1. Introduction
Differential equations have been a significant area of pure and applied mathematics since their establishment in the middle of the 17th century. Despite their extensive study in the past, it remains an important field for research with the arrival of new connections with other branches of mathematics, the fruitful interaction with applied fields, the interesting reformulation of fundamental issues and theories in various eras, the new perspectives in the twentieth century, and so on. Ordinary differential equations (ODEs) have several applications in mathematics and other fields, but when they are used to explain certain phenomena, including natural phenomena, we find that they contain delay times in their modeling, which leads to the so-called delay differential equations (DDEs). DDEs are a type of differential equation that takes into account time delays in the dynamics of the system. This indicates that the delay differential equation can directly represent any event that happened in the past, which gives it the ability to capture and analyze the behavior of systems where time delays play a critical role. Therefore, it is easy to see how these equations are utilized in physics, engineering, biology, and other sciences (see references [
1,
2]). A subtype of delay differential equations is known as neutral delay differential equations (NDDEs), where the highest-order derivative of the unknown function appears on the solution both with and without delay, and the development of NDDEs involves past values of the time and state variables. The delay differential equation solution requires information about the state at a certain time in the past in addition to the current state. There are numerous applications for neutral delay differential equations (NDDEs) in science and engineering. They are employed in the modeling of systems with delayed feedback, including control systems, neural networks, chemical reactions, and populations, as highlighted in references [
3,
4]. One of the fundamental goals of oscillation theory is to find sufficient conditions to ensure that all differential equation solutions oscillate. The first monograph that dealt with oscillation theory was that of Ladas et al. [
5], which covered the results until 1984. There has been a lot of research done in the last few years on the oscillation and the oscillatory properties of differential equations (see references [
6,
7,
8,
9,
10]). In recent years, there have been numerous studies on the oscillation and non-oscillation of solutions to various kinds of neutral functional differential equations (see references [
11,
12]). Numerous authors have examined the oscillations of fourth-order differential equations, and a number of methods for generating oscillatory criteria for these equations [
13,
14].
In this paper, we pay particular attention to the oscillatory behavior of solutions to the fourth-order neutral differential equation
where
,
is called the corresponding function of the solution
x. We will assume the following conditions:
- (H)
, ;
- (H)
, , and , .
Via a solution of (
1), we mean a function
for
, which has the property
, and satisfies (
1) on
. We only take into account the solutions
x of (
1) that satisfy Sup
for all
T ≥
oscillatory.
Definition 1. If solution x for (1) is ultimately positive or negative, it is said to be non-oscillatory; if not, it is said to be oscillatory. If all of an equation’s solutions oscillate, the equation itself is said to be oscillatory. One of the most important motivations for conducting this research is the importance of neutral differential equations, which have many uses in technology and natural science. They are often employed, for instance, in the study of distributed networks with lossless transmission lines (see reference [
15]), therefore, their qualitative characteristics are crucial. Besides the importance of fourth-order differential equations in mathematical representations of several physical, biological, and chemical phenomena, fourth-order differential equations are frequently encountered. Their applications include, for example, elasticity issues, structure distortion, or soil settlement (see reference [
16]). Complementary to the motives behind this paper is the fact that one of the conditions for oscillation is to find a condition in the form of a Kneser-type oscillation. The Kneser oscillation theorem states that a second-order linear differential equation of the form
is oscillatory if
where one finds that the above condition ensures the oscillatory behavior of all solutions while there is a positive solution in the case
Thus, such conditions are more accurate and effective for the oscillation test. Therefore, the aim of this study was to extend the results obtained in the second- and fourth-order delay equations to neutral. Moreover, there are a number of related results that inspired our study in particular; Chatzarakis et al. [
17] analyzed the oscillation behavior of the following fourth-order differential equations
under the canonical case
Li et al. [
11] studied the oscillatory behavior of the fourth-order nonlinear differential equation:
Bazighifan et al. [
18] study the oscillatory properties of solutions of the following equation:
for
, under the canonical case
In this work, we investigate the oscillatory behavior of solutions of fourth-order differential equations with neutral-delay arguments. We establish new monotonic properties for some cases of positive solutions of the studied equation and improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The solutions of any equation are classified as positive, negative, and oscillatory. Most of the techniques used in studying oscillation to find oscillation standards are based on excluding positive and negative solutions. In this paper, we are interested in finding conditions that exclude positive solutions only, and this is based on the fact that every negative value of a positive solution to the studied equation is also considered a solution, or what is called symmetry between positive and negative solutions. As usual, Euler-type differential equations are used to highlight the improvement over the previous results from the literature. We will organize our paper as follows. In
Section 2.1, we introduce the essential notations and the base of the method established in the sequel. In
Section 2.2, we introduce the we introduce a number of lemmas that iteratively enhance the monotonic properties of the positive solutions. In
Section 2.3, we present the main results and our main oscillations, that is, a single-oscillation criterion for (
1) based on a series of lemmas. In the end, we highlight the importance of our results by comparing them with previous results in the literature.
