Abstract
The main purpose of this study is to establish new improved conditions for testing the oscillation of solutions of second-order neutral differential equation where and . By optimizing the commonly used relationship , we obtain new criteria that give sharper results for oscillation than the previous related results. Moreover, we obtain criteria of an iterative nature. Our new results are illustrated by an example.
1. Introduction
In this study, we consider the neutral delay differential equation (NDDE) with second-order of the formwhere and . Throughout the results, we always suppose , is a nonnegative constant, , , , is not congruently zero for large enough, , , , and
By a solution of equation (1), we mean a for , which has the feature and satisfies (1) on . We only take into account those solutions that achieve the advantage , for all . If the solution of (1) is neither ultimately positive nor ultimately negative, then it is called an oscillatory solution; otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions oscillate.
In real-world life problems, the NDDEs have interesting applications. The NDDEs appear in the modeling of the networks containing lossless transmission lines, in the study of vibrating masses attached to an elastic bar, in the theory of automatic control, and others, see [1]. It is easy—in recent times—to observe the great development in the theory of oscillation for differential equations of different orders.
In the following, we review some of the works that contributed to the development of the oscillation theory of second-order NDDEs were the motivation for this work.
At studying the oscillatory behavior of NDDEs with canonical Case (2), the relationship between the solution and the corresponding functionhas been commonly used in the literature. For canonical case (2), by using the Riccati technique, Xu and Meng [2] presented some oscillation criteria for (1) when . In the case, where , Baculikova and Dzurina [3] established the oscillation criteria for (1).
Theorem 1 (see Corollary 2 in [3]). Let , ,Ifthen (1) is oscillatory, where .
Using relationship (3), Grace et al. [4] studied the oscillatory behavior of solutions of (1) when and . Moreover, they improved previous results in the literature.
Theorem 2 (see Theorems 3 and 6 in [4]). Iforthen (1) is oscillatory, where , , , and
Recently, Moaaz et al. [5, 6] generalized and complemented the results in [4]. They established the following criteria for oscillation of (1) with .
Theorem 3 (see Theorem 2 in [6]). Let . Ifthen (1) is oscillatory, whereand and are defined as in (8).
The similar results as those above have been extended for even-order NDDEs in [7–11]. For the works that dealt with the noncanonical case, that is,see, for example, [12–14].
The objective of this paper is to establish new oscillation criteria for the NDDE (1) by improving (3). The new relationship enables us to(i)Create more effective criteria for studying neutral equations in both cases and (ii)Essentially take into account the influence of the delay argument that has been careless in all related results(iii)Exclude some restrictions that are usually imposed on the coefficients of the neutral equations in the case where
Moreover, we use an iterative technique to establish new oscillation criteria for the NDDE (1) when and . One purpose of this paper is to further improve Theorems 2 and 1. The results reported in this paper generalize, complement, and improve those in [3–6]. To show the importance of our results, we provide an example.
2. Main Results I: New Relationship between and
For simplicity, we just write the functions without the independent variable, such as and . Moreover, assumingwhere and are positive constants, the set of all eventually positive solutions of (1) is denoted by .
Lemma 1 (see Lemma 3 in [3]). Let . Then,for , where is sufficiently large.
The following lemma is a direct observation from the Proof of 2.1 in [5].
Lemma 2. If , then , eventually.
Lemma 3. Let and , and there exists an even positive integer such thatThen,
Proof. Suppose that . Thus, , , and are positive for all , where is sufficiently large. From Lemma 1, we see that (13) holds. Since , we have thatfor all . Using the definition of , we obtainRepeating this procedure, we obtainfor , where is sufficiently large, and any even positive integer . Taking (16) and into account, we obtainfor . Combining (18) and (19), we obtainThis completes the proof.
Lemma 4. Let and . Then,for any odd positive integer , where
Proof. Proceeding as in the proof of Lemma 3, we arrive at (16). From the definition of , we have thatRepeating this procedure, we obtainfor , where is sufficiently large, and any odd . Since , we see thatfor , which with (24) givesFrom (16), we findwhich with (26) givesThis completes the proof.
