Comparison Theorems for Oscillation of Higher-Order Neutral Delay Differential Equations

Abstract

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons.

1. Introduction

Functional differential equations (FDEs), in which the largest derivative is formed in the solution with and without delay, are known as neutral differential equations (NDEs). This type of equation has many applications, especially second-order ones. Second-order neutral equations appear in applications such as issues with lossless transmission lines in electric networks; see [1,2,3,4].

Qualitative theory is of great importance in the applied sciences. Numerous significant details concerning nonlinear mathematical models, including their stability and oscillatory behavior, as well as the examination of their features like symmetry, periodicity, bifurcation, etc., are provided by this theory. The area of qualitative theory known as oscillation theory examines the asymptotic and oscillatory characteristics of differential equation. This theory is distinguished by its diversity of mathematical methods and analytical techniques. Studying the oscillatory features of differential equations helps understand the behavior of the solutions of the examined applied models; see [5,6,7,8]. Most studies in oscillation theory rely on the symmetry feature between non-oscillatory solutions to the oscillation conditions by excluding only positive solutions.

Extensive research has been conducted on the oscillation of solutions to several kinds of FDEs. Monographs [9,10,11] and papers [3,12,13] show interest in this topic.

This study investigates the oscillatory features of solutions to a class of $n^{th}$-order NDEs with multiple delays. The practical importance of NDEs and their richness with interesting analytical problems are the main motivations for interest in studying this type of equation. In the remaining part of the introduction, we present the studied equation and the basic hypotheses of the study, and we also review some related previous works. The main results are divided into two sections: preliminary lemmas and oscillation results. In the preliminary lemmas section, we prove some properties of positive solutions to the examined equation. Taking into account the even and odd orders, we offer several new conditions that test the oscillation of the examined equation in the oscillation results section. In the next section—examples and discussions—we apply our criteria to special cases of the considered equation. We also compare our results with some previous related ones to illustrate the importance of the new conditions.

1.1. Higher-Order NDE

This work studies the asymptotic features of solutions to the higher-order NDE

$${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+\sum _{i=1}^m{q}_i\left(u\right)x^{\beta }\left({\sigma }_i\left(u\right)\right)=0,$$

(1)

where $u\in \left[u_0,\infty \right)$, n and m are natural numbers, $n\ge 3$, and $z\left(u\right):=x\left(u\right)+p\left(u\right)x\left(\tau \left(u\right)\right)$. Throughout the investigation, the following presumptions are made:

(H₁) 

$\alpha$ and $\beta$ are the quotient of odd natural numbers, $r\in \mathbf{C}\left(\left[u_0,\infty \right),\left(0,\infty \right)\right)$, and

$${\int }_{u_0}^{\infty }\frac{\mathrm{d}s}{r^{1/\alpha }\left(s\right)}=\infty ;$$

(2)

(H₂) 

$\tau ,$${\sigma }_i\in \mathbf{C}\left(\left[u_0,\infty \right),\mathbb{R}\right),$${\sigma }_i\left(u\right)\le u,$${\lim }_{u\to \infty }\tau \left(u\right)=\infty$, and ${\lim }_{u\to \infty }{\sigma }_i\left(u\right)=\infty$ for all $i=1,2\dots ,m$,

(H₃) 

$p\in \mathbf{C}\left(\left[u_0,\infty \right),\left[0,p_0\right)\right)$, and $q_i\in \mathbf{C}\left(\left[u_0,\infty \right),\left(0,\infty \right)\right)$, where $p_0\in {\mathbb{R}}^{+}$.

For a solution of (1), we mean a $x\in {\mathbf{C}}^1\left(\left[u_x,\infty \right),\mathbb{R}\right)$ for $u_x\ge u_0$, which has the property $r{\left(z^{\left(n-1\right)}\right)}^{\alpha }\in {\mathbf{C}}^{\left(n-1\right)}\left(\left[u_x,\infty \right),\mathbb{R}\right)$, and satisfies (1) on $\left[u_x,\infty \right)$. We consider only solutions of (1) that have the property $sup\left\{\right|x\left(u\right)|:u\ge u_1\}>0$ for all $u_1\ge u_0$.

Definition 1. 

A solution of (1) is called oscillatory if it has arbitrarily large zeros on $[u_0,\infty )$; otherwise, it is called non-oscillatory. If every solution to Equation (1) is oscillatory, then the equation is considered oscillatory.

1.2. Review of Relevant Literature

In the previous few years, research on finding criteria for testing the oscillatory and asymptotic features of FDEs has been ongoing, see [14,15,16,17]. Among the many published studies on this topic, we present below some related works.

In 1976, Dahiya [18] tested the oscillatory features of a class of even-order delay differential equations (DDEs). He was able to provide conditions that guarantee the oscillation of all bounded solutions. By establishing a criterion that guarantees the oscillation of all solutions, Grace and Lalli [19] tested the oscillation of DDEs using the Riccati technique. Later, Zhang and Ladde [20] extended the results in [19] to equations with a middle term. For half-linear equations, Agarwal et al. [21,22] used Riccati and comparison techniques to verify the oscillation of DDEs. These works were only concerned with the canonical case of delay equations, while studies [23,24,25,26] were interested in extending their results to the noncanonical case.

Studying neutral equations often requires more inequalities and analytical tools than delay equations. In the case where $p\left(t\right)<1$, Zhang et al. [27] developed conditions of an iterative nature to study the oscillation of NDEs in the canonical case. Using the principle of comparison with first-order DDEs, Baculíková et al. [28] and Baculíková and Džurina [29] tested the oscillatory characteristics of NDE when $p\left(t\right)\le p_0<\infty$. Meanwhile, Li and Rogovchenko [30] considered the case $p\left(t\right)>1$ and presented criteria that improved the results in [28,29] for linear NDEs. Recently, Salah et al. [17] extended the results of [30] to the nonlinear case.

In order to study the oscillation of more general NDEs, Rath et al. [31] considered the NDEs with positive and negative coefficients; Moaaz et al. [32] studied the NDEs with distributed deviating arguments; Zhang et al. [33] were interested in the DDEs with p-Laplacian-like operators; and Liu et al. [34] took into account the damped DDEs. Furthermore, by improving some of the inequalities used in the study, Nabih et al. [35] recently developed criteria that investigate the oscillation of NDEs with several delays.

