Some New Oscillation Criteria of Even-Order Quasi-Linear Delay Differential Equations with Neutral Term

: The neutral delay differential equations have many applications in the natural sciences, technology, and population dynamics. In this paper, we establish several new oscillation criteria for a kind of even-order quasi-linear neutral delay differential equations. Comparing our results with those in the literature, our criteria solve more general delay differential equations with neutral type, and our results expand the range of neutral term coefﬁcient. Some examples are given to illustrate our conclusions.


Introduction
Up to now, many academics have made essential contributions to the delay differential equations, because such equations have various applications in natural science and social science [1][2][3][4][5]. For example, when studying the London-Yorke model of measles transmission [6], some scholars considered the following delay equatioṅ where σ > 0.
In the study of lossless transmission lines in electrical networks, the following equation [7].
is derived. Since the delay differential equation is obviously different from the ordinary differential equation, the method of studying ordinary differential equation cannot be used to that. Thus, researchers have focused attention on the qualitative theory of delay differential equations, where the oscillation and asymptotic behavior of the equation is an important branch.
A proper solution of (3) is oscillatory if it has infinitely many zeros on [t y , ∞). That is, for any t 1 ≥ t y , there exists t 2 ≥ t 1 , s.t. y(t 2 ) = 0. Otherwise, it is called non-oscillatory. Equation (3) is oscillatory if all its proper solutions are oscillatory.
Based on the above results of previous scholars, in this article, we are concerned with the following quasi-linear neutral delay differential equations of the form (i.e., Equation (3) where n is even. The study of quasi-linear differential equations has numerous applications, such as in the study of p-Laplace equations, porous medium problems, chemotaxis models, and so forth; see, e.g., the papers [25][26][27] for more details. We establish some oscillation criteria of (8) by using the Riccati transformation technique and comparison method. Compared with the second-order results of [4,[8][9][10][11][12][13][14][15][16], we extend Equations (4) and (5) to Equation (8), where n is even. For the results of [17][18][19][20][21][22][23][24][28][29][30][31][32][33][34], we get the oscillation criteria of the more general equations. In other word, a(t) may not be 1, η ∈ S and b(t) is not only bounded, but also can be unbounded. Therefore, we complement and extend upon some results reported in literature. At the end of this paper, some examples are provided to exhibit our conclusions.

Auxiliary Lemmas
Throughout this paper, we will analyze the following situations of a(t), b(t), g(t), and h(t): ; Before starting our major criteria, the following lemmas are needed.
). If f (n) (t) is eventually of one sign for all large t, then there exist a t x ≥ t 0 and an integer l, 0 ≤ l ≤ n with n + l even for f (n) (t) ≥ 0, or n + l odd where a and b are positive constants, β ∈ S. Then, g attains its maximum value on + at u * = ( βa (β+1)b ) β and
If 0 < η < 1, the argument is analogous to that in the above discussion, so is omitted. This completes the proof.

Corollary 1. Let the hypotheses of Theorem 1 hold. If the following inequality
has no eventually positive solution, then (8) is oscillatory.
Proof. Similar to the proof of Theorem 1, we have (11), (15) and lim t→∞ χ(t) = 0. Thus, by Lemma 2, there exists t 2 ≥ t 1 , such that That achieves It is straightforward to know that Φ(t) is positive and satisfies (19). The proof is complete. 20) has no eventually positive solution, then (8) is oscillatory.

Proof. Suppose that
First, we consider case (I). Then, lim t→∞ χ (t) = 0. From that and Lemma 2, we achieve By h(t) ≤ t and the fact that χ (n−1) (t) is not increasing, we obtain Owing to χ > 0 and the definition of χ, we have Let Thus, µ(t) > 0 on [t 1 , ∞) and set Then By Lemma 3, we get
It is easy to verify that ν(t) > 0 on [t 1 , ∞). Since χ is decreasing and h(t) ≤ t, according to Lemma 3, we get This implies that for any r large enough. This contradicts our assumption (22), which completes the proof.
Proof. Just as the proof of Theorem 2, by the above assumptions, χ satisfies either case (I) or case (I I).

Examples
For t ≥ 1, the following examples are given to verify our criteria.
Author Contributions: Writing-original draft, R.G., Q.H. and Q.L. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.