1. Introduction
This paper deals with the oscillatory behavior of solutions to a third-order linear neutral delay differential equation of the form
where
. We make the following assumptions:
Hypothesis 1 (H1)
. , ;
Hypothesis 2 (H2)
. , q does not vanish eventually, and ;
Hypothesis 3 (H3)
. is strictly increasing, , and ; , , and ;
Hypothesis 4 (H4)
. .
For the sake of simplicity, we define the operators
and assume without further mention that
is of noncanonical type, that is,
Under a solution of Equation (
1), we mean a nontrivial function
with
, which has the property
,
, and satisfies (
1) on
. We restrict our attention to only those solutions of (
1) which exist on some half-line
and satisfy the condition
We tacitly assume that (
1) admits such a solution. A solution
y of (
1) is called oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. The equation itself is termed oscillatory if all its solutions oscillate. Following the seminal work of Kiguradze and Kondrat’ev (see, e.g., [
1]), we say that (
1) has
property A if any solution
x of (
1) is either oscillatory or satisfies
.
Being aware of numerous indications of the practical importance of third-order differential equations as well as a number of mathematical problems involved [
2], the area of the qualitative theory for such equations has attracted a large portion of research interest in the last three decades. The asymptotic properties of equations of type (
1) with
were extensively investigated in the literature, see, e.g., [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14] and the references cited therein. Most of the papers have been devoted to the examination of so-called canonical equations, where conditions opposite to (
2) hold, namely,
The advantage and usefulness of a noncanonical representation of linear disconjugate operators in the study of the oscillatory and asymptotic behavior of (
1) was recently shown in [
15]. In 2018, Džurina and Jadlovská [
16] considered a particular case of Equation (
1) in nonocanical form with
and established various oscillation criteria for Equation (
1). Their method simplifies the process and reduces the number of conditions required in previously known results. A further improvement of these results was presented in [
17].
Depending on various ranges of
p, a variety of results for property A of (
1), its generalizations or particular cases, exist in the literature, see, e.g., [
13,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27] and the references cited therein. Among them, only the work in [
13] considered (
1) in noncanonical form with
.
The question concerning the study of sufficient conditions which guarantee that all solutions of (
1) in the presence of a neutral term oscillate was open until recently. First such results were established in [
28] for (
1) in canonical form, under the crucial requirement of
and the commutativity of the delayed arguments
. To the best of the authors’ knowledge, there is nothing known regarding property A or the oscillation of all solutions of (
1) under the assumptions (H
)–(H
). Motivated by this observation, we attempt to fill this gap by extending the ideas exploited in [
16], based on comparisons with associated first and second-order delay differential inequalities, for more general neutral equations in noncanonical form.
Hopefully, our contribution should be of interest to the reader as, contrary to the majority of results in the literature, we attain the oscillation of all solutions of (
1), and find the conditions of all theorems very simple and easy to verify.
2. Main Results
For the reader’s convenience, we list the functions to be used in the paper. That is, for
, we put
where
is a constant satisfying
Here, we remark that the function
h from (H
) is nondecreasing with
, which follows from
and
respectively. As usual, all functional inequalities are assumed to hold eventually, that is, they are satisfied for all
t large enough.
We start with the following auxiliary result, which can be considered as a slight extension of [
16] (Lemma 1) given for (
1) with
.
Lemma 1. Suppose that (
H)
–(
H)
are satisfied and x is an eventually positive solution of Equation (1). Thenand the corresponding function y belongs to one of the following classes
eventually.
Proof. Choose
such that
and
on
. From the definition of
y, we see that
and
for
. Obviously,
is nonincreasing on
, since
Hence,
and
are of one sign eventually, which implies that four cases
–
are possible for
y. □
Next, we state an auxiliary criterion for the nonexistence of positive increasing solutions of (
1). As will be shown later, this condition is already included in those eliminating solutions from the class
. In the proof, we will take advantage of the useful fact
which follows from (
2).
Lemma 2. Suppose that (
H)
–(
H)
are satisfied. Ifthen . Proof. For the sake of contradiction, let (
7) be satisfied but
. Choose
such that
,
, and
on
.
Assume first that
. Since
is decreasing, we have
Thus,
Therefore,
is nonincreasing on
and moreover, this fact yields
for
. Consequently,
is nonincreasing on
as well, since
From
, we have
Using this in (
5), we find that
By virtue of (H
) and (
6), there is a
such that for any constant
and
,
which implies
Combining (
9) with (
1), we have
where we used the fact that
y and
h are increasing, and set
. Integrating (
10) from
to
t, we obtain
On the other hand, from (
2) and (
7), it follows that
which in view of (
11) contradicts the positivity of
.