Conversely, odd-order FDEs have not received the same amount of attention compared to even-order equations. Das [36] used comparison theorems to look at how solutions to a canonical odd-order DDE oscillate and found necessary and sufficient oscillation conditions. For third-order DDEs with a first-order middle term, Agarwal et al. [37] investigated the asymptotic properties by using the comparison method. They established conditions for solutions to oscillate or tend to zero as t $\to \infty$. Li and Rogovchenko [38] extended the results in [37] to half-linear odd-order DDEs.

For linear NDEs, Gopalsamy et al. [39] established adequate conditions for every solution to oscillate, which are derived along with some comparison results. Candan [40] studied the oscillatory properties of mixed NDEs with distributed deviating arguments. In the case where $p\ge 1$, Graef et al. [41] obtained adequate conditions for solutions of a class of third-order NDEs to be either oscillatory or to converge to zero. Muhib et al. [42] studied the asymptotic features of solutions to half-linear NDEs of odd order. For NDEs with several delays, Masood et al. [43] examined the asymptotic properties of solutions.

In the noncanonical case, Chatzarakis et al. [44] studied the oscillatory properties of solutions to a class of third-order NDEs. Recently, Tunc and Sarig [45] studied the oscillation behavior of DDEs with unbounded neutral coefficients.

Remark 1. 

The following assumes that every functional inequality that occurs eventually holds, meaning that it is satisfied for every u that is large enough. As usual, and without losing generality, we are restricted to dealing with eventually positive solutions of (1).

2. Preliminary Lemmas

For simplicity, we define the next functions:

$$\sigma \left(u\right):=min\left\{{\sigma }_i\left(u\right):i=1,2,...,m\right\},$$

$${\tilde{q}}_i\left(u\right):=min\left\{q_i\left(u\right),q_i\left(\tau \left(u\right)\right)\right\},\mathrm{for}i=1,2,...,m,$$

$${\mathrm{\Lambda }}_1\left(u\right):={\int }_{u_1}^u{\left(\frac{{\left(u-s\right)}^{n-2}}{r\left(s\right)}\right)}^{1/\alpha }\mathrm{d}s,$$

$${\mathrm{\Lambda }}_2\left(u\right):={\int }_{u_1}^u{\left(\frac{{\left(s-u_1\right)}^{n-2}}{r\left(s\right)}\right)}^{1/\alpha }\mathrm{d}s,$$

and

$$\delta :=\left\{\begin{array}{cc}1\hfill & \alpha \in \left(0,1\right),\hfill \\ 2^{\alpha -1}\hfill & \alpha \ge 1,\hfill \end{array}\right.$$

where $u_1\ge u_0$ is large enough.

Below, we review some lemmas that help us in our study, which were proven in [46] and [47], respectively.

Lemma 1. 

Assume that $V,$$W\in \left[0,\infty \right)$. Then,

$${\left(V+W\right)}^{\alpha }\le \delta \left(V^{\alpha }+W^{\alpha }\right).$$

Lemma 2. 

Assume that $w\in {\mathbf{C}}^n\left(\left[t_0,\infty \right),\left(0,\infty \right)\right)$. If $w^{\left(n\right)}$ has eventually one sign for any large u, then there is a $u_x$ such that $u_x\ge u_0$ and an integer $\kappa ,0\le \kappa \le n,$ with $n+\kappa$ even for $w^{\left(n\right)}\left(u\right)\ge 0$, or $n+\kappa$ odd for $w^{\left(n\right)}\left(u\right)\le 0$ such that

$$\kappa >0\mathit{implies}{w}^{\left(j\right)}\left(u\right)>0,j=0,1,...,\kappa -1,$$

and

$$\kappa \le n-1\mathit{implies}{\left(-1\right)}^{\kappa +j}{w}^{\left(j\right)}\left(u\right)>0,j=\kappa ,\kappa +1,...,n-1,$$

for $u\ge u_x$.

By employing the previous lemma, we can get the following classification of the corresponding function to the positive solutions of the studied equation.

Lemma 3. 

Assume that $z\in {\mathbf{C}}^1\left(\left[u_0,\infty \right),{\mathbb{R}}^{+}\right)$, $r{\left(z^{\prime }\right)}^{\alpha }\in {\mathbf{C}}^{n-1}\left(\left[u_0,\infty \right),\mathbb{R}\right)$, and

$${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}<0.$$

Then, there is $\kappa \in \mathbb{Z}$, with $\kappa \in \left[0,n-1\right]$ and $n+\kappa$ odd, such that

$${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(i\right)}>0,i=0,1,...,\kappa -1,\mathit{for}\kappa \ge 1,$$

(3)

and

$${\left(-1\right)}^{\kappa +j-1}{\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(j\right)}>0,j=\kappa ,\kappa +1,...,n-2,$$

(4)

for $u\ge u_1$, where $u_1$ is large enough.

Definition 2. 

The corresponding function z is said to be of degree κ if $z\in {\mathbf{C}}^1\left(\left[u_0,\infty \right),{\mathbb{R}}^{+}\right)$, $r{\left(z^{\prime }\right)}^{\alpha }\in {\mathbf{C}}^{n-1}\left(\left[u_0,\infty \right),\mathbb{R}\right)$, ${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}<0$, and (3) and (4) hold.

Lemma 4. 

Assume that z is of degree κ. Then,

$$z^{\alpha }\left(u\right)\ge \frac{1}{\left(n-2\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\mathrm{\Lambda }}_{\ast }^{\alpha }\left(u\right),$$

(5)

eventually, where

$${\mathrm{\Lambda }}_{\ast }\left(u\right):=\left\{\begin{array}{cc}{\mathrm{\Lambda }}_1\left(u\right)\hfill & \kappa =1,\hfill \\ {\mathrm{\Lambda }}_2\left(u\right)\hfill & \kappa \ge 1.\hfill \end{array}\right.$$

Proof. 

