Improved Properties of Positive Solutions of Higher Order Differential Equations and Their Applications in Oscillation Theory

Abstract

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results.

1. Introduction

Delay differential equations (DDE) are differential equations (DE) that take into account the effect of different times. Therefore, they are a better way to model natural phenomena in engineering and physical problems. It is easy to note the recent increase in research into the qualitative theory of DDEs. This is not only due to their practical importance, but also because they are rich in analytical problems and interesting open issues.

One of the most important branches of qualitative theory is oscillation theory, which studies the asymptotic and oscillatory behavior of solutions of DEs. Finding adequate conditions to guarantee that all DE solutions oscillate is one of the main objectives of oscillation theory. Ladas et al. [1] is one of the earliest monographs addressing oscillation theory, including the findings up until 1984. The main objective of this monograph is to investigate the deviating arguments on the oscillation of solutions; neutral delay equations are not discussed in this monograph. The monograph by Gyori and Ladas [2], which made significant contributions to the creation of linearized oscillation theory and the relationship between the oscillation of all solutions and the distribution of the roots of characteristic equations, is one of the key works in the theory of oscillation. Additional topics that are crucial to the theory of oscillation are covered in [3], including determining the conditions for the existence of solutions with particular asymptotic properties and calculating the separation between zeros of oscillatory solutions. The monographs [4,5,6,7,8,9] covered and summarized many of the results known in the literature up to the past ten years for further results, approaches, and references.

In addition to the theoretical importance and many interesting analytical problems, delay differential equations have many vital applications in engineering and physics, as they appear when modeling many phenomena that are fundamentally time dependent. For example, we find that such equations appear in the modeling of electrical networks that contain lossless transmission lines (such as high-speed computers). Understanding the qualitative properties and behavior of equation solutions greatly helps in studying and developing the studied models.

It is easy to notice the research movement that aims to improve and develop the criteria for oscillation of solutions of DDEs, especially of the second-order, which is led by the Slovakian school; see, for example, the papers of Baculíková, Džurina and Jadlovská [10,11,12,13].

Mostly, we find that the study of the oscillation of solutions of DDEs of different orders adopts one of two approaches, either substituting Riccati or comparing with equations of lower orders, often first-order. In 1999, Koplatadze et al. [14], with a different approach than the traditional one, studied the asymptotic and oscillatory behavior of solutions to the DE

$$\frac{{\mathrm{d}}^{n}x}{\mathrm{d}{u}^n}+q·\left[x\circ g\right]=0,$$

(1)

where $u>0$, $n\ge 2$, and $\left[x\circ g\right]\left(u\right)=x\left(g\left(u\right)\right)$. They considered the even- and odd-order of this equation. One of their results was to ensure that solutions of DDE (1) oscillate under the conditions $g\left(u\right)\le u,$ $g^{\prime }\left(u\right)\ge 0$, and

$$\begin{array}{ccc}& & \underset{u\to \infty }{limsup}\left[g\left(u\right){\int }_u^{\infty }{g}^{n-2}\left(\upsilon \right)q\left(\upsilon \right)\mathrm{d}\upsilon +{\int }_{g\left(u\right)}^u{g}^{n-1}\left(\upsilon \right)q\left(\upsilon \right)\mathrm{d}\upsilon \right.\hfill \\ & & \left.+\frac{1}{g\left(u\right)}{\int }_0^{g\left(u\right)}\upsilon g^{n-1}\left(\upsilon \right)q\left(\upsilon \right)\mathrm{d}\upsilon \right]>\left(n-1\right)!.\hfill \end{array}$$

Before and after that, many researchers also verified the oscillation of the higher order DDEs solutions in the canonical case by using traditional methods; see, for example, [15,16,17,18,19,20].

In the non-canonical case, the oscillation conditions for solutions of the DDE

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·{\left(\frac{{\mathrm{d}}^{n-1}x}{\mathrm{d}{u}^{n-1}}\right)}^{\alpha }\right)+q·\left[f\circ x\circ g\right]=0$$

(2)

were established by Baculíková et al. [21] and Moaaz et al. [22] by using the comparison technique. Using the Riccati substitution, Zhang et al. [23] and Moaaz and Muhib [24] studied oscillation of the DDE

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·{\left(\frac{{\mathrm{d}}^{n-1}x}{\mathrm{d}{u}^{n-1}}\right)}^{\alpha }\right)+q·\left[x^{\beta }\circ g\right]=0.$$

The results of the second-order equation were most recently expanded to the even-order equations in the non-canonical case by Moaaz et al. [25]. In order to develop iteratively new oscillation criteria, they developed an approach that involved obtaining new monotonic properties for positive decreasing solutions.