Theorem 4. Assume that . If there exists a function such thatthen (1) is oscillatory, where
Proof. Assume the contrary that is a nonoscillatory solution of (1). Without loss of generality, we suppose that . Thus, , , and are positive for all , where is sufficiently large. Using Lemma 4, we have that (21) holds. Using (1) and (21), we obtainUsing the chain rule and simple computation, we findwhich with (31) givesIntegrating (33) from to , we obtainSince , we haveThus, (24) becomesthat is,Integrating from to , we findthat is,Next, we define the functionClearly, for all andIt follows from (31) and (39) thatfrom definition , we haveUsing the inequality (see Lemma 1.2 in [5]),with , and , we obtainIntegrating this inequality from to , we obtainwhich contradicts (29). This completes the proof.
Theorem 5. Assume that (14) holds for some even positive integer . If there exists a function such thatthen (1) is oscillatory, where
Proof. To prove this theorem, it suffices to use (15) instead of (21) in the proof of Theorem 4.
3. Main Results II: Iterative Technique
Lemma 5. Assume that , , and . Then,for , where and
Proof. Suppose that . Thus, , , and are positive for all , where is sufficiently large. From Lemma 1, we see that (13) holds. Now, we will prove (49) using induction.
For , proceeding as in proof of Theorem 4, we obtain that (31) and (37) hold. From (37), we obtainNext, we assume that (49) holds at , that is, . Thus, since , we findwhich with (31) givesIf we set , (53) becomesApplying the Grönwall inequality in (54), we obtainfor , soIntegrating this inequality from to , we obtainThis completes the proof.
Theorem 6. Assume that and . Iffor some integers , then (1) is oscillatory, where and are defined as in (22) and (50), respectively.
Proof. Assume the contrary that is a nonoscillatory solution of (1). Without loss of generality, we suppose that . Thus, , , and are positive for all , where is sufficiently large. From Lemma 5, we have that (49) holds. Proceeding as in the proof of Theorem 4, we arrive at (31). Combining (53) and (49), we obtainIf we set , we have that is a positive solution of the delay differential inequality:Using Theorem 1 in [15], the associated DDEhas also a positive solution. However, condition (58) ensures oscillation of (61), which is a contradiction. This completes the proof.
Theorem 7. Assume that and . If there exists a function such thatfor some integers , then (1) is oscillatory, whereand and are defined as in (22) and (50), respectively.
Proof. Assume the contrary that is a nonoscillatory solution of (1). Without loss of generality, we suppose that . Thus, , , and are positive for all , where is sufficiently large. Now, we define the function . Thus, andFrom Lemma 5, we have that (49) holds. By replacing (37) with (49) in the proof of Theorem 4, this part of proof is similar to that of Theorem 4, so we omit it.
Now, we give an example to illustrate our main results.
Example 1. Consider the NDDE:where and . It is easy to verify that , , , and , whereUsing Theorem 4, we see that (65) is oscillatory if andUsing Theorem 5, we see that (65) is oscillatory if
Remark 1. The best-known criteria for oscillation of NDDE (65) arefor and , respectively.
Giving values for the parameters , , and , we can determine the lower bound of the parameter to ensure that every solution of (65) is oscillatory. Table 1 shows the lower boundaries of the parameter in different special cases of (65) when by using conditions (69) and (67).
Let another particular case of (66), namely,Conditions (70) and (68) reduce to and , respectively.
So, our results improve the related results in [3, 4].
Remark 2. Using the boundedness condition , it will be easy to infer results similar to ours if is a function in .
4. Conclusion
This article is concerned with oscillatory behavior of a class of the neutral delay differential equation (NDDE) with second-order. By optimizing the commonly used relationship , we obtained new criteria that give sharper results for oscillation than the previous related results. Moreover, we obtained criteria of an iterative nature. Furthermore, in the future work, we can try to get some oscillation criteria of (1) under case of , .
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors contributed equally to this article.
Acknowledgments
Taif University Researchers Supporting Project number (TURSP-202020), Taif University, Taif, Saudi Arabia.