An Improved Oscillation Result for a Class of Higher Order Non-canonical Delay Differential Equations

In this work, by obtaining a new condition that excludes a class of positive solutions of a type of higher order delay differential equations, we were able to construct an oscillation criterion that simplifies, improves and complements the previous results in the literature. The adopted approach extends those commonly used in the study of second-order equations. The simplification lies in obtaining an oscillation criterion with two conditions, unlike the previous results, which required at least three conditions. In addition, we illustrate the improvement with the new criterion, applying it to some examples and comparing the results obtained with previous results in the literature.

The many applications of DDEs in different sciences were and continue to be the motivation behind the growing interest in studying the qualitative behavior of the solutions of these equations.
In the non-canonical case, it is easy to see how much research has progressed on the oscillatory behavior of the solutions of second-order DDEs. This progress can be traced through the recent results of Baculí ková [2,3] and Džurina and Jadlovská [4,5]. They provided improved techniques and sharper criteria for the oscillation of second-order DDE solutions.
In the study of oscillatory behavior, there are two common techniques: Riccati substitution and comparison with first-order equations. In the noncanonical case, Baculíková et al. [6] used the comparison technique to establish the oscillation conditions for the solutions of the DDE On the other hand, Zhang et al. [7] used the Riccati substitution to establish criteria for deciding that all the solutions of the DDE are oscillatory, where α and β are ratios of odd positive integers. Below, we present two results obtained from the literature, to which we will refer to later.
is oscillatory for some 1 , 2 ∈ (0, 1), where We note here that the linear delay equation has been studied by Koplatadze et al. [8]. They took into account the oddand even-order cases of this equation. Very recently, Moaaz et al. [9] extended the results about the secondorder equations to even-order equations in the non-canonical case. They adopted a strategy that involved new monotonic properties for positive decreasing solutions and used those properties to iteratively develop new oscillation criteria.
This study aims to establish a new criterion to determine the oscillation of all solutions of Eq. (1.1) in the non-canonical case. The approach followed is an extension of the approach used by Koplatadze et al. [8] and later by Baculíková [2] to obtain an effective oscillation criterion for second-order equations. The new criterion ensures that Eq. (1.1) is oscillatory without the need to check the additional condition (1.4), which has traditionally been imposed on all previous related results. In addition, the new criterion also introduces a measure of oscillation that is sharper than previous results in the literature.

Main Results
Using Lemma 2.2.1 in [10], we can classify the positive solutions of (1.1) as follows: Lemma 3] Suppose that υ is an eventually positive solution of (1.1). Then, a · υ (n−1) (t) ≤ 0, and there are eventually the following three cases:

Lemma 2.2.
Suppose that υ is an eventually positive solution of (1.1) and Proof. Since υ is an eventually positive solution of (1.1) and satisfies (C 3 ), there is a t 1 ≥ t 0 such that υ (t) > 0 and υ (g (t)) > 0 for all t ≥ t 1 . We also have Furthermore, according to Lemma 2.1 it is a · υ (n−1) (t) ≤ 0, and thus we have Integrating this inequality over [t, ∞), we obtain Integrating this inequality n − 3 times over [t, ∞), and taking into account the behavior of the derivatives of υ in (C 3 ), we conclude that and From ( Integrating this inequality over [t, ∞), we arrive at Integrating (1.1) from t 1 to t, we find that From (2.5) and (2.6), we obtain Thus, we have On the other hand, from (2.2), we get This implies Repeating the same procedure n − 3 times, we obtain that υ An−2 (t) ≥ 0.
Proof. We proceed by contradiction. Let us assume that υ is an eventually positive solution of (1.1). Then, there is a t 1 ≥ t 0 , such that υ (t) > 0 and υ (g (t)) > 0 for all t ≥ t 1 . It follows from Lemma 2.1 that υ satisfies one of the cases (C 1 ) − (C 3 ). Assume that case (C 1 ) holds. Proceeding similarly as in the proof of Theorem 1 in [2], we can prove that (2.10) implies that (2.12) Integrating (1.1) from t 1 to ∞ and using the fact that υ (n−1) is a decreasing positive function, we find that Since υ (t) > 0 and υ (t) > 0, there is a t 2 ≥ t 1 , such that υ (g (t)) > l for t ≥ t 1 . Then, from (H3), we arrive at which contradicts (2.12).
It follows from Corollary 1 in [12] that the DDE (2.11) also has a positive solution, which is a contradiction. Assume that case (C 3 ) holds. From Lemma 2.2, we have that (2.1) holds, which contradicts the hypothesis (2.10). The proof is complete.