Assume that z is of degree $\kappa$. Now, if $\kappa <n-1$, then

$${\int }_v^u{\left[r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}\mathrm{d}s\le -{\left(r\left(v\right){\left(z^{\prime }\left(v\right)\right)}^{\alpha }\right)}^{\left(n-3\right)}.$$

for $v\in \left[u_1,u\right]$. Since ${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}<0$, we obtain

$${\left(r\left(s\right){\left(z^{\prime }\left(s\right)\right)}^{\alpha }\right)}^{\left(n-3\right)}\le -{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}\left(u-s\right),\mathrm{for}s\in \left[u_1,u\right].$$

(6)

Integrating (6) from v to u, we arrive at

$${\left(r\left(v\right){\left(z^{\prime }\left(v\right)\right)}^{\alpha }\right)}^{\left(n-4\right)}\ge \frac{1}{2!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\left(u-v\right)}^2.$$

By repeating this procedure $n-\kappa -3$ times, we infer that

$${\left(r\left(v\right){\left(z^{\prime }\left(v\right)\right)}^{\alpha }\right)}^{\left(\kappa -1\right)}\ge \frac{1}{\left(n-\kappa -1\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\left(u-v\right)}^{n-\kappa -1},\mathrm{for}v\in \left[u_1,u\right].$$

(7)

For $\kappa =1$, (7) reduces to

$$r\left(v\right){\left(z^{\prime }\left(v\right)\right)}^{\alpha }\ge \frac{1}{\left(n-2\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\left(u-v\right)}^{n-2}.$$

Hence, we find

$$z^{\prime }\left(s\right)\ge {\left[\frac{1}{\left(n-2\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}\frac{{\left(u-s\right)}^{n-2}}{r\left(s\right)}\right]}^{1/\alpha },$$

(8)

Integrating (8) from $u_1$ to u, we find

$$\begin{array}{ccc}\hfill z\left(t\right)& \ge & z\left(u_1\right)+\frac{1}{{\left[\left(n-2\right)!\right]}^{1/\alpha }}{\int }_{u_1}^u{\left[{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}\frac{{\left(u-s\right)}^{n-2}}{r\left(s\right)}\right]}^{1/\alpha }\mathrm{d}s\hfill \\ & \ge & \frac{1}{{\left[\left(n-2\right)!\right]}^{1/\alpha }}{\left({\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}\right)}^{1/\alpha }{\mathrm{\Lambda }}_1\left(u\right).\hfill \end{array}$$

Next, we assume that $\kappa >1$. Integrating (7) from $u_1$ to u, we see that

$$\begin{array}{ccc}\hfill {\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(\kappa -2\right)}& \ge & {\int }_{u_1}^u{\left(r\left(v\right){\left(z^{\prime }\left(v\right)\right)}^{\alpha }\right)}^{\left(\kappa -1\right)}\mathrm{d}v\hfill \\ & \ge & \frac{1}{\left(n-\kappa -1\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\int }_{u_1}^u{\left(u-v\right)}^{n-\kappa -1}\mathrm{d}v\hfill \\ & =& \frac{1}{\left(n-\kappa \right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\left(u-u_1\right)}^{n-\kappa }.\hfill \end{array}$$

Integrating this inequality $k-2$ times from $u_1$ to u, we see that

$$r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\ge \frac{1}{\left(n-2\right)!}{\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right]}^{\left(n-2\right)}{\left(u-u_1\right)}^{n-2}.$$

By completing as in the case $\kappa =1$, we obtain (5).

The proof is now complete. □

Lemma 5. 

Assume that z is of degree $\kappa \ge 1$. Equation (1) can be written in one of two forms

$${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+{\left(1-p_0\right)}^{\beta }{z}^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{q}_i\left(u\right)\le 0,\mathrm{if}{p}_0<1,$$

(9)

and

$${\left[r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}r\left(\tau \left(u\right)\right){\left(z^{\prime }\left(\tau \left(u\right)\right)\right)}^{\alpha }\right]}^{\left(n-1\right)}\le -\frac{1}{\delta }{z}^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right),$$

(10)

if ${\tau }^{\prime }\left(u\right)={\tau }_0>0$ and $\tau \left({\sigma }_i\left(t\right)\right)={\sigma }_i\left(\tau \left(t\right)\right)$.

Proof. 

Assume that z is of degree $\kappa$.

Let $p_0<1$. In this case, we have $z^{\prime }\left(t\right)>0$, and so

$$x\left(u\right)=z\left(u\right)-p\left(u\right)x\left(\tau \left(u\right)\right)\ge z\left(u\right)-p\left(u\right)z\left(\tau \left(u\right)\right)\ge z\left(u\right)\left(1-p_0\right).$$

(11)

Thus, Equation (1) reduces to (9).

Let ${\tau }^{\prime }\left(u\right)={\tau }_0>0$ and $\tau \left({\sigma }_i\left(t\right)\right)={\sigma }_i\left(\tau \left(t\right)\right)$. From (1), we obtain

$$\frac{1}{{\tau }_0^{n-1}}{\left(r\left(\tau \left(u\right)\right){\left(z^{\prime }\left(\tau \left(u\right)\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+\sum _{i=1}^m{q}_i\left(\tau \left(u\right)\right)x^{\beta }\left({\sigma }_i\left(\tau \left(u\right)\right)\right)=0.$$

(12)

Combining (1) and (12), we arrive at

$$\begin{array}{cc}& {\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}{\left(r\left(\tau \left(u\right)\right){\left(z^{\prime }\left(\tau \left(u\right)\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}\hfill \\ =& -\sum _{i=1}^m{q}_i\left(u\right)x^{\beta }\left({\sigma }_i\left(u\right)\right)-p_0^{\beta }\sum _{i=1}^m{q}_i\left(\tau \left(u\right)\right)x^{\beta }\left({\sigma }_i\left(\tau \left(u\right)\right)\right)\hfill \\ \le & -\sum _{i=1}^m{\tilde{q}}_i\left(u\right)\left[x^{\beta }\left({\sigma }_i\left(u\right)\right)+p_0^{\beta }{x}^{\beta }\left(\tau \left({\sigma }_i\left(u\right)\right)\right)\right].\hfill \end{array}$$

It follows from Lemma 1 that

$$\begin{array}{ccc}\hfill {\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}{\left(r\left(\tau \left(u\right)\right){\left(z^{\prime }\left(\tau \left(u\right)\right)\right)}^{\alpha }\right)}^{\left(n-1\right)}& \le & -\frac{1}{\delta }\sum _{i=1}^m{\tilde{q}}_i\left(u\right)z^{\beta }\left({\sigma }_i\left(u\right)\right)\hfill \\ & \le & -\frac{1}{\delta }{z}^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right).\hfill \end{array}$$

The proof is now complete. □

3. Oscillation Results

Using a comparison method with first-order DDEs, we provide conditions that assure the oscillation of every solution of the equation under study. We indicate the category of every eventually positive solution of (1) by $S^{\ast }$ for convenience.

