Neutral Delay Differential Equations: Oscillation Conditions for the Solutions

The purpose of this article is to explore the asymptotic properties for a class of fourth-order neutral differential equations. Based on a comparison with the differential inequality of the first-order, we have provided new oscillation conditions for the solutions of fourth-order neutral differential equations. The obtained results can be used to develop and provide theoretical support for and to further develop the study of oscillation for a class of fourth-order neutral differential equations. Finally, we provide an illustrated example to demonstrate the effectiveness of our new criteria.


Introduction
In recent years, there has been growing interest in exploring neutral differential equations, with applications in many areas, including fluid dynamics, physics, chemistry, and biology. Problems in these areas have often guided researchers and physicists to expend great efforts to investigate interesting phenomena, such as the effect of vibrating systems fixed to an elastic bar. Examples of such problems can be found in the Euler equations and the Taylor-Goldstein equation in fluid dynamics, and the perturbed vertical velocity in stratified flow with the effect of viscosity.
The oscillatory properties of neutral differential equations also play crucial roles in mechanical engineering, civil engineering, and application-oriented research-which can support research with the potential of developing the ship building, airplane, and rocket industries-along with the fabrication of microelectromechanical systems (MEMS) and gyroscopes; see [1][2][3]. In this paper, we aim to effectively study the oscillation criteria of the following equation.
The operator L x is the canonical form if Here, our significant novel outcomes are obtained by considering the following conditions: S1: r ∈ C[x 0 , ∞), 0 ≤ r(x) < r 0 < ∞, S4: p i > 1, i = 1, 2 are constants and By a solution of Equation (1) we mean a function w ∈ C 3 [x, ∞), x ≥ x 0 , which has the property ξ(x)(y (x)) α ∈ C 1 [x 0 , ∞), and satisfies Equation (1) on [x 0 , ∞). (1) represents oscillatory behavior if it has arbitrarily large zeros on [x 0 , ∞). Otherwise, a solution can be known as nonoscillatory. (1) is known as oscillatory if each solution of it is oscillatory.

Definition 3.
If the highest-order derivative of the unknown function occurs with and without delay, then the differential equation is defined as neutral.
Over the past few years, the asymptotic behavior of solutions of differential equations has become a key research area in different disciplines. In particular, in mathematics, many researchers investigated the oscillatory properties of solutions to neutral differential equations; see [4][5][6][7][8][9][10][11][12]. For differential equations of neutral type, we present the following results that are closely related to our work. For example, Bazighifan [13], obtained oscillation conditions for solutions of Equation (1). In [14], the authors considered the equation where F = |s| p−2 s and obtained some new oscillation conditions. In [6,15] the authors studied oscillation for equations of Emden-Fowler neutral type where a criterion of Kamenev-type oscillation was found.
The authors in [16,17] considered the equation and proved that it is oscillatory if and where K(t) := z r−1 (x)(1 − b(z(x)))q(x) and r is an even number. In [18,19], the authors considered the equation and it was found to be oscillatory if and lim inf where η ∈ C 1 ([x 0 , ∞), R) and q(x) := min q z −1 (x) , q z −1 (δ(x)) . Inspired by the work of [16][17][18], we concentrate on establishing the oscillatory properties of solutions of (1). We aim also to provide some examples that can ensure the validity of the proposed criteria.

Main Results
This section aims to present the main contribution in this article which lies in its potential to empower systematic analysis and understanding the oscillatory properties of solutions of (1).
These are some of the important hypotheses of Equation (1): w is an eventually positive solution of Equation (1).

Hypothesis 2. The inequality
is oscillatory where

Hypothesis 3. The inequality
is oscillatory.
It can be noted that the first order of equations (H2) and (H3) has been discussed previously by different authors-for example, [10,22,23], and they proved that these equations are oscillatory. In our current study, we use them as a point of comparison so that (H2) and (H3) can be assumed to be oscillatory.
Proof. Let x be a non-oscillatory solution of Equation (1) By using Lemma 2 and the definition of y in (2), we get From Equations (12) and (13), we obtain which with (10) gives Since y (x) > 0, we find lim x→∞ y(x) = 0, and by Lemma 3 we obtain Combining (14) and (15), we obtain If we set , then it is easy to see that From (16), we find which is a contradiction. Therefore, this contradiction completes the proof.
Proof. It is known that (11) holds in the proof of Theorem 1. By the definition of y in (2), we find which with Equation (1) gives From Lemma 3, we obtain Combining (19) and (20), we get If we set ψ := ξ(y ) (p 1 −1) , then the inequality In view of ([10] Corollary 1), Equation (18) has a positive solution, which is a contradiction. This completes the proof.
Thus, we find This completes the proof.
Proof. It is known that (11) and (19) hold in the proof of Theorem 3. From (11), we have that y is of one sign. From Lemma 4, we find that (22) holds.
Since y (x) > 0, there exists a x 2 ≥ x 1 and a constant M > 0 such that for all x ≥ x 2 . From the inequality . This implies that which contradicts (31). From Lemma 5, we find that (26) holds. This implies that Then, we obtain which contradicts (32). This completes the proof.

Applications
This section presents some interesting examples and applications which are addressed based on above hypotheses to show some interesting results in this paper. Example 1 is presented to show how to investigate the problem of practical interest to the specific conditions (0 ≤ r(x) < r 0 < 1), whereas Example 2 extends the study to be under this condition (0 ≤ r(x) < r 0 < ∞).

Conclusions
This paper explored the oscillatory behavior of the fourth-order neutral differential equation. In particular, we addressed and investigated the oscillation criteria of Equation (1). We obtained different forms of conditions to expand the application area by using different methods. These new conditions complement several results in the literature. Additionally, some interesting examples were presented to investigate the proposed criteria. Further research based on the results of this article could extend the analysis herein to investigate a philos type oscillation criteria to ensure that every solution of the desired Equation (1) is oscillatory . Moreover, some oscillation criteria for (1) if