More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations

: In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results.


Introduction
We direct our attention during this work to studying the oscillatory behavior of the solutions of the neutral delay differential equation (NDDE): a · (u + p · (u • τ)) β (t) + q · (u • σ) β (t) = 0, t ≥ t 0 (1) in the non-canonical case, that is, when: Furthermore, we assume that β is a ratio of odd positive integers, a, τ, σ, p and q are in C[t 0 , ∞), a is positive, a , p and q are non-negative, p < 1, q = 0 on any half line [t * , ∞) for all t * ≥ t 0 , τ(t) ≤ t, σ(t) ≤ t, lim t→∞ τ(t) = ∞, lim t→∞ σ(t) = ∞, ( f • g)(t) = f (g(t)) and: A solution u of the Equation (1) means a function in C([t * , ∞), R), which satisfies: and also satisfies (1) on [t * , ∞). We will only consider solutions that are not identically zero eventually. A solution u of (1) is called oscillatory if it is neither positive nor negative, ultimately; otherwise, it is called non-oscillatory. Differential equations with a neutral argument have interesting applications in problems of real-world life. In the networks containing lossless transmission lines, the neutral differential equations appear in the modeling of these phenomena as is the case of highspeed computers. In addition, second order neutral equations appear in the theory of automatic control and in aeromechanical systems in which inertia plays an important role. Moreover, second order delay equations play an important role in the study of vibrating masses attached to an elastic bar, as the Euler equation, see: [1][2][3].
To the best of our knowledge, the number of works dealing with the study of higherorder neutral differential equations in the non-canonical case is much smaller than those that deal with equations in the canonical case (see [4][5][6][7][8][9][10][11][12][13][14][15][16]). On the other hand, it is easy to find many works that have dealt with non-canonical higher-order equations with delay but not neutral (see for example [17][18][19][20]).
When studying the oscillation of the NDDEs in (1) in the non-canonical case, one of the most interesting goals is to find criteria that ensure the non-existence of Kneser solutions (solutions which satisfy (−1) k (u + p · (u • τ)) (k) (t) > 0 for k = 0, 1, 2, 3, t ∈ [t 0 , ∞)). This is because most of the relationships commonly used are not valid in this case.
For second-order equations, in an interesting work, Bohner et al. [21] addressed this problem, obtaining the following restriction for the solution and a related function: where u is a Kneser-type solution. This relationship allowed the authors to find many new criteria that simplified and improved their previous results in the literature. The first interesting problem was how to extend Bohner's results in [21] to the even-order equations.
Recently, by using comparison techniques, Li and Rogovchenko [22] studied the oscillatory behavior of the even-order neutral delay differential equation: where n ≥ 4 is an even number. However, the results in [22] depend on the existence of three unknown functions that satisfy certain conditions, and there is no general rule on how to choose these functions. So another interesting problem is how to find criteria that do not include unknown functions.
In this work, we will address all the interesting problems above by obtaining a new relationship between the solution and a related function (as an extension of Bohner's results in [21]). Furthermore, the new criteria ensure the oscillation of all the solutions of (1), and are distinguished by the following: -They do not require unknown functions; -They do not need condition (3).
In order to prove our main results, we will use the following lemmas.

Lemma 3 ([21] Lemma 2.6).
Assume that K i is a real number for i = 1, 2, 3, K 2 > 0, and β is a ratio of odd positive integers. Then, for all w ∈ R :

Main Results
First, we will proceed to classify the set of positive solutions of (1) according to the behavior of its derivatives. To facilitate the calculations, we adopt the following notations: z := u + p · (u • τ), and: We assume that u is a positive solution of (1). Note that from the definition of z, we have that z(t) > 0; moreover, from (1) it is a(t)(z (t)) β ≤ 0. This implies that a(t)(z (t)) β is non-increasing and of constant sign, and thus, since a(t) > 0, we have that (z (t)) β is of constant sign, and so is z (t).

Lemma 4. If u(t) is a Kneser solution of
Proof. Based on the positivity of the solution u, it follows from (1) that a(t)(z (t)) β is non-increasing. Then, taking into account that we are in case D3, we have that: which leads to: Therefore, we have that z /A 0 is an increasing function, and thus: which implies that: By using a similar approach, it is easy to conclude that −A 1 (t)z(t) ≤ z (t)A 2 (t), and so z(t)/A 2 (t) is an increasing function.

Proof.
We proceed by contradiction. Assuming that u is a Kneser solution of (1) on [t 1 , ∞), where t 1 ≥ t 0 . As in the proof of Lemma 4, we arrive at (4). Integrating (4) from t to ∞ and taking into account the behavior of the derivatives of z, we obtain: and integrating again, we obtain: By Lemma 4, we have that z(t)/A 2 (t) is an increasing function, and hence z(τ(t)) ≤ (A 2 (τ(t))/A 2 (t))z(t). Thus, it follows from the definition of z that: which together with (1) gives: Now, we define the function: It follows readily from (7) that T(t) ≥ 0 for t ≥ t 1 . Moreover, we have that: Now, using the inequalities in (6) and (8), we obtain that: Using Lemma 3 with K 1 := θ /θ, K 2 := βA 1 θ −1/β , K 3 := θ A −β 2 and w :=T, we obtain: Integrating the above inequality from t 1 to t, we have: From (7), we see that −a(z ) β z −β ≤ 1/A β 2 and so (10) becomes: The obtained inequality (11) conflicts with the condition (5), and this contradiction ends the proof.

Discussion and Examples
In the following theorem, we present sufficient conditions for the oscillation of all solutions of (1).
Proof. Assume that (1) has a positive solution u. From (1), we have: According to Lemma 1 and taking into account the order of the equation in (1), we eventually obtain the following three exclusive cases D1-D3.
Proof. Using Theorem 2.1.1 in [25], we obtain that Equation (12) is oscillatory under the condition (25). Therefore, the proof is the same as that of Theorem 3.

Conflicts of Interest:
The authors declare no conflict of interest.