The following theorems cover different study areas in terms of constraints on functions such as $p,$$\tau$ and $\sigma$. Therefore, we need to impose the following conditions:

(H_a) 

$\tau \left(u\right)\le u$ and $p_0<1$

(H_b) 

${\tau }^{\prime }\left(u\right)={\tau }_0>1$, ${\sigma }^{\prime }\left(u\right)\ge 0$, and $\tau \left({\sigma }_i\left(t\right)\right)={\sigma }_i\left(\tau \left(t\right)\right)$ for $i=1,2,\dots ,m.$

(H_c) 

${\tau }^{\prime }\left(u\right)={\tau }_0<1$, $\tau \left(u\right)\ge \sigma \left(t\right)$, ${\sigma }^{\prime }\left(u\right)\ge 0$, and $\tau \left({\sigma }_i\left(t\right)\right)={\sigma }_i\left(\tau \left(t\right)\right)$ for $i=1,2,\dots ,m.$

3.1. Comparison Theorems (I)

Theorem 1. 

Assume that n is even, and (H_a) holds. If the DDEs

$${\varphi }^{\prime }\left(u\right)+\frac{{\left(1-p_0\right)}^{\beta }}{{\left[\left(n-2\right)!\right]}^{\beta /\alpha }}{\varphi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{q}_i\left(u\right)=0$$

(13)

and

$${\varphi }^{\prime }\left(u\right)+\frac{{\left(1-p_0\right)}^{\beta }}{{\left[\left(n-2\right)!\right]}^{\beta /\alpha }}{\varphi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_2^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{q}_i\left(u\right)=0$$

(14)

are oscillatory, then (1) oscillates.

Proof. 

Suppose, on the contrary, that $x\in S^{\ast }$. Therefore, there is a $u_1\ge u_0$ such that $x\left(\tau \left(t\right)\right)>0$ and $x\left(\sigma \left(t\right)\right)>0$ for $u\ge u_1$. From Lemma 3, we infer that $z\left(u\right)>0$ and is of degree $\kappa \ge 1$ (for n even).

Consider $\kappa \ge 1$. Then, $z^{\prime }\left(u\right)>0$ for $u\ge u_1$. From Lemma 5, we have (9) holds. Setting $\varphi :$$={\left(r{\left(z^{\prime }\right)}^{\alpha }\right)}^{\left(n-2\right)}>0$, Equation (9) reduces to

$${\varphi }^{\prime }\left(u\right)+{\left(1-p_0\right)}^{\beta }{z}^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{q}_i\left(u\right)\le 0.$$

(15)

Now, let $k=1$. It follows from Lemma 4 that

$$z\left(u\right)\ge {\left[\frac{1}{\left(n-2\right)!}\varphi \left(u\right){\mathrm{\Lambda }}_1^{\alpha }\left(u\right)\right]}^{1/\alpha }.$$

Consequently, (15) becomes

$${\varphi }^{\prime }\left(u\right)+\frac{{\left(1-p_0\right)}^{\beta }}{{\left[\left(n-2\right)!\right]}^{\beta /\alpha }}{\varphi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{q}_i\left(u\right)\le 0.$$

We note that $\varphi >0$ is a solution to this inequality. Based on Theorem 1 in [48], we also find that Equation (13) has a positive solution, and this contradicts the assumptions of the theorem.

In the case that $\kappa >1$, we find—in the same way as before—that Equation (14) has a positive solution, and this is a contradiction.

The proof is now complete. □

Theorem 2. 

Assume that n is odd, and (H_a) holds. Assume also that the DDEs (13) and (14) are oscillatory. If

$${\int }_{u_1}^{\infty }{r}^{-1/\alpha }\left(v\right){\left({\int }_v^{\infty }{\left(s-v\right)}^{n-2}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}v=\infty ,$$

(16)

for $u_1\ge u_0$, then all solutions of (1) are oscillatory or converge to zero.

Proof. 

Suppose, on the contrary, that $x\in S^{\ast }$. Therefore, there is a $u_1\ge u_0$ such that $x\left(\tau \left(t\right)\right)>0$ and $x\left(\sigma \left(t\right)\right)>0$ for $u\ge u_1$. From Lemma 3, we infer that $z\left(u\right)>0$ and is of degree $\kappa \ge 0$.

First, we find that the oscillation of the DDEs (13) and (14) leads to the exclusion of the possibility that $\kappa \ge 1$, as in the proof of Theorem 1.

Consider $\kappa =0$. Since $z\left(u\right)>0$ and $z^{\prime }\left(u\right)<0$, we obtain ${\lim }_{u\to \infty }z\left(u\right)=c\ge 0$. If $c>0$, then we find $c+ϵ>z\left(u\right)>c$, for all $ϵ>0$. Hence,

$$x\left(u\right)=z\left(u\right)-p\left(u\right)x\left(\tau \left(u\right)\right)\ge z\left(u\right)-p\left(u\right)z\left(\tau \left(u\right)\right),$$

and so

$$x\left(u\right)\ge c-p\left(c+ϵ\right)=\frac{c-p\left(c+ϵ\right)}{\left(c+ϵ\right)}\left(c+ϵ\right)=k\left(c+ϵ\right)>kz\left(u\right),$$

(17)

where $k=\left(c-p\left(c+ϵ\right)\right)/\left(c+ϵ\right)>0$. By integrating (1) from u to ∞ we obtain

$${\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-2\right)}\ge {\int }_u^{\infty }\sum _{i=1}^m{q}_i\left(s\right)x^{\beta }\left({\sigma }_i\left(s\right)\right)\mathrm{d}s=0.$$

