Simplified and improved criteria for oscillation of delay differential equations of fourth order

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

If v is either positive or negative, eventually, then v is called nonoscillatory; otherwise it is called oscillatory. Equation (1.1) itself is termed oscillatory if all its solutions are oscillatory.
Zhang et al. [25] considered the higher-order DDE where κ, γ are a ration of odd integers and 0 < γ ≤ κ. Moreover, Zhang et al. [23] studied the oscillation of solutions for (1.3) and improved the results [25]. For the convenience of the reader, we present some of their results below at κ = γ and n = 4.
The objective of this paper is to improve and simplify the oscillation criteria of the fourth-order DDE (1.1) in the noncanonical case. In the noncanonical case, it is usual to have oscillation criteria in the form of at least three independent conditions; however, in Sect. 2, we obtain only two independent conditions that guarantee the oscillation of all solutions. In Sect. 3, we take an approach that creates improved criteria for oscillation.

Simplified criteria for oscillation
) is a solution of (1.1). Then (a(l)(v (l)) κ ) ≤ 0, and one of the following cases holds, eventually: From (1.1) and Lemma 1.1, there exist three possible cases (a), (b), and (c) for l ≥ l 1 , l 1 large enough. The proof is complete.
Let us define Proof Assume on the contrary that v ∈ C(I 0 , (0, ∞)) is a solution (1.1) and satisfies either case (a) or case (c). First, we suppose that (c) holds on which is If we divide (2.3) by a 1/κ and then integrate from lto , we find Letting → ∞, we get Integrating (2.7) from l 1 to l, we obtain At l → ∞, we arrive at a contradiction with (2.1). Finally, let case (a) hold on I 1 . On the other hand, it follows from (2.1) and (1.2) that l l 1 h(s)ψ κ 2 (s) ds must be unbounded. Further, since ψ 2 (s) < 0, it is easy to see that Integrating (1.1) from l 2 to l, we get From (2.9) and (2.10), we get a contradiction with the positivity of a(l)(v (l)) κ . This completes the proof.
Proof Assume on the contrary that v ∈ C(I 0 , (0, ∞)) is a solution (1.1) and satisfies case (a) or case (c). First, we suppose that (c) holds on I 1 . Then (2.14) From (2.14) and (2.16), we find Dividing both sides of inequality (2.17) by a(l)(v (l)) κ and taking the limsup, we arrive at we arrive at a contradiction with (2.11). Next, we suppose that case (a) holds on I 1 . From (2.11) and the fact that ψ 2 (l) < ∞, we get that (2.9) holds. Then, this part of the proof is similar to that of Theorem 2.1. This completes the proof.
( Remark 2.4 Note that, we used two conditions only for testing the oscillation of the fourthorder DDEs. Moreover, our results can also be applied to ordinary DEs when g(l) = l.

Improved criteria for oscillation
is oscillatory, then the solution v does not satisfy case (c).
Proof Suppose the contrary that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.12) and (2.15) hold. From (2.12), we have Thus, we get that Using (3.2), we obtain that Now, integrating (2.15) from l to ∞ and using (3.3), we get Thus, it is easy to see that v is a positive solution of the first-order delay differential inequality v (l) + 1 Using [22], we have that (3.1) has also a positive solution, a contradiction. This completes the proof. Proof Using [22], we note that condition (3.5) ensures the oscillation of (3.1). This completes the proof.
Proof Suppose that v satisfies case (c). Then we obtain that lim l→∞ v(l) = c ≥ 0. We claim that lim l→∞ v(l) = 0. Suppose the contrary that c > 0. Thus, there exists l 1 ≥ l 0 such that v(g(l)) ≥ c for l ≥ l 1 , and hence for l ≥ l 1 . Integrating (3.7) twice from l 1 to l, we obtain Letting l → ∞ and using (3.6), we obtain that lim l→∞ v (l) = -∞, which contradicts v (l) > 0. Thus, the proof is complete. Lemma 3.2 Assume that (3.6) holds, v ∈ C(I 0 , (0, ∞)) is a solution of (1.1), and case (c) holds. If there exists a constant μ ≥ 0 such that Proof Suppose that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.13) holds. Integrating (1.1) from l 1 to l and using v (l) < 0, we find (3.10) Using Lemma 3.1, we get that lim l→∞ v(l) = 0. Thus, there is l 2 ≥ l 1 such that h(s) ds < 0 for every l ≥ l 2 , which, with (3.10), gives Next, we have that .
is oscillatory, then the solution v does not satisfy case (c).
Proof Assume on the contrary that (1.1) has a positive solution v which satisfies case (c). Using Theorem 2.2 and Lemma 3.2, we get that (2.13) and (3.9) hold, respectively. Integrating (3.9) from g(l) to l, we obtain v g(l) ≥ ψ 2 (g(l)) which with (1.1) gives (3.14) Integrating (2.13) from l to ∞ provides v(l) ≥ -a 1/κ (l)v (l)ψ 2 (l). (3.15) Next, we define From (3.14) and (3.16), we conclude that which, in view of (2.13), gives In view of [6], differential equation ( hold, then the solution v does not satisfy case (c). Proof Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists l 1 ∈ [l 0 , ∞) such that v(l) > 0 and v(g(l)) > 0 for l ≥ l 1 . Using Lemma 2.1, there exist three possible cases (a)-(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.1, we get a contradiction with condition (3.5). This completes the proof. Proof Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists l 1 ∈ [l 0 , ∞) such that v(l) > 0 and v(g(l)) > 0 for l ≥ l 1 . Using Lemma 2.1, there exist three possible cases (a)-(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.2, we get a contradiction with condition (3.19). This completes the proof. It is easy to verify that ψ 2 (l 0 ) < ∞,  Thus, we note that Theorem 3.4 provides a better criterion for the oscillation of (3.20). Moreover, our oscillation criteria take into account the influence of g(l), which has not been taken care of in the related results [18,25].

Conclusion
In this work, we simplified and improved the oscillation criteria for a class of even-order delay differential equations. In the noncanonical case, it always sets three conditions to check the oscillation of even-order DDEs. First, we obtained a criterion with only two conditions to check the oscillation. Furthermore, we improved the three-condition oscillation criteria by creating a better estimate of the ratio v(g(l))/v(l). Through the example, we compared our results with the previous results and explained the importance of our new oscillation criteria. It will be interesting to extend our results of this study to the neutral and mixed case.