Now, assume that
for
. Using the monotonicity of
, we find
Thus, one can see that
which implies that
in nonincreasing on
. Hence,
As in the first case, we use (
6) to arrive at (
10), which holds for any
and
for
sufficiently large. Integrating (
10) from
to
t, we have
Integrating the above inequality again from
to
t, we find that
Letting
t to ∞, changing the order of integration, and using (
7), we obtain
a contradiction. The proof is complete. □
Theorem 1. Suppose that (
H)
–(
H)
are satisfied. Ifthen (
1)
has property A. Proof. Assume that
x is a nonoscillatory solution of (
1). Without loss of generality, we may make it eventually positive, i.e., we suppose that
,
and
on
, where
is large enough. By conclusion of Lemma 1,
,
for
. First, it is easy to see that in view of (
2), condition (
12) implies
Thus, by Lemma 2,
and so either
or
. Using (H
) and the fact that
y is decreasing in (
5), we have
On the other hand, since
, there is a constant
such that
If
, there exists a
such that
for
. Hence, from (
13), we see that
Using this in (
1), we find
If we assume that
, then by integrating (
14) from
to
t, we obtain
Integrating the above inequality over
again, we have
Integrating (
15) over
, letting
t to ∞ and changing the order of integration in the resulting inequality, and taking (
12) into account, we obtain
a contradiction. Hence,
in this case.
If we take
, then by integrating (
14) from
to
t and using (
12), we arrive at
which contradicts the positivity of
and so
. Since
, we find
. The proof is complete. □
In the following auxiliary result, we present a criterion for nonexistence of
type solutions, based on comparison of the studied Equation (
1) with an associated first-order delay differential inequality. The given criterion also excludes solutions from classes
and
.
Lemma 3. Suppose that (
H)
–(
H)
are satisfied. Ifthen . Proof. For the sake of contradiction, let (
18) be satisfied but
. Choose
such that
,
, and
on
.
Assume first that
. As in the proof of Theorem 1, we arrive at (
13), which in view of (
1) yields
Define the function
From
and
we see that
is a strictly decreasing eventually positive function, say on
, for
. Using the definition of
w in (
19), we have
Hence,
w is a solution of the second-order delay differential inequality
Similarly as before, we define the function
u by
From
and
we conclude that
u is eventually positive and nonincreasing. Using the definition of
u in (
22), it is easy to see that
u satisfies the first-order delay differential inequality
However, by [
29] (Theorem 2.1.1), condition (
18) ensures that the above inequality does not possess a positive solution, which is a contradiction.
To show that also
, it suffices to note that (
12) is necessary for validity of (
18) since otherwise, the left-hand side of (
18) would equal zero. The conclusion then immediately follows from Theorem 1. The proof is complete. □
Using the Riccati transformation technique, we propose a result serving as an alternative of Lemma 3, which is applicable also in the case when .
Lemma 4. Suppose that (
H)
–(
H)
are satisfied and (7) holds. If, for any large enough,then . Proof. For the sake of contradiction, let (
18) be satisfied but
. Choose
such that
,
, and
on
.
Assume first that
. We proceed as in the proof of Lemma 3 to obtain (
22), where
w is given by (
20). Consider the function
defined by
Clearly,
on
. From (
23), it is easy to see that
Differentiating
and using (
22) together with (
26), we have
Multiplying both sides of (
28) by
and integrating the resulting inequality from
to
t, we have
However, in view of (
27), this inequality contradicts (
25). Hence,
. By Lemma 2,
due to (
7). The proof is complete. □
Corollary 1. Suppose that (
H)
–(
H)
are satisfied and (7) holds. If there is a constant such thatthen . To attain the oscillation of all solutions, it remains to eliminate the solutions of type.
Lemma 5. Suppose that (
H)
–(
H)
are satisfied. Ifthen . Proof. For the sake of contradiction, let (
30) be satisfied but
. Choose
such that
,
, and
on
. Using (
13) in (
1), we obtain
Using the monotonicity of
, one can easily verify that
for
. Integrating the latter inequality from
u to
, we obtain
Setting
and
,
in (
33), we find
On the other hand, integrating (
31) from
to
t and using (
34), we see that
Dividing the above inequality by
and taking the lim sup on both sides of the resulting inequality as
, we obtain a contradiction with (
30). The proof is complete. □
Here, a further improvement of Lemma 5 is made.
Lemma 6. Suppose that (
H)
–(
H)
are satisfied and let γ be a constant satisfying (4) eventually. Ifthen . Proof. Setting
and
in (
33), we obtain
By (
4), (
31), and (
36), we have
That is,
is eventually nonincreasing. From here, we obtain that
for
. Proceeding as in the proof of Lemma 5 with (
32) replaced by (
37), one arrives at a contradiction with (
35). The proof is complete. □
As a main result of the paper, we have the following oscillatory criterion for (
1), which is a simple consequence of Lemmas 3–6.
Theorem 2. Suppose that (
H)
–(
H)
are satisfied. If (18) (or (25)) and (30) (or (35)) hold, then (1) is oscillatory. Example 1. Consider the third-order neutral differential equationwhere , , , , , and the delay functions τ and σ satisfy (
H)
. Using thatit is easy to verify that condition (12) reduces toHence, ifall assumptions of Theorem 1 are satisfied and (38) has property A. Example 2. Consider the third-order neutral differential equationwhere , , , , and . Here, Since (12) is satisfied, by Theorem 1, (39) has property A. By some computations, conditions (18) and (25) reduce toandrespectively. Finally, condition (30) takes the formBy Theorem 2, we conclude that (39) is oscillatory if (40) (or (41)) and (42) hold.