Using (17), we see that

$$\begin{array}{ccc}\hfill {\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-2\right)}& \ge & k^{\beta }{\int }_u^{\infty }\sum _{i=1}^m{q}_i\left(s\right)z^{\beta }\left({\sigma }_i\left(s\right)\right)\mathrm{d}s\hfill \\ & \ge & {\left(kc\right)}^{\beta }{\int }_u^{\infty }\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s.\hfill \end{array}$$

Integrating again from u to ∞, we obtain

$$\begin{array}{ccc}\hfill -{\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-3\right)}& \ge & {\left(kc\right)}^{\beta }{\int }_u^{\infty }{\int }_v^{\infty }\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s\mathrm{d}v\hfill \\ & =& {\left(kc\right)}^{\beta }{\int }_u^{\infty }\left(s-u\right)\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s.\hfill \end{array}$$

By repeating this procedure $n-3$ times, we conclude that

$$-r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\ge \frac{{\left(kc\right)}^{\beta }}{\left(n-2\right)!}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s.$$

Integrating this inequality from $u_1$ to ∞, we obtain

$$z\left(u_1\right)\ge \frac{c{k}^{\beta /\alpha }}{{\left(\left(n-2\right)!\right)}^{1/\alpha }}{\int }_{u_1}^{\infty }{r}^{-1/\alpha }\left(v\right){\left({\int }_v^{\infty }{\left(s-v\right)}^{n-2}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}v,$$

which contradicts (16). Thus, $c=0$, and so z and x converge to zero.

The proof is now complete. □

Corollary 1. 

Assume that $\alpha =\beta$, (H_a) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{\sigma \left(u\right)}^u{\mathrm{\Lambda }}_j^{\alpha }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s>\frac{\left(n-2\right)!}{{\left(1-p_0\right)}^{\alpha }\mathrm{e}},j=1,2.$$

(18)

Assume also that (16) is achieved for n odd. Then,

$\left(\mathrm{a}\right)$ (1) is oscillatory if n even.

$\left(\mathrm{b}\right)$ every solution of (1) is oscillatory or converges to zero if n odd.

Proof. 

Using Theorem 2 in [49], we find that the conditions in (18) guarantee the oscillation of Equations (13) and (14). Thus, from Theorems 1 and 2, what is required is achieved. □

Corollary 2. 

Assume that $r^{\prime }\left(t\right)\ge 0$, $\alpha =\beta$, (H_a) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{\sigma \left(u\right)}^u\frac{{\left(\sigma \left(s\right)\right)}^{\alpha +n-2}}{r\left(\sigma \left(s\right)\right)}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s>\frac{\left(n-2\right)!{\left(n+\alpha -2\right)}^{\alpha }}{{\alpha }^{\alpha }{\left(1-p_0\right)}^{\alpha }\mathrm{e}}.$$

Assume also that (16) is achieved for n odd. Then,

$\left(\mathrm{a}\right)$ (1) is oscillatory if n even.

$\left(\mathrm{b}\right)$ every solution of (1) is oscillatory or converges to zero if n odd.

3.2. Comparison Theorems (II)

Theorem 3. 

Assume that n is even, and (H_b) holds. If the DDEs

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\left(1+{\tau }_0^{1-n}{p}_0^{\beta }\right)\right]}^{\beta /\alpha }}{\psi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)=0$$

(19)

and

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\left(1+{\tau }_0^{1-n}{p}_0^{\beta }\right)\right]}^{\beta /\alpha }}{\psi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_2^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)=0$$

(20)

are oscillatory, then (1) oscillates.

Proof. 

Suppose, on the contrary, that $x\in S^{\ast }$. Therefore, there is a $u_1\ge u_0$ such that $x\left(\tau \left(t\right)\right)>0$ and $x\left(\sigma \left(t\right)\right)>0$ for $u\ge u_1$. From Lemma 3, we infer that $z\left(u\right)>0$ and is of degree $\kappa \ge 1$ (for n even). From Lemma 5, we obtain that (10) holds. Setting $\varphi :={\left(r{\left(z^{\prime }\right)}^{\alpha }\right)}^{\left(n-2\right)}$, (10) reduces to

$${\left[\varphi \left(u\right)+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\varphi \left(\tau \left(u\right)\right)\right]}^{\prime }\le -\frac{1}{\delta }{z}^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right),$$

(21)

Consider $\kappa =1$. It follows from Lemma 4 that

$$z\left(u\right)\ge {\left[\frac{1}{\left(n-2\right)!}\varphi \left(u\right){\mathrm{\Lambda }}_1^{\alpha }\left(u\right)\right]}^{1/\alpha }.$$

Consequently, (21) becomes

$${\left[\varphi \left(u\right)+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\varphi \left(\tau \left(u\right)\right)\right]}^{\prime }\le -\frac{1}{\delta {\left[\left(n-2\right)!\right]}^{\beta /\alpha }}{\varphi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right).$$

(22)

Now, let

$$\psi \left(u\right):=\varphi \left(u\right)+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\varphi \left(\tau \left(u\right)\right).$$

(23)

Since $\tau \left(u\right)\ge u$, we find

$$\psi \left(u\right)\le \left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)\varphi \left(u\right),$$

and so

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\right]}^{\beta /\alpha }}{\left(\frac{1}{1+{\tau }_0^{1-n}{p}_0^{\beta }}\right)}^{\beta /\alpha }{\psi }^{\beta /\alpha }\left(\sigma \left(u\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)\le 0.$$

We note that $\varphi$ is a positive solution to this inequality. Based on Theorem 1 in [48], we also find that Equation (19) has a positive solution, and this contradicts the assumptions of the theorem.

In the case that $\kappa >1$, we find—in the same way as before—that Equation (20) has a positive solution, and this is a contradiction.

The proof is now complete. □

Theorem 4. 

Assume that $\beta \le \alpha$, n is odd, and (H_b) holds. Assume also that the DDEs (19) and (20) are oscillatory. If

$$\underset{u\to \infty }{\lim sup}{\int }_{\sigma \left(u\right)}^u{\left[\frac{1}{r\left(u\right)}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s\right]}^{1/\alpha }\mathrm{d}u>{\left[\frac{\delta \left(n-2\right)!}{M^{\beta -\alpha }}\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)\right]}^{1/\alpha },$$

(24)

for all $M>0$, then (1) oscillates.

