Abstract
This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results.
1. Introduction
This study is concerned with developing new iterative criteria to test the oscillation of solutions of neutral delay differential equations NDDE of third order: where , is a corresponding function of , is a quotient of odd positive integers, , as , , is a positive real number, , , , and .
By a solution of (1), we mean a nontrivial real function for all , which has the properties , , and , and satisfies (1) on . We only consider those solutions of (1) which exist on some half-line and satisfy the condition for any .
If the solution is either ultimately positive or ultimately negative, then is called nonoscillatory; otherwise, it is called an oscillatory solution. The equation itself is termed oscillatory if all its solutions oscillate. Solutions whose corresponding function satisfies are called Kneser solutions. We denote the class of all Kneser solutions of (1) with the symbol . Otherwise, denote to the class of all positive solutions of (1) whose satisfies .
Delay differential equations as a subclass of functional differential equations take into account the dependence on the system’s past history, which results in predicting the future in a more reliable and efficient way. Neutral delay differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays, or in the theory of automatic control and in neuromechanical systems in which inertia plays a major role, see [1–6].
For interesting methods, techniques, and results which are concerned with the study of oscillation of third-order NDDEs, we refer the reader to the works [7–16]. Furthermore, the studies [17–20] are concerned with the study of odd-order DDEs.
At the beginning of any study of the oscillatory properties of solutions of differential equations, it is easy to notice the importance of classifying signs of derivatives of nonoscillatory solutions. For the positive solutions, based on the canonical condition as , it follows from ([21], Lemma 1) that and there are two possible cases for : either or . By creating conditions that ensure and are empty sets, we can directly set the criteria for oscillation.
There are numerous results interested in finding conditions that ensure class is empty, which include Hille and Nehari types and Philos type. Baculikova and Dzurina [7] established a condition of Hille and Nehari type and proved that if where and , then . By comparison principles, Baculikova and Dzurina [8] proved that if the first-order DDE is oscillatory, then . We can easily notice that the delay argument has been neglected in (2) and (3). Otherwise, by using the Riccati transformation, Thandapani and Li [16] guaranteed that class is empty if where , , and . All previous results focused on the class only and proved that every solution that belongs to tends to zero.
On the other hand, by establishing conditions for the nonexistence of Kneser solutions (), Dzurina et al. [12] attained the oscillation of all solutions of (1) in the linear case . They proved that if (4) and hold, then equation (1) is oscillatory, where satisfying .
One purpose of this study is to further complement and improve the well-known results reported in the literature. In Section 2, by using an iterative technique, we get analogous iterative estimates for Kneser solutions of (1). These iterative estimates enable us to establish new criteria that ensure the nonexistence of Kneser solutions. Further, criteria of an iterative nature help check the oscillation, even when the other criteria fail to apply. In Section 3, we use the Riccati transformation method and comparison principles to obtain different criteria which guarantee that . Examples illustrating the new results are also given.
For the sake of ease and assistance in presenting the main results, we provide the following notations and lemmas:
Lemma 1 (see [7], Lemma 1). All eventually positive solutions of (1) have the following properties:
and are positive, is of fixed sign, and is nonincreasing, for large enough.
Lemma 2 (see [16]). Let . Then, for all .
2. Main Results 1: Iterative Technique
Lemma 3. Assume that belongs to and there is a function with the property
If , then for , where , and
Proof. Suppose belongs to . Thus, there is a satisfying and and are positive for . As a direct result of Lemma 1, achieves property . Using induction, we will prove the iterative relationship (8).
At , since , we obtain
which in turn leads to
As a result of integrating (11) over , we get
Next, we will prove (8) at based on the assumption that it is correct at , that is,
First, we have from (1) that
By exploiting Lemma 2 with and , we obtain
which, with (1) and (14), gives
Bringing (13) with and , and combining it with (16), we get
Secondly, we define as
Since , we see that
Therefore,
From (17) and (20), it follows that
It is easy to note from (17) that and hence . Thus, (21) becomes
Using the Grönwall inequality, (22) becomes
From (19), we have
Integrating (24) twice over , we get
This completes the proof.
Theorem 4. Assume that there exists a function with the properties and . If then .
