results of these works only apply when .
Example 1. Consider the DDE:
![](/media/common/202112/blobid0-61c32883b8c23.png)
(16)
where
and
. By choosing
the condition (
12) becomes:
(17)
Using Corollary 1, Equation (
16) is oscillatory if (
17) holds.
Remark 2. To apply Theorems 3 and 4 on (
16), we must stipulate that
. Let a special case of (
16), namely,
![](/media/common/202112/blobid2-61c3298aa86db.png)
A simple computation shows that (
16) is oscillatory if:
(18)
or:
(19)
or:
(20)
Consider the following most specific special case:
(21)
Note that (
18)–(
20) fail to apply. However, (
17) reduces to:
![](/media/common/202112/blobid7-61c32a53bc68e.png)
which ensures the oscillation of (
21).