Proof. 

Suppose, on the contrary, that $x\in S^{\ast }$. Therefore, there is a $u_1\ge u_0$ such that $x\left(\tau \left(t\right)\right)>0$ and $x\left(\sigma \left(t\right)\right)>0$ for $u\ge u_1$. From Lemma 3, we infer that $z\left(u\right)>0$ and is of degree $\kappa \ge 0$.

First, we find that the oscillation of the DDEs (19) and (20) leads to the exclusion of the possibility that $\kappa \ge 1$, as in the proof of Theorem 3.

Consider $\kappa =0$. As in the proof of Theorem 3, we arrive at (21). Integrating (21) from u to ∞, we obtain

$$\begin{array}{ccc}\hfill \frac{1}{\delta }{\int }_u^{\infty }{z}^{\beta }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s& \le & \varphi \left(u\right)+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\varphi \left(\tau \left(u\right)\right)\hfill \\ & \le & \left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right){\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-2\right)}.\hfill \end{array}$$

Integrating again from u to ∞, we obtain

$$\begin{array}{ccc}\hfill -\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right){\left(r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\right)}^{\left(n-3\right)}& \ge & \frac{1}{\delta }{\int }_u^{\infty }{\int }_v^{\infty }{z}^{\beta }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s\mathrm{d}v\hfill \\ & =& \frac{1}{\delta }{\int }_u^{\infty }\left(s-u\right)z^{\beta }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s.\hfill \end{array}$$

By repeating this procedure $n-3$ times, we conclude that

$$-\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)r\left(u\right){\left(z^{\prime }\left(u\right)\right)}^{\alpha }\ge \frac{1}{\delta \left(n-2\right)!}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}{z}^{\beta }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s,$$

or

$$-{\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)}^{1/\alpha }{z}^{\prime }\left(u\right)\ge \frac{1}{{\left[\delta \left(n-2\right)!\right]}^{1/\alpha }}{\left[\frac{1}{r\left(u\right)}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}{z}^{\beta }\left(\sigma \left(s\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s\right]}^{1/\alpha }$$

Integrating this inequality from $\sigma$ to u, we see that

$$\begin{array}{cc}& {\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)}^{1/\alpha }z\left(\sigma \left(u\right)\right)\hfill \\ \ge & \frac{1}{{\left[\delta \left(n-2\right)!\right]}^{1/\alpha }}{z}^{\beta /\alpha }\left(\sigma \left(u\right)\right){\int }_{\sigma \left(u\right)}^u{\left[\frac{1}{r\left(u\right)}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s\right]}^{1/\alpha }\mathrm{d}u.\hfill \end{array}$$

(25)

Since z is positive decreasing function, there is a constant $M>0$ such that $z\left(u\right)\le M$ for $u\ge u_1$. Thus, $z^{\beta /\alpha }\left(u\right)\ge M^{\beta /\alpha -1}z\left(u\right)$. Therefore, (25) yields

$${\left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)}^{1/\alpha }\ge \frac{M}{{\left[\delta \left(n-2\right)!\right]}^{1/\alpha }}{\int }_{\sigma \left(u\right)}^u{\left[\frac{1}{r\left(u\right)}{\int }_u^{\infty }{\left(s-u\right)}^{n-2}\sum _{i=1}^m{\tilde{q}}_i\left(s\right)\mathrm{d}s\right]}^{1/\alpha }\mathrm{d}u,$$

which contradicts with (24).

The proof is now complete. □

Corollary 3. 

Assume that $\beta =\alpha$, and (H_b) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{\sigma \left(u\right)}^u{\mathrm{\Lambda }}_j^{\alpha }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)>\frac{\delta \left(n-2\right)!}{\mathrm{e}}\left(1+{\tau }_0^{1-n}{p}_0^{\alpha }\right),j=1,2.$$

(26)

Assume also that (24) is achieved for n odd. Then, Equation (1) is oscillatory.

Proof. 

Using Theorem 2 in [49], we find that the conditions in (26) guarantee the oscillation of Equations (19) and (20). Thus, from Theorems 3 and 4, what is required is achieved. □

Corollary 4. 

Assume that $\beta =\alpha$, and (H_b) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{\sigma \left(u\right)}^u\frac{{\left(\sigma \left(s\right)\right)}^{\alpha +n-2}}{r\left(\sigma \left(s\right)\right)}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s>\frac{\left(n-2\right)!{\left(n+\alpha -2\right)}^{\alpha }}{{\alpha }^{\alpha }\mathrm{e}}\left(1+{\tau }_0^{1-n}{p}_0^{\alpha }\right).$$

(27)

Assume also that (24) is achieved for n odd. Then, Equation (1) is oscillatory.

3.3. Comparison Theorems (III)

Theorem 5. 

Assume that n is even, and (H_c) holds. If the DDEs

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\left(1+{\tau }_0^{1-n}{p}_0^{\beta }\right)\right]}^{\beta /\alpha }}{\psi }^{\beta /\alpha }\left({\tau }^{-1}\left(\sigma \left(u\right)\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)=0$$

(28)

and

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\left(1+{\tau }_0^{1-n}{p}_0^{\beta }\right)\right]}^{\beta /\alpha }}{\psi }^{\beta /\alpha }\left({\tau }^{-1}\left(\sigma \left(u\right)\right)\right){\mathrm{\Lambda }}_2^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)\le 0$$

(29)

are oscillatory, then (1) oscillates.

Proof. 

Suppose, on the contrary, that $x\in S^{\ast }$. Therefore, there is a $u_1\ge u_0$ such that $x\left(\tau \left(t\right)\right)>0$ and $x\left(\sigma \left(t\right)\right)>0$ for $u\ge u_1$. From Lemma 3, we infer that $z\left(u\right)>0$ and is of degree $\kappa \ge 1$ (for n even).

