Improved criteria for oscillation of noncanonical neutral differential equations of even order

In this work, we aim at studying the asymptotic and oscillatory behavior of even-order neutral delay noncanonical differential equations. To the best of our knowledge, most of the related previous works are concerned only with neutral equations in the canonical case. Our new oscillation criteria essentially improve, simplify, and complement related results in the literature, especially those from a paper by Li and Rogovchenko (Abstr. Appl. Anal. 2014:395368, 2014). Some examples are presented that illustrate the importance of the new criteria.


Introduction
Neutral delay differential equations (NDDEs) have many interesting applications in various branches of applied science. It is well known that the modeling of many natural and technological phenomena can be carried out using differential equations, often of a higher order (see [1,2]). The study of half-linear/Emden-Fowler differential equations with deviating arguments has numerous applications in physics and engineering (e.g., half-linear/Emden-Fowler differential equations arise in the study of p-Laplace equations, porous medium problems, chemotaxis models, and so forth); see, e.g., the papers [3,4] for more details, the papers [5][6][7] for the oscillation of half-linear differential equations, and the papers [3,[8][9][10] for the oscillation and asymptotic behavior of half-linear/Emden-Fowler differential equations with different neutral coefficients.
A solution u(l) of (1.1) is called oscillatory if it is neither positive nor negative and presents arbitrarily large zeros on [l 0 , ∞); otherwise, it is called nonoscillatory.
Although there are many works that have dealt with the oscillation of solutions of morder neutral differential equations, as far as we know, most of them are concerned only with the canonical operator, that is, when r(l) verifies that On the other hand, in the noncanonical case, when the studied equations have the so-called Kneser's solutions. The sign of one of such solutions differs from the sign of its first derivative, that is, u(l)u (l) < 0. Moreover, in case of even-order differential equations, the assumption (1.2) has been commonly used in the literature to ensure that any possible positive solution u satisfies u > (1p)ν, which does not generally hold in the case of (1.3). This results in the difficulty of studying the case when u(l)u (l) < 0, using the usual techniques (see [11][12][13]). From 1969 until recently, the asymptotic behavior of a DDE of the form with the canonical condition in (1.2), has attracted the interest of several authors (see [14][15][16][17] [23] improved and complemented the results in [19][20][21]. The authors in [24][25][26][27][28][29] were interested in studying and developing the oscillation theory of even-order neutral equations of the form To see other oscillation criteria of more general neutral differential equations considering the canonical operator, one can see the references [30,31]. Li and Rogovchenko [32] obtained some results on the oscillatory and asymptotic behavior of the solutions of (1.1) under the condition (1.3). For the reader's convenience, we present the following result which appeared in [32]. Theorem 1.1 Let m ≥ 4 be even and 0 < α = β ≤ 1. Assume that 0 ≤ p(l) ≤ p 0 < ∞ for some constant p 0 , τ ≥ τ * > 0, and τ • σ = σ • τ , and there exist three functions η 1 , η 2 , η 3 ∈ C([l 0 , ∞), R) such that Suppose also that where Q(l) = min{q(l), q(τ (l))}. Then (1.1) is oscillatory.
In this work, we establish new oscillation criteria for neutral delay differential equations of even order. Unlike most of the previous related works, we are interested in studying the behavior of the solutions of equation (1.1) in the noncanonical case. As far as we know, the unique work related with oscillations in the noncanonical case of (1.1) is [32]. However, in [32], there is no detailed guideline about how to choose the functions η i , i = 1, 2, 3, fulfilling the forcing conditions, an intriguing issue is how to build up oscillation criteria without requiring the presence of the obscure functions η i . Here, we will address this topic and introduce some new oscillation criteria. Some examples are provided to illustrate the new results.
The following lemmas are needed in the proofs of our main results.

, ∞)) and f (m) (l) is eventually of one sign for all large l. Then, there exists a nonnegative integer h ≤ m, with m + h even for f
eventually.

Main results
In order to facilitate the calculation, let us define the following: where c 1 , c 2 , c 3 , and c 4 are any positive constants.
Proof Assume that u is an eventually positive solution of (1.1) and that ν satisfies (B) for l ≥ l 1 . Let us consider different possibilities.
Using the decreasingness property of r(ν (m-1) ) α , we obtain, for l ≥ l 1 , Multiplying (2.1) by r -1/α (l) and integrating it on [l, L], we get Letting L → ∞, we get Thus, we see that Therefore, The proof is complete.
The proof of the theorem is complete.
Remark Combining Theorem 2.1 and the results reported in the papers [37,38] for equation (2.6), one can obtain various oscillation criteria for equation (1.1) in the case where α = β.
Theorem 2.2 Let us assume that the first-order DDE (2.6) is oscillatory for some λ 0 ∈ (0, 1) and that (2.7) holds for some λ 1 ∈ (0, 1). If Proof We argue by contradiction. Assume to the contrary that there is a nonoscillatory solution u of (1.1). Then, we can assume u(l), u(τ (l)), and u(σ (l)) are positive for l ≥ l 1 ≥ l 0 . It follows from Lemma 2.1 that there are three possible cases for the behavior of ν and its derivatives.
The proofs of the cases in which (A) or (B) is fulfilled are similar to those of Theorem 2.1.
Therefore, it is easy to verify that δ i (l) = e -l for i = 0, 1, 2.