Oscillatory Properties of Solutions of Even-Order Differential Equations

: This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using Riccati transformation and the integral averaging technique, we obtain a new oscillation criteria. This new theorem complements and improves some known results from the literature. An example is provided to illustrate the main results.


Introduction
Neutral differential equations are used in numerous applications in technology and natural science. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale [1], and therefore their qualitative properties are important.
Delay differential equations are often studied in one of two cases ∞ t 0 which are said to be in the canonical or noncanonical form, respectively, see [12]. For the canonical form, many authors in [13][14][15][16] studied the asymptotic behavior of the solutions of equation In the noncanonical form, Li and Rogovchenko [17] studied the asymptotic properties of solutions of higher-order neutral differential (2) under the assumptions that allow applications to even-and odd-order equations with delayed and advanced arguments. This paper is motivated by several recent studies [3,7,9,10] of such higher order equations. Using the integral averaging technique and the Riccati transformation, we study the asymptotic properties of solutions of even order neutral delay differential equations of the form where n is an even natural number and m is a natural number. In this paper, we assume that α i are positive integers, During the following results, for clarity of presentation, we study only the case where m = 3. Moreover, we denote, for convenience, that and We say that a function, y, is a solution of (3), we mean a non-trivial real function z(t) ∈ C n−1 [t y , ∞) , t y ≥ t 0 , satisfying (3) on [t y , ∞) and which has the property r (t) z (n−1) (t) α 1 ∈ C [t y , ∞) . We consider only those solutions y of (3) which satisfy sup{|y(t)| : t ≥ T} > 0, for any T ≥ t y . A solution of (3) is called oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory. Studying the functional differential equations, the continuity of all functions, and r(t)>0 are sufficient conditions for the existence of one or more solutions of the equation.
To establish our main results, we make use of the following lemmas: ). Let C and D nonnegative real numbers. Then where the equality holds, if and only if, C = D.
is eventually of one sign for all large t, then there exist a t y > t 1 for some t 1 > t 0 and an integer m, 0 ≤ m ≤ n with n + m even for h (n) (t) ≥ 0 or n + m odd for all sufficiently large t.
This paper is concerned with the oscillatory behavior of a class of even-order neutral differential equations with multi-delays. Firstly, by using the Riccati transformations, we obtain a new oscillation criteria for this equation. Secondly, using the integral averaging technique, we establish a Philos type oscillation criterion. This new theorem complements and improves some known results in the literature. Finally, an example is provided to illustrate the main results.

Oscillation Criteria
In this section, we establish new oscillation results for Equation (3) using the Riccatti transformation.
then all solutions of (3) are oscillatory.
In Theorem 1, we can obtain different conditions for oscillation of all solutions of Equation (3) with different choices of ρ (t). If we set ρ(t) := B α 1 (σ(t)), then we obtain the following corollary.
then all solutions of (3) are oscillatory.

Kamenev-Type Criteria
In the section theorem, we establish new oscillation results for Equation (3) using the integral averaging technique to establish the Philos-type.
Proof. Proceeding as in the proof of Lemma 1, we obtain (11). Multiplying (11) by H (t, s) δ (s) and integrating from t 1 to t, we find Using Lemma 1 with which contradicts (14). Theorem 2 is proved.

Conclusions
In this work, by using the generalized Riccati transformation technique and the integral averaging technique, we establish a new oscillation criteria for (3) under (1). Further, in future work, we can attempt to find some oscillation criteria of Equation (3), if z (t) = y (t) − a (t) y (τ (t)).

Author Contributions:
The authors have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.

Funding:
The authors received no direct funding for this work.