Consider $\kappa =1$. As in the proof of Theorem 3, we obtain that (21) and (22) hold. Now, assume that $\psi$ is defined as in (23). Since $\tau \left(t\right)\le t$, we have

$$\psi \left(u\right)\le \left(1+\frac{p_0^{\beta }}{{\tau }_0^{n-1}}\right)\varphi \left(\tau \left(u\right)\right),$$

which with (22) gives

$${\psi }^{\prime }\left(u\right)+\frac{1}{\delta {\left[\left(n-2\right)!\left(1+{\tau }_0^{1-n}{p}_0^{\beta }\right)\right]}^{\beta /\alpha }}{\psi }^{\beta /\alpha }\left({\tau }^{-1}\left(\sigma \left(u\right)\right)\right){\mathrm{\Lambda }}_1^{\beta }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)\le 0.$$

We note that $\varphi$ is a positive solution to this inequality. Based on Theorem 1 in [48], we also find that Equation (28) has a positive solution, and this contradicts the assumptions of the theorem.

In the case that $\kappa >1$, we find—in the same way as before—that Equation (29) has a positive solution, and this is a contradiction.

The proof is now complete. □

Using the approach followed in the previous sections, we directly get the next results:

Theorem 6. 

Assume that n is odd, and (H_c) holds. Assume also that the DDEs (28) and (29) are oscillatory. If (16) for $u_1\ge u_0$, then all solutions of (1) oscillate or tend to zero.

Corollary 5. 

Assume that $\alpha =\beta$, (H_c) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{{\tau }^{-1}\left(\sigma \left(u\right)\right)}^u{\mathrm{\Lambda }}_j^{\alpha }\left(\sigma \left(u\right)\right)\sum _{i=1}^m{\tilde{q}}_i\left(u\right)>\frac{\delta \left(n-2\right)!}{\mathrm{e}}\left(1+{\tau }_0^{1-n}{p}_0^{\alpha }\right),j=1,2.$$

Assume also that (16) is achieved for n odd. Then,

(a) 

(1) oscillates if n even.

(b) 

all solutions of (1) oscillate or tend to zero if n odd.

Corollary 6. 

Assume that $r^{\prime }\left(t\right)\ge 0$, $\alpha =\beta$, (H_c) holds, and

$$\underset{u\to \infty }{\lim \inf }{\int }_{{\tau }^{-1}\left(\sigma \left(u\right)\right)}^u\frac{{\left(\sigma \left(s\right)\right)}^{\alpha +n-2}}{r\left(\sigma \left(s\right)\right)}\sum _{i=1}^m{q}_i\left(s\right)\mathrm{d}s>\frac{\left(n-2\right)!{\left(n+\alpha -2\right)}^{\alpha }}{{\alpha }^{\alpha }\mathrm{e}}\left(1+{\tau }_0^{1-n}{p}_0^{\alpha }\right).$$

Assume also that (16) is achieved for n odd. Then,

(a) 

(1) oscillates if n even.

(b) 

all solutions of (1) oscillate or tend to zero if n odd.

4. Examples and Discussions

In this part, we apply our findings to special cases of the studied equation. We also compare our results with previous relevant results to illustrate their novelty and importance.

Example 1. 

Consider the NDE

$${\left(u^2{\left[{\left(x\left(u\right)+p_{0}x\left(\lambda u\right)\right)}^{\prime }\right]}^3\right)}^{\left(n-1\right)}+\frac{q_0}{u^n}\sum _{i=1}^m{x}^3\left({\mu }_{i}u\right)=0,u\ge 1,$$

(30)

where ${\mu }_i\in \left(0,1\right)$ and $q_0>0$. We see that $\alpha =3,$$r\left(u\right)=u^2,$$\tau \left(u\right)=\lambda u,$${\sigma }_i\left(u\right)={\mu }_{i}u,$ and $q_i\left(u\right)=\frac{q_0}{u^n}$ for $i=1,2,...,m$. Assume that $\mu :=min\left\{{\mu }_i,i=1,2,\dots ,m\right\}$.

For n even, we assume that $\lambda \in \left(0,1\right]$. Applying Corollary 2, we obtain

$$\underset{u\to \infty }{\lim \inf }m{\mu }^{n-1}{q}_0{\int }_{\mu u}^u\frac{\mathrm{d}s}{s}>\frac{\left(n-2\right)!{\left(n+1\right)}^3}{27{\left(1-p_0\right)}^3\mathrm{e}},$$

which ensures the oscillation of Equation (30) if $p_0<1$ and

$$q_0>\frac{\left(n-2\right)!{\left(n+1\right)}^3}{m{\mu }^{n-1}{\left(1-p_0\right)}^3\mathrm{e}ln\left(1/\mu \right)}.$$

(31)

On the other hand, Corollary 6 guarantees the oscillation of Equation (30) if

$$m{q}_0{\mu }^{n-1}\underset{u\to \infty }{\lim \inf }{\int }_{\left(\mu /\lambda \right)u}^u\frac{1}{s}\mathrm{d}s>\frac{\left(n-2\right)!{\left(n+1\right)}^3}{3^3\mathrm{e}}\left(1+{\lambda }^{1-n}{p}_0^3\right),$$

which is achieved when $\lambda >\mu$ and

$$q_0>\frac{\left(n-2\right)!{\left(n+1\right)}^3}{3^{3}m{\mu }^{n-1}\mathrm{e}ln\left(\lambda /\mu \right)}\left(1+{\lambda }^{1-n}{p}_0^3\right).$$

(32)

For n odd, it is easy to see that (16) is satisfied. Then, for $\lambda \in \left(0,1\right]$, all solutions of (30) oscillate or tend to zero. For $\lambda >1$, conditions (24) and (27) reduce, respectively, to

$$q_0{\left[ln\left(\frac{1}{\mu }\right)\right]}^3>\frac{4\left(n-1\right)!}{m}{\lambda }^n\left(1+\frac{p_0^3}{{\lambda }^{n-1}}\right)$$

(33)

and

$$q_{0}ln\left(\frac{1}{\mu }\right)>\frac{\left(n-2\right)!{\left(n+1\right)}^3}{3^3\mathrm{e}m{\mu }^{n-1}}\left(1+\frac{p_0^3}{{\lambda }^{n-1}}\right).$$

(34)

Therefore, using Corollary 4, Equation (30) is oscillatory if (33) and (34) hold.

Remark 2. 