Lemma 1 ([
19])
. Let . If is eventually of one sign for all large s, say, , then there exists a and an integer l, 0 , with even for , or odd for such thatand implies that for, . 2. Main Results
In this section, we will establish some important lemmas that we will use in the proof to illustrate the main results of the research.
Notation 1. Firstly, we will display the important notation used in this paper. Our results are dependent on the necessity of positive stated byalso, let us definewhere are positive because of (H) and (H). We will use, in our proof, the statement that there is a sufficiently large , such thaton , where for arbitrary but fixedfor , and for . In the following lemma, we classify the signs of the derivatives of non-oscillatory solutions to study the oscillatory features of solutions.
Lemma 2. Assume that x is a positive solution of (1), then there are eventually only two possible cases for z Proof. Let
x be a positive solution of (
1), we get
from (
1). By using Lemma 1, we obtain case
, case
, and their derivatives. □
Notation 2. We will eventually refer to the class of positive solutions whose corresponding function to Case by and whose corresponding function to Case by
Lemma 3. Suppose that x is a positive solution of (1), then Proof. Suppose that
x is a positive solution of (
1), it follows that there exists
such that
,
and
for
. From the definition of
z, we obtain
with which with (
1), we obtain (
5). The proof is achieved. □
2.1. The Properties of the Solution in
We will proceed to the first lemma, which analyses and provides details regarding the behavior of the positive solutions
Lemma 4. Let and x is a positive solution of (1) belonging to the class . Then, eventually: - (A)
converges to 0 for ;
- (A)
is decreasing;
- (A)
is decreasing;
- (A)
is decreasing;
Proof. Suppose x to the class , then for there is , and for .
(A
): Since we have
, which is a non-increasing positive function, then
as
, if
then
, so
for
. From (
4) and (
5), we see that
From (
7) into (
8), we get
It is obvious that there exists
such that
for
, so we find from (
9)
for
. By integrating the above inequality from
to
s, we obtain
We find that there is a contradiction, therefore
. We see when
that
as
, and also
for
is increasing such that
as
. Then, according to L’Hôpital’s rule, we find that (A
) is satisfied.
(A
): Since
is non-increasing in
, we see that
where by (A
) there is
such that
for
.
So
for
, then
is decreasing, which proves (A
).
(A
): From (A
) and (A
),
decreases and tends to zero. Then, we find
for
. Hence
for
. We arrive at (A
).
(A
): Likewise, since
is decreasing and tends to zero, we obtain
for
, so
for
. That proves (A
).
As a result, the proof of the lemma is complete. □
In this lemma, we will establish some additional properties of the behavior of positive solutions in
Lemma 5. Assume x is a solution a positive of (1) belonging to the class and let . Then, for s large enough and every : - (A)
is decreasing;
- (A)
;
- (A)
, ;
- (A)
is decreasing;
- (A)
is decreasing.
Proof. Assume
x is a positive solution of (
1) to the class
, then for
there is
,
and
for
, and (A
)–(A
) of Lemma 4 holds for
.
(A
): Define the positive function
by differentiating
and employing (
8), in addition to having
decrease in (A
), we get
From (A
) and (A
) respectively, in the above inequality, we see
By integrating the above inequality from
to
s, and using (A
) and (A
), we obtain
for
, which is
it follows from (
11) that
for
, and hence
We observe that
is decreasing, thus (A
) holds.
(A): Since is increasing, and from (A) we find that , (A) thus holds.
(A
): For
, to display that
it suffices to prove that there is
such that, for large
s,
which if
, then
We notice that there is a contradiction. We find that by using (
14) for any
, then there is
large enough to obtain
for
, by employing this in (
13), we have
Integrating the above inequality from
to
s and from (A
) this forms
for
, from (
11) in the above inequality; for
we arrive at
so, from this, we can see that (
16) is satisfied with
For
the other limits in (A
) are obtained from (
15) and from using L’Hôpital’s rule.
