Oscillation Results for Nonlinear Higher-Order Differential Equations with Delay Term

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained in the literature. Some examples are presented to demonstrate the main results.


Introduction
Nowadays, analysis of the oscillation properties of partial differential equations is attracting considerable attention from the scientific community due to numerous applications in several contexts such as biology, physics, chemistry, and dynamical systems (see [1][2][3]). For some details related to recent studies on the oscillation properties of the equations under consideration, we refer the reader to [4,5]. Moreover, the oscillation of partial equations contributes to many applications in economics, medicine, engineering, and biology.
In 2011, Run et al. [6] established new oscillation criteria for second-order partial differential equations with a damping term. Agarwal et al. [7] obtained some oscillation criteria for solutions of second-order neutral partial functional differential equations.
In this article, we investigate the oscillation of the higher-order delay differential equations Our novel outcomes are obtained by considering the following suppositions: ., n, j is even, γ is a quotient of odd positive integers.
The following condition is satisfied: Our main purpose for studying this work is to extend the results in [16]. We will use different methods to obtain these results.
In [16] the authors obtained oscillation criteria for fourth-order delay differential equations with middle term Bazighifan et al. [17,18] obtained some oscillation conditions for the equation Zhang et al. in [19] investigated some oscillation properties of the equation Bazighifan and Ramos [20] studied the following delay differential equations: Liu et al. [21] derived oscillation theorems for the equations where n is even and used the integral averaging technique. Grace et al. [22] discussed the equation Zhang et al. [23] considered the even-order equation and used the comparison technique. The aim of this paper is to give several oscillatory properties of Equation (1). New criteria extend the results in [16].
In the following, we mention some notations.

Main Results
Here we present the following lemmas.
We say that a function H ∈ C(D, R) belongs to the class w if H, H * have a nonpositive continuous partial derivative ∂H/∂s, ∂H * /∂s on D 0 with respect to the second variable, and there exist functions and Theorem 1. Let j ≥ 4 be even. Let Equations (3) and (4) hold. If there exist functions δ 1 , for some constant µ ∈ (0, 1) and then Equation (1) is oscillatory.
Proof. Let w be a nonoscillatory solution of Equation (1), then w(z) > 0. From Lemma 3, we have two possible cases: Let case (C 1 ) hold. Define the function y 1 (z) by Then y 1 (z) > 0 for z ≥ z 1 and By Lemma 2, we get Using Equations (7) and (8), we obtain By Lemma 1, we find w(z) w (z) ≥ z j − 1 .
Thus we obtain that w/z j−1 is nonincreasing and so From Equations (1) and (9), we get From Equations (10) and (11), we obtain It follows from Equation (12) that Replacing z by s, multiplying two sides by H(z, s)A(s), and integrating the resulting inequality from z 1 to z, we have Here Putting the resulting inequality into Equation (13), we obtain for some µ ∈ (0, 1), which contradicts Equation (5).
Thus we obtain that w/z is nonincreasing and so From Equation (15) and integrating Equation (1) from z to ∞, we obtain Integrating Equation (16) from z to ∞ for a total of (j − 3) times, we obtain Now, define Then y 1 (z) > 0 for z ≥ z 1 and It follows from Equations (17) and (18) that Replacing z by s, multiplying two sides by H * (z, s)A * (s), and integrating the resulting inequality from z 1 to z, we have From Lemma 2, we obtain for all ∈ (0, 1). Set Using Equation (21) in Equation (1), we obtain the inequality That is, x is a positive solution of the inequality in Equation (19), which is a contradiction. Thus the theorem is proved.

Conclusions
In this article, we give several oscillation criteria of even-order differential equations with damped. These criteria that we obtained complement some oscillation theorems for delay differential equations with damping. In future work, we will discuss the oscillatory behavior of these equations by using a comparing technique with second-order equations under the condition Funding: The authors received no direct funding for this work.
Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.