Proof. Suppose belongs to . As a direct result of Lemma 2, we get that (8) holds. By following the same approach as in proof of Lemma 2, we get the relationships from (14) to (21). Now, assume is defined as in (18). From Lemma 1, we have that , and hence, for . Then, the delay inequality (21) has a positive solution. From Theorem 1 in [22], the associated equation of (21) is has also a positive solution. However, it is well known from ([23], Theorem 2) that (34) implies oscillation of (27), a contradiction. This completes the proof.
Example 1. Assume the following NDDE of third order: where , and and are positive. First, we need to calculate , , and . Some careful calculations shows that , , for , where , ,
Next, if we set , then applying Theorem 4 requires that
Then, it is easy to verify that
Now, we set
In the following particular case, we note that , , and
Therefore, condition (26) is not satisfied when , but satisfies when . Thus, (34) has no Kneser solutions.
Remark 5. Very recently, Dzurina et al. ([12], Example 1) proved that (28) is oscillatory if
In the particular case (34), condition (36) reduces to (not fulfilled). However, by using our results, the oscillation condition is (fulfilled). Thus, equation (34) is oscillatory.
3. Main Results 2: An Improved Approach
Lemma 6. Assume that belongs to on , and . Then, for all .
Proof. Assume that belongs to on . Then, it follows from Lemma 1 that there is a , such that , , and are positive and is nonpositive, for all . It is easy to conclude that which, with the fact that , gives Now, we define From (39), we get Thus, we have that is an increasing function with , and so, for all . Therefore, from the definition of , we get the required directly.
Lemma 7. Assume that belongs to on , , and (constant). Then, for any odd positive integer and for all where large enough and
Proof. Assume that belongs to on . From the definition of , we obtain Resuming this procedure, we get, for any odd positive integers, for , where is sufficiently large. Since and for all , (45) becomes From Lemma 6, we get for all . By integrating this inequality from , we arrive at for all , which with (46) gives This completes the proof.
By replacing the commonly used relationship with the new relationship (42), we directly get the following results.
Theorem 8. Assume that and (constant). If then , where
Proof. The proof of this theorem is similar to that of ([7], Corollary 1), and hence, we omit it.
Theorem 9. Assume that (constant) and is defined as (50). If then .
Proof. Assume the contrary that belongs to on . From Lemma 7, we get that (42) holds. Combining (1) and (42), we find for . Now, we define Then, for all , and From Lemma 6, we get for all . By integrating this inequality from , we arrive at for . Since , we have From (52)–(56), we obtain Next, we define Then, , and so attains its maximum value on at and Combining (57) and (59), we find By integrating this inequality from , we arrive at which contradicts (51). This contradiction completes the proof.
Theorem 10. Assume that , (constant), and is defined as (50). If is oscillatory, then .
Proof. Assume the contrary that belongs to on . Proceeding as in the proof of Theorem 9, we obtain (52) and (56) hold for . Integrating (56) from and taking into account the fact that , we get which with (52) gives Thus, if we set , then we note that is a positive solution of the differential inequality Thus, it follows from ([22], Theorem 1) that the corresponding equation (62) also has a positive solution, which is a contradiction. This contradiction completes the proof.
Example 2. Consider the third-order NDDE, where and . Then, it is easy to verify that
To apply Theorem 8, condition (49) reduces to
On the other hand, using Theorem 10, the class of (66) is empty if the first-order DDE is oscillatory. Based on Theorem 2.1 in [24], the first-order DDE (69) is oscillatory if
Remark 11. In order to ensure that the class of (66) is empty, Table 1 compares between our criteria and the previous related one:
It is easy to notice that Theorem 10 supports the most efficient condition. Thus, our results improve the results in [7, 8].
4. Conclusion
Most studies investigating the asymptotic behavior of solutions of DDEs provided only sufficient criteria. This was a major motivation for the multiplicity of methods and techniques used in the study, so that a larger area of models could be covered. The aim of this study is to develop new criteria that have an iterative nature, so that we can apply them when the other criteria fail. First, we established a criterion to ensure that there are no Kneser solutions of (1) based on the comparison with a first-order DDE. Further, we obtained various criteria ensuring that is empty. By combining the results in Sections 2 and 3, we can obtain new conditions for the oscillation of all solutions of (1). It would be interesting to extend our results to the NDDE
It is also interesting to eliminate some restrictions that are usually imposed on the coefficients of the NDDEs studied.
Data Availability
No data were used to support this study.
Conflicts of Interest
There are no competing interests.
Acknowledgments
This study was supported by the Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.