To compare criteria (31) and (32), we consider the following special case of Equation (30):

$${\left(u^2{\left[{\left(x\left(u\right)+p_{0}x\left(\frac{1}{4}u\right)\right)}^{\prime }\right]}^3\right)}^{\prime \prime \prime }+\frac{q_0}{u^4}\sum _{i=1}^2{x}^3\left({\mu }_{i}u\right)=0.$$

(35)

Figure 1 shows the lower bounds of the $q_0$–values in conditions (31) and (32) when $p_0=0.5$ and $\mu \in \left(0,0.25\right)$. Figure 2 shows the lower bounds of the $q_0$–values in conditions (31) and (32) when $\mu =0.2$. We note that criterion (32) provides better results for the oscillation of Equation (35). We also note that condition (32) takes into account the effect of parameter λ, while condition (31) does not. However, Condition (31) is distinguished in that it does not require the constraint $\lambda >\mu$, so it provides criteria in the if case where $\lambda \le \mu$.

Figure 1. The lower bounds of $q_0$-values in conditions (31) and (32) when $p_0=0.5$.

Figure 2. The lower bounds of $q_0$-values in conditions (31) and (32) when $\mu =0.2$.

Example 2. 

Assume the NDE

$${\left[x\left(u\right)+p_{0}x\left(\lambda u\right)\right]}^{\left(n\right)}+\frac{q_0}{u^n}x\left(\mu u\right)=0,u\ge 1,$$

(36)

where $n\ge 4$ is even, $\mu \in \left(0,1\right)$, and $q_0>0$. We see that $\alpha =1$, $m=1$, $r\left(u\right)=1$, $\tau \left(u\right)=\lambda u,$$q\left(u\right)=\frac{q_0}{u^n}$, and $\sigma \left(u\right)=\mu u$.

For $\lambda \in \left(\mu ,1\right]$, from Corollary 6, Equation (36) is oscillatory when

$$q_0>\frac{\left(n-1\right)!}{\mathrm{e}{\mu }^{n-1}ln\left(\lambda /\mu \right)}\left(1+\frac{p_0}{{\lambda }^{n-1}}\right).$$

(37)

For $\lambda >1$, from Corollary 4, Equation (36) is oscillatory when

$$q_0>\frac{\left(n-1\right)!}{\mathrm{e}{\mu }^{n-1}ln\left(1/\mu \right)}\left(1+\frac{p_0}{{\lambda }^{n-1}}\right).$$

(38)

Remark 3. 

Baculikova and Džurina [28], Li and Rogovchenko [30], and Agarwal et al. [50] investigated the oscillatory behavior of solutions of even-order NDEs using different techniques and considered different cases of the functions p and τ.

By applying Corollary 2.1 in [30], Equation (36) is oscillatory if

$$q_0>\frac{6^{n-1}\left(n-1\right)!{\left[p_0{\lambda }^{n-1}\right]}^2}{\mathrm{e}\left({\lambda }^{n-1}{p}_0-1\right)ln\left(\frac{\lambda }{\mu }\right)},$$

(39)

where ${\lambda }^{n-1}{p}_0>1$ and $\lambda >\mu$. Using Corollary 1 in [28], Equation (36) oscillates if

$$q_0>\frac{6\left(\lambda +p_0\right){\lambda }^{n-1}}{\mu \mathrm{e}ln\left(1/\mu \right)},$$

(40)

where $\lambda \ge 1$. While Theorem 2.1 in [50] confirms the oscillation of Equation (36) if

$$q_0>\frac{\left(n-1\right)!\left(n-1\right)}{4\left(1-p_0\right){\mu }^{n-1}},$$

(41)

where $p_0<1$ and $\lambda <1$. Table 1 and Figure 3, Figure 4 and Figure 5 compare our criteria (37) and (38) with criteria (39)–(41) by applying them to three different cases of Equation (36) when $n=4$.

Table 1. Comparison of the oscillation conditions of Equation (36) in three different cases.

	Case (I)		Case (II)		Case (III)
	${\mathit{p}}_{\mathbf{0}}=\mathbf{2},$$\mathit{\lambda }=\mathbf{0.9}$		${\mathit{p}}_{\mathbf{0}}=\mathbf{0.5},$$\mathit{\lambda }=\mathbf{2}$		${\mathit{p}}_{\mathbf{0}}=\mathbf{0.5},$$\mathit{\lambda }=\mathbf{0.9}$
Criterion	(37)	(39)	(38)	(40)	(37)	(41)
	$\frac{\left(5458/243\right)}{\mathrm{e}{\mu }^{3}ln\left(9/10\mu \right)}$	$\frac{6015.3}{\mathrm{e}ln\left(9/10\mu \right)}$	$\frac{51}{8\mathrm{e}{\mu }^{3}ln\left(1/\mu \right)}$	$\frac{120}{\mu \mathrm{e}ln\left(1/\mu \right)}$	$\frac{\left(262/27\right)}{\mathrm{e}{\mu }^{3}ln\left(9/10\mu \right)}$	$\frac{9}{{\mu }^3}$
Preference $\mu$–values	$\left[0.155,\lambda \right)$	$\left(0,0.155\right]$	$\left[0.23,1\right)$	$\left(0,0.23\right]$	$\left(0,0.6\right]$	$\left[0.6,\lambda \right)$
Figures	Figure 3		Figure 4		Figure 5

Figure 3. The lower bounds of $q_0$-values in conditions (37) and (39).

Figure 4. The lower bounds of $q_0$-values in conditions (38) and (40).

Figure 5. The lower bounds of $q_0$-values in conditions (37) and (41).

Remark 4. 

In the case where $\alpha =\beta$ and $m=1$, the results of Section 3.1 and Section 3.2 are reduced to results in [51]. Therefore, our findings are an extension and generalization of the findings in [51]. Moreover, we considered Case (H_c), which was not studied in [51].

5. Conclusions

In this study, oscillatory conditions are constructed for a class of n-order NDEs. By applying the comparison method with first-order DDEs, we were able to create new standards that ensure that every solution of (1) is oscillatory if n is even, or oscillatory or converge to zero if n is odd. We considered three different cases of functions p and $\tau$. Through examples and remarks, we explained the novelty and importance of the results. Accordingly, it can be said that our findings complement, generalize and improve the related results in [28,30,50,51].

It is an interesting problem to develop new conditions that ensure that every solution of (1) is oscillatory in the noncanonical case.