Oscillation Criteria of Higher-order Neutral Differential Equations with Several Deviating Arguments

: This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using the technique of Riccati transformation and comparison principles with the second-order differential equations, we obtain a new Philos-type criterion. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.

Having in mind such applications, for instance, in electrical engineering, we cite models that describe electrical power systems, see [18]. Neutral differential equations also have wide applications in applied mathematics [31,32], physics [33], ecology [34] and engineering [35].
Xing et al. [28] proved that (1) is oscillatory if where Hence, [28] improved the results in [29,30]. In our paper, by carefully observing and employing some inequalities of different type, we provide a new criterion for oscillation of differential Equation (1). Here, we provide different criteria for oscillation, which can cover a larger area of different models of fourth order differential equations. We introduce a Riccati substitution and comparison principles with the second-order differential equations to obtain a new Philos-type criteria. Finally, we apply the main results to one example.

Some Auxiliary Lemmas
We shall employ the following lemmas: . Let β be a ratio of two odd numbers, V > 0 and U are constants. Then

Main Results
In this section, we give the main results of the article. Here, we define the next notation:

Lemma 5 ([8], Lemma 1.2).
Assume that u is an eventually positive solution of (1). Then, there exist two possible cases: Lemma 6. Let u be an eventually positive solution of (1) and Then Proof. Let u be an eventually positive solution of (1) on [t 0 , ∞). From the definition of z (t), we see that Repeating the same process, we obtain which yields Thus, (11) holds. This completes the proof.

Theorem 1. Let
and lim sup where then (1) is oscillatory.
Proof. Let u be a non-oscillatory solution of (1) on [t 0 , ∞). Without loss of generality, we can assume that u is eventually positive. It follows from Lemma 5 that there exist two possible cases (S 1 ) and (S 2 ). Let (S 1 ) hold. From Lemma 7, we arrive at (12). Next, we define a function ξ by Differentiating and using (12), we obtain This yields It follows from Lemma 4 that for all µ 1 ∈ (0, 1) and every sufficiently large t. Thus, by (21), (22) and (23), we get Hence, Multiplying (24) by H 1 (t, s) and integrating the resulting inequality from t 1 to t; we find that which contradicts (19).
On the other hand, let (S 2 ) hold. Using Lemma 7, we get that (13) holds. Now, we define 1 which contradicts (20). This completes the proof.
In the next theorem, we establish new oscillation results for (1) by using the theory of comparison with a second order differential equation.

Theorem 2.
Assume that the equation are oscillatory, then every solution of (1) is oscillatory.

Conclusions
The aim of this article was to provide a study of asymptotic nature for a class of even-order neutral delay differential equations. We used a generalized Riccati substitution and the integral averaging technique to ensure that every solution of the studied equation is oscillatory. The results presented here complement some of the known results reported in the literature.
A further extension of this article is to use our results to study a class of systems of higher order neutral differential equations as well as of fractional order. For all these there is already some research in progress.