Abstract
This paper aims to study a class of neutral differential equations of higher-order in canonical form. By using the comparison technique, we obtain sufficient conditions to ensure that the studied differential equations are oscillatory. The criteria that we obtained are to improve and extend some of the results in previous literature. In addition, an example is given that shows the applicability of the results we obtained.
MSC:
34K11; 34C10
1. Introduction
Consider the even-order DE with the neutral term
where
and is an even positive integer. We also suppose the following:
- (M1)
- ℓ and i are quotients of odd positive integers;
- (M2)
- and under the canonical form, that is
- (M3)
- , , and
- (M4)
- ;
- (M5)
Under the solution of (1), we mean a non-trivial function , which has the properties ,, and x satisfies (1) on . Our attention is restricted to those solutions of (1) satisfying for all , and we assume that (1) possesses such solutions.
A solution x to (1) is referred to as oscillatory or non-oscillatory depending on whether it is essentially positive or negative. If all the solutions to an equation oscillate, the equation is said to be oscillator.
Differential equations have played a critical role in different sciences for a long time, and they are expected to continue being indispensable for future investigations. However, they often provide only an initial estimate of the systems being studied. To create more realistic models, the past states of these systems must be taken into account, necessitating the use of differential equations (DEs) with time delays.
In recent times, there has been a growing interest in the theory of oscillation in functional differential equations (FDEs) due to their numerous applications in various fields of science. As a result, we recommend that readers refer to [,,,,,,,,,,] to learn about the various contributions to the study of oscillatory and non-oscillatory behaviour of DEs with different orders.
It is known that the neutral differential equation (NDE) has many applications in various sciences, but as a general rule, we find that they have specific properties, thus studying them is difficult in both aspects of ideas and techniques. These difficulties explain the relatively small number of works devoted to the investigation of the oscillatory properties of solutions to this type of equation.
Several researchers have investigated the oscillatory behaviour of even-order DEs under various conditions. For more information, see [,,,,,,,,]. We mention in some detail:
Dzurina et al. [] investigated the oscillatory properties of the DE
where and is positive.
For DEs of the form
some oscillation criteria were established by Bazighifan et al. [], where and (3) hold.
The oscillatory behaviour of NDEs
was the focus of research by Agarwal et al. []. They introduced some new conditions that ensure that (6) is oscillatory. For the convenience of the reader, we mention one of the theorems.
Theorem 1.
Let be even, (M3) and (M4) hold, and
Assume that such that, for some
and
where
and
Then (6) is oscillatory.
Muhib et al. [] took into account the oscillatory behaviour of the NDE
where and are ratios of odd positive integers with , and , and there exists such that . For the convenience of the reader, we mention one of the theorems.
Theorem 2.
Assume that
holds. If there exists such that
for all , and for some , , then (7) is oscillatory, where
and
Based on the literature mentioned earlier, our objective is to establish criteria for oscillation in (1) by comparing it to first-order delay DEs with known oscillatory properties. In the final part of the paper, we use an example to show how our conditions improve some of the relevant findings that have been published in the literature.
2. Preliminary Lemmas
The following lemmas are needed in order to arrive at our result:
Lemma 1
([]). Let . Assume that the derivative is of fixed sign and not identically zero on a sub-ray of , and there exists a for all such that . If , then for every there exists such that
for all
Lemma 2
([]). Let the function be as in Lemma 1 for , and be assigned to by Lemma 1. Then there exists a such that
for every .
3. Main Results
The oscillation criteria for (1) will now be presented.
Theorem 3.
Let conditions (M1)–(M5) and hold. Assume that there exists such that
Proof.
Assume that (1) possesses an eventually positive solution , say for . From (2), we find
and so
where . Suppose (I) holds. Since , using Lemma 2, we have
Now,
Since and , is increasing, and therefore . Thus,
From (M5), there exists an , such that
Since and , inequality (23) becomes
Since and on , there exists a and a constant such that
From (24), (25) and , we find
has a positive solution . That is, (13) also possesses a solution that is positive, and thus we arrive at a contradiction.
Next, suppose (II) holds. Since , using Lemma 2, we have
from which we obtain
Since and , (32) takes the form
Theorem 4.
Proof.
Assume that (1) possesses an eventually positive solution , say for .
Suppose (I) holds. By applying the same processes used in the proof of Theorem 3, we obtain (17) and (24). By (17), there exists and a constant such that
and so
That is, (35) possesses a solution that is positive, and thus we arrive at a contradiction.
Next, suppose (II) holds. By applying the same processes used in the proof of Theorem 3, we obtain (28) and (33). By (28), there exists and a constant such that
and so
That is, (36) possesses a solution that is positive, and thus we arrive at a contradiction. Here, the proof ends. □
Theorem 5.
Proof.
Assume that (1) possesses an eventually positive solution , say for .
Suppose (I) holds. By applying the same processes used in the proof of Theorem 3, we obtain (26). Now, by Lemma 1, we have
and so
for some . From (44), (26), we obtain
If we set , then is a positive solution of
It follows from [] that (41) also possesses a solution that is positive, and thus we arrive at a contradiction with (41).
Next, suppose (II) holds. By applying the same processes used in the proof of Theorem 3, we obtain (27) and (34). Integrating (34) from to ∞ gives
and so
If we set , then is a positive solution of
for every . We complete the proof in the same way as in case (I). Here, the proof ends. □
Corollary 1.
Proof.
Theorem 6.
Proof.
Assume that (1) possesses an eventually positive solution , say for .
Suppose (I) holds. By applying the same processes used in the proof of Theorem 4, we obtain (38). Now, by Lemma 1, we see that (43) holds. Using (43) in (38) we have
for . If we set , then is a positive solution of
It follows from [] that (52) also possesses a solution that is positive, thus arriving at a contradiction with (52).
Corollary 2.
Proof.
The proof is similar to Corollary 1, and therefore the details are omitted. □
We use the example below to illustrate our results.
Example 1.
Let us consider the NDE
It is easy to verify that
Choosing , where and , then (12) holds. We also find that
By choosing , we find that (4) holds.
Now, we note that Condition (50) is satisfied, where
Moreover, by a simple computation, we have that
and
Thus, we note that Condition (51) is satisfied, where
Thus, using Corollary 1, we find that (60) is oscillatory.
Remark 1.
Remark 2.
Remark 3.
It is easy to see that Equation (60) in Example 1 oscillates at any value of . Furthermore, through this paper, we were able to extend previous results in the literature, which can be applied more widely compared to [,,].
4. Conclusions
The focus of this paper was to investigate the oscillatory behaviour of NDEs of even-order under the Condition (3). By using comparison principles with the first-order DEs, we offer some new sufficient conditions which ensure that any solution to (1) oscillates. Further, the results in [,] cannot apply to the example. In future studies, we aim to establish further criteria for the oscillation of Equation (1) when
Author Contributions
Conceptualization, A.K.A., A.M. and S.K.E.; Methodology, A.K.A., A.M. and S.K.E.; Validation, A.K.A., A.M. and S.K.E.; Formal analysis, A.M.; Investigation, A.M.; Writing—original draft, A.M.; Writing—review & editing, A.K.A., A.M. and S.K.E.; Visualization, S.K.E.; Supervision, A.M.; Project administration, A.K.A.; Funding acquisition, A.K.A. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia. [Grant No. 3792].
Data Availability Statement
No data sharing (where no datasets are produced).
Acknowledgments
The authors acknowledge the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia, under project Grant No. 3792.
Conflicts of Interest
The authors declare no conflict of interest.
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