(A
): From (
16) into (
17), we see
for
, and for
we arrive at
for
, when it is obvious that
is decreasing, then (A
) holds.
(A
): We notice from (A
) and (A
) that
is decreasing and tends to zero, then
for
, and for
we obtain
for
. It follows that (A
) holds, thus Lemma 5 is proved. □
The next lemma is a result of (A).
Lemma 6. Assume that and then .
Proof. Suppose the opposite is true and assume
x is a positive solution of (
1) to the class
; then, for
there is
,
and
for
. From (
4) in (
12), and taking into consideration (A
), we get
which is
Furthermore, from (A
) and (A
), respectively, we obtain
By integrating the above inequality from
to
s and from (A
) and (A
) in Lemma 5, with the definition of
in (
11) for
, we arrive at
for
, where
can be arbitrarily large, we chose it in such a way that
This indicates that
, where
and
are positive, this causes a contradiction and completes the proof of Lemma 6. □
From (A) and Lemma 6 we can assume that so
The following lemma can be considered as an iterative version of Lemma 5.
Let us define a sequence
(that is needed in the next lemma) as follows:
for
, where
denotes
or
.
By induction, it is simple to demonstrate that, if
for
, then
holds, such that
where the definitions of
is as follows:
We need to specify the sequence
as follows:
The value of
is arbitrary and determined by the value of
, where
is defined in (
4). It is simple to show that
Lemma 7. Assume x is a positive solution of (1) belonging to the class and let . Then, for s large enough and - (A)
is decreasing;
- (A)
;
- (A)
, ;
- (A)
is decreasing;
- (A)
is decreasing,
where
Proof. Assume
x is a positive solution of (
1) belonging to the class
; then, for
there is
,
and
for
. This Lemma will be proved by induction on
m. For
, it holds from Lemma 5 with
. After that, assume that (A
)
(A
)
hold for
and
. We will display that they all hold
.
(A
)
; by using (
4) and (A
) in (
12) we see
from (A
)
and (A
)
we obtain
By integrating the above inequality from
to
s and from (A
)
and (A
)
, we find that there exists
such that
which is
from the definition of
, it follows that
and we obtain that
That is the prove of (A
)
.
(A): We have increasing and, from (A), we arrive at proving (A).
(A
)
: To prove this case, it suffices to prove that there is
, as done in the case
, such that for
From (
23), we find that there is
sufficiently large such that
which is
for any
. If we merge the above inequality with (
21) we obtain
by integration from
to
s and from (A
)
we receive
for
. Since
, we can choose
such that
, from (A
) and the definition of
we see that (
16) is satisfied; the other limits are the same as those for
.
(A
)
: By using (A
)
in (
25), we obtain
Then, (A
)
is satisfied.
(A
)
: From (A
) and (A
), we obtain
which indicates (A
)
holds and submits the lemma’s proof. □
The following lemma can be easily deduced from the aforementioned arguments.
Lemma 8. Suppose that andwhereThen = ⌀. Proof. Suppose the opposite is true, that
We claim that
From the case (A
)
we have
. Since
can be picked arbitrarily, set
where
is given by (
19). Then
which supports the claim. From (
28) we conclude that the sequence
is increasing and bounded from above, which is defined by (
18), which means that
where
is a root of the following equation:
However, from (
26) we find that (
29) has no positive solutions. As a result,
, which is the end of the proof and the Lemma. □
2.2. The Properties of the Solution in
In this section, we will show results similar to the previous results of the section of solutions in the class
Lemma 9. Assume x is a positive solution of (1) relating to the class and let . Then, for s large enough: - (A)
converges to 0 for ;
- (A)
is decreasing.
Proof. Suppose
x is a positive solution of (
1) belonging to the class
, then, for
, there is
,
and
for
.
(A
): As a result of the fact that
is decreasing positive function,
as
if
then
, so
for
. By (
4) and (
5) we obtain
From (
30) in the above inequality we obtain
we have obtained this by integrating twice from
s to
∞
Integrating from
to
s we arrive at
there is, as we found, a contradiction. Therefore
. We can show that (A
) is satisfied by applying L’Hôpital’s rule.
