A Philos-Type Oscillation Criteria for Fourth-Order Neutral Differential Equations

: Some sufﬁcient conditions are established for the oscillation of fourth order neutral differential equations of the form (cid:0) r ( t ) ( z (cid:48)(cid:48)(cid:48) ( t )) α (cid:1) (cid:48) + q ( t ) x β ( σ ( t )) = 0, where z ( t ) : = x ( t ) + p ( t ) x ( τ ( t )) . By using the technique of Riccati transformation and integral averaging method, we get conditions to ensure oscillation of solutions of this equation. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. Moreover, the importance of the obtained conditions is illustrated via some examples.


Introduction
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 in high-speed computers; see [1]. The theory of oscillation is an important branch of the qualitative theory of differential equations. In recent years, there has been a great deal of interest in studying oscillatory behavior of solutions to differential equations; see .
In the following, we show some previous results in in the literature which related to this paper: In 2019, Moaaz et al. [22] studied the oscillation of the even-order equation q (t, s) f (x (σ (t, s))) ds = 0 and prove that it is oscillatory if also they used the technique of comparison with first order delay equations, Xing et al. [25] proved that the equation where 0 ≤ p (t) < p 0 < ∞ and q (t) := min q σ −1 (t) , q σ −1 (τ (t)) . Very recently, Chatzarakis et al. [10] established some oscillation criteria for neutral differential equation by using the only Riccati transformations, prove that it is oscillatory if This paper is concerned with the oscillatory behavior of the fourth-order neutral delay differential equation where t ≥ t 0 and z (t) := x (t) + p (t) x (τ (t)). Throughout this paper, we assume the following conditions to hold: (S 1 ) α and β are quotient of odd positive integers; Moreover, we study (2) under the condition (1). (2) is called oscillatory if it has arbitrarily large zeros on [t x , ∞), and otherwise is called to be nonoscillatory. (2) is said to be oscillatory if all its solutions are oscillatory.

Definition 4.
A neutral delay differential equation is a differential equation in which the highest-order derivative of the unknown function appears both with and without delay.
A kernel function H i ∈ C (D, R) is said to belong to the function class , written by H ∈ , if, for i = 1, 2, (ii) H i (t, s) has a continuous and nonpositive partial derivative ∂H i /∂s on D 0 and there exist functions and In this work, by using the Riccati transformations and the integral averaging technique, we establish a new oscillation criterion for a class of fourth-order neutral delay differential equations (2). Our results improve and complement the results in [10]. Some examples are provided to illustrate the main results.
Here, we define the next notations:

Some Auxiliary Lemmas
We shall employ the following lemmas: Assume that x (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists a t 1 ≥ t 0 such that x (n−1) (t) x (n) (t) ≤ 0 for all t ≥ t 1 .

Lemma 3 ([14]
). Let α be a ratio of two odd numbers, V > 0 and U are constants. Then
for t ≥ t 1 , where t 1 ≥ t 0 is sufficiently large.

Lemma 5.
Assume that x is an eventually positive solution of (2). then where Proof. Let x be an eventually positive solution of (2) on [t 0 , ∞). From definition of z, we get which with (2) gives Using Lemma 1, we see that Combining (6) and (7), we find Thus, (5) holds. This completes the proof.

Lemma 6.
Assume that x is an eventually positive solution of (2) and and where ξ (t) := δ (t) r (t) (z (t)) α z α (t) (10) and Proof. Let x be an eventually positive solution of (2) on [t 0 , ∞). It follows from Lemma 4 that there exist two possible cases (N 1 ) and (N 2 ). Assume that Case (N 1 ) holds. From the definition of ξ (t), we see that ξ (t) > 0 for t ≥ t 1 , and using (6), we obtain From Lemma 2, we have that z (t) ≥ t 3 z (t), and hence, It follows from Lemma 1 that for all µ 1 ∈ (0, 1) and every sufficiently large t. Thus, by (12)-(14), we get Since z (t) > 0, there exist a t 2 ≥ t 1 and a constant A 1 > 0 such that Thus, we obtain which yields Thus, (8) holds. Assume that Case (N 2 ) holds. Integrating (6) from t to u, we obtain From Lemma 2, we get that z (t) ≥ tz (t), and hence, For (16), letting u → ∞ and using (17), we get Integrating this inequality again from t to ∞, we get From the definition of ϕ (t), we see that ϕ (t) > 0 for t ≥ t 1 , and using (15) and (18), we find Since z (t) > 0, there exist a t 2 ≥ t 1 and a constant A 2 > 0 such that Thus, we obtain Thus, (9) holds. This completes the proof.

Philos-Type Oscillation Result
In the section, we employ the integral averaging technique to establish a Philos-type oscillation criteria for (2) for all µ 2 ∈ (0, 1) , and lim sup then (2) is oscillatory.
Proof. Let x be a non-oscillatory solution of (2) on [t 0 , ∞). Without loss of generality, we can assume that x is eventually positive. It follows from Lemma 4 that there exist two possible cases (N 1 ) and (N 2 ). Assume that (N 1 ) holds. From Lemma 6, we get that (8) holds. Multiplying (8) by H (t, s) and integrating the resulting inequality from t 1 to t; we find that which contradicts (20). Assume that (N 2 ) holds. From Lemma 6, we get that (9) holds. Multiplying (9) by H 2 (t, s) and integrating the resulting inequality from t 1 to t, we obtain and we can try to get some oscillation criteria of (2) in future work.

Author Contributions:
The authors claim to have contributed equally and significantly in this paper. All authors read and approved the final manuscript.

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