A New Approach in the Study of Oscillation Criteria of Even-Order Neutral Differential Equations

: Based on the comparison with ﬁrst-order delay equations, we establish a new oscillation criterion for a class of even-order neutral differential equations. Our new criterion improves a number of existing ones. An illustrative example is provided.


Introduction
In the last decade, many studies have been carried out on the oscillatory behavior of various types of functional differential equations, see  and the references cited therein. As a result of numerous applications in technology and natural science, the issue of oscillation of nonlinear neutral delay differential equation has caught the attention of many researchers, see [1,[3][4][5]8,12,17,19,[22][23][24]. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines, see [11].
In this paper, we are concerned with improving the oscillation criteria for the even-order neutral differential equation of the form where t ≥ t 0 , n ≥ 4 is an even natural number and z (t) := x (t) + p (t) x (τ (t)). In this work, we assume that α is a quotient of odd positive integers, By a solution of (1) we mean a function x ∈ C 3 [t y , ∞), t y ≥ t 0 , which has the property , and satisfies (1) on [t y , ∞). We consider only those solutions x of (1) which satisfy sup{|x (t)| : t ≥ T} > 0, for all T ≥ t y . A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. A neutral delay differential equation is a differential equation in which the highest-order derivative of the unknown function appears both with and without delay.
In the following, we briefly review some important oscillation criteria obtained for higher-order neutral equations which can be seen as a motivation for this paper.
In 1998, based on establishing comparison theorems that compare the nth-order equation with only one first-order delay differential equations, Zafer [23] proved that the even-order differential equation In a similar approach, Zhang and Yan [24] proved that (2) is oscillatory if either It's easy to note that (n − 1)! < (n − 1) 2 (n−1)(n−2) for n > 3, and hence results in [24] improved results of Zafer in [23]. For nonlinear equation, Xing et al. [22] proved that (1) is oscillatory if where q (t) := min q σ −1 (t) , q σ −1 (τ (t)) . If we apply the previous results to the equation then we get that (6) is oscillatory if Hence, Xing et al. [22] improved the results in [23,24]. By establishing a new comparison theorem that compare the higher-order Equation (1) with a couple of first-order delay differential equations, we improve the results in [22][23][24]. An example is presented to illustrate our main results.
In order to discuss our main results, we need the following lemmas: Assume that x (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists a t 1 ≥ t 0 such that x (n−1) (t) x (n) (t) ≤ 0 for all t ≥ t 1 .

Main Results
Here, we define the next notation:

Lemma 4 ([20] Lemma 1.2).
Assume that x is an eventually positive solution of (1). Then, there exist two possible cases: for t ≥ t 1 , where t 1 ≥ t 0 is sufficiently large.
Proof. Let x be a non-oscillatory solution of (1) on [t 0 , ∞). Without loss of generality, we can assume that x is eventually positive. It follows from Lemma 4 that there exist two possible cases (I 1 ) and (I 2 ).
Assume that Case (I 1 ) holds. From the definition of z (t), we see that By repeating the same process, we find that Using Lemma 1, we get z (t) ≥ 1 (n−1) tz (t) and hence the function t 1−n z (t) is nonincreasing, which with the fact that τ (t) ≤ t gives Combining Equations (11) and (12), we conclude that From Equations (1) and (13), we obtain Since η (t) ≤ σ (t) and z (t) > 0, we get Now, by using Lemma 2, we have for some µ ∈ (0, 1). It follows from (14) and (15) that, for all µ ∈ (0, 1) , Thus, if we set ψ (t) = r (t) z (n−1) (t) α , then we see that ψ is a positive solution of the first-order delay differential inequality It is well known (see [21] (Theorem 1)) that the corresponding Equation (9) also has a positive solution, which is a contradiction.