(A
): we have
non-increasing and by (A
) we see that
for
, where
is sufficiently large such that
for
. Then
for
, the completion of the proof follows. □
Lemma 10. Assume x is a positive solution of (1) belonging to the class and let . Then, for s large enough and any : - (A)
is decreasing;
- (A)
;
- (A)
;
- (A)
is non-decreasing.
Proof. Suppose
x is a positive solution of (
1) belonging to the class
then for
there is
,
and
for
.
(A
): From (A
) we have
, which is decreasing; in (
31) we obtain
by integrating from
s to
∞ twice such that
is increasing, it goes as follows:
Let us define a positive function
by differentiating and from (
32) we find
Integrating, again,
to
s, and we have
decreasing and tending to zero, and we see
for
, where
is sufficiently large such that
for
. From the definition of
we obtain
and
, then (A
) holds.
(A): This simply implies from (A), and from the case that x is increasing.
(A
): The proof is identical to the proof for class
, and it is sufficient to show that
For
, we can derive from (
35) into (
34), finding that there exists
, such that
for any
, from that we deduce that
It is now obvious that (
36) holds with
.
(A
): By integrating (
32) from
s to
∞ we see
and so
which is proof that
is increasing. The lemma’s proof is now accomplished. □
Lemma 11. Assume that and then .
Proof. Suppose
x is a positive solution of (
1) belonging to the class
, then for
there is
,
and
for
. From (
4) into (
31), and taking into account (A
), we arrive at
By twice integrating from
s to
∞, with the assumption that
x is increasing, we obtain
From the above inequality into (
33)
Using integration as in (
34), and replacing
by
, we obtain
where we can choose
in such a way that it can be arbitrarily large, so that
, this shows
—a contradiction. This illustrates the lemma. □
Now, from Lemma 10, we obtain an iterative.
Lemma 12. Assume x is a positive solution of (1) belonging to the class and let . Then, for s large enough and any : - (A)
is decreasing;
- (A)
;
- (A)
;
- (A)
is non-decreasing;
where .
Proof. Assume
x is a positive solution of (
1) belonging to the class
then for
there is
,
and
for
by induction on
m. For
, it holds from Lemma 5 that
. After that, assume that (A
)
(A
)
hold for
and
. We will show that (A
)
holds.
(A
)
: From (
31) and (A
)
we find
Integrate the above inequality from
s to
∞, and taking into account that
is increasing, yield
Repeating this process, we obtain
By incorporating this into (
33), we have
By integrating from
to
s and from (A
)
and (A
)
, we find
That is the proof of (A
)
. The other parts’ proofs of the lemma are the same as those in the case where
. □
Lemma 13. Suppose that andwherefor then = ⌀. Proof. The proof is similar to the proof of Lemma 8 and defines
as in (
18). □
Now, in the next theorem, we offer the fundamental result in this work by combining the results from the previous two sections.
2.3. Oscillation Results
Theorem 1. Suppose thatwherewhere then (1) is oscillatory. Proof. From (
41) observe that
, and since
we find that
. Now, if
∞, then Lemmas 6 and 11 imply that
. For
, from Lemmas 8 and 13, the same conclusion is derived. This illustrates the theorem. □
Corollary 1. Let with . Ifthen (1) is oscillatory. 2.4. Application and Discussion
In the next section, we provide an example to highlight our study results.
Example 1. Now, consider the fourth-order Euler delay differential equationfor , where and , By applying condition (41), we obtainwhich is,where Thus, by applying Theorem 1, we can guarantee that all solutions of Equation (42) are oscillatory if condition (43) is satisfied. Remark 1. If we consider the special case and , the condition (43) reduces toBy checking the result of the oscillation constants for Equation (42) in references [20] and [21], respectively, with and we seeand Example 2. Consider the NDE as the following:where , and , To check the oscillation of Equation (47), we will apply condition (41) of Theorem 1 in the previous section and see thatwhich isApplying results in both [22] and [23] to Equation (47), we get, respectively,andWe notice that the condition (48) improves the condition (49) and (50). It also improves results (45) and (46). Remark 2. It can be easily observed that condition (44) and (48) improve conditions (45), (46), (49), and (50). In addition to this improvement, there is something that distinguishes our results from other results in [20] and [21] is that their results require constraints , but Theorem 1 does not need them. This leads to the conclusion that Theorem 1 improves many previous results in the literature, even without the usual restrictive suppositions on the diverging argument. Remark 3. We make a simulation experiment for Example 1, by considering the ODE of (42)where and By using Theorem 1, Equation (51) is oscillatory ifwhich is