For the neutral equations, which the higher derivative appears on the solution with and without delay, Li and Rogovchenko [26] related oscillation of solutions of the DDE

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·{\left(\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\left(x+p·\left[x\circ h\right]\right)\right)}^{\alpha }\right)+q·\left[x^{\beta }\circ g\right]=0$$

(3)

to three equations of the first-order by using comparison techniques. In [27], a criterion for ruling out the existence of so-called Kneser solutions to DDE (3) was developed. The results in [27] are more accurate and effective than those in [26] since they do not rely on unknown functions. Very recently, Elabbasy et al. [28] studied the asymptotic behavior of the solutions of the DDE (3). Let us review the following theorem, which gives the conditions ensuring that non-oscillatory solutions of (3) tend to zero.

On the other hand, for neutral equations of the second order, the study of oscillation of these equations has been developed with many improved techniques; see, for example, [29,30,31].

In this article, we consider the neutral DDE of the form

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\left(x+p·\left[x\circ h\right]\right)\right)+q·\left[F\circ x\circ g\right]=0,$$

(4)

where $u\ge u_0$, $n\ge 4$ is an even natural number. We also suppose the following:

(H1)

$a,$ $p,$ $q\in C\left(\left[u_0,\infty \right),\left[0,\infty \right)\right),$ $a\left(u\right)>0,$ $p\left(u\right)\le {\mathcal{E}}_0\left(u\right)/{\mathcal{E}}_0\left(h\left(u\right)\right)$ and ${\mathcal{E}}_{n-2}\left(u_0\right)<\infty$, where

$${\mathcal{E}}_0\left(u\right):={\int }_u^{\infty }{a}^{-1}\left(\upsilon \right)\mathrm{d}\upsilon ,\mathrm{and}{\mathcal{E}}_k\left(u\right):={\int }_u^{\infty }{\mathcal{E}}_{k-1}\left(\upsilon \right)\mathrm{d}\upsilon ,$$

for $k=1,2,\dots ,n-2.$

(H2)

$h,$ $g\in C\left(\left[u_0,\infty \right),\mathbb{R}\right),$ $h\left(u\right)\le u,$ $g\left(u\right)\le u,$ $g^{\prime }\left(u\right)>0,$${lim}_{u\to \infty }h\left(u\right)=\infty$ and ${lim}_{u\to \infty }g\left(u\right)=\infty .$

(H3)

$F\in C\left(\mathbb{R},\mathbb{R}\right),$ $F^{\prime }\left(x\right)\ge 0,$ $xF\left(x\right)>0$ for $x\ne 0,$ and $F\left(xy\right)\ge F\left(x\right)F\left(y\right)$ for $xy>0.$

First, we define the corresponding function of solution x as $\omega :=x+p·\left[x\circ h\right]$. By a solution of (4), we mean a function $x\in C^{n-1}\left(\left[u_x,\infty \right),\mathbb{R}\right)$, for $u_x\ge u_0$, which $a·{\omega }^{\left(n-1\right)}\in C^1\left(\left[u_x,\infty \right),\mathbb{R}\right)$ and x satisfies (4) for all $u_x\ge u_0$. We consider only those solutions of (4) that do not eventually vanish. A solution of DDE (4) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. DDE (4) is called oscillatory if all its solutions are oscillatory.

The aim of the study is to provide new conditions for determining the oscillation parameters of all solutions of Equation (4) in the non-canonical case. We also aim to develop the oscillation theorems of higher-order neutral delay differential equations by deriving new oscillation parameters characterized by an iterative nature. The method employed is an extension of the method Koplatadze et al. [14] and, later, by Baculková [10].

2. Preliminary Lemmas

Notation 1. 

The class of all eventually positive solutions of DDE (4) is denoted by the symbol ${\mathcal{P}}_s$.

Notation 2. 

To facilitate the presentation of the results, we define the function $\mathcal{Q}$ as

$$\mathcal{Q}:=q·\left[F\circ \left(1-\left[p\circ g\right]·\frac{\left[{\mathcal{E}}_{n-2}\circ h\circ g\right]}{\left[{\mathcal{E}}_{n-2}\circ g\right]}\right)\right].$$

Lemma 1 

(Lemma 3, [32]). Suppose that $x\in {\mathcal{P}}_s$. Then,

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)\le 0,$$

eventually. Moreover, one of the following conditions is satisfied eventually:

(D1)

$\omega ,$ ${\omega }^{\prime }$ and ${\omega }^{\left(n-1\right)}$ are positive, and ${\omega }^{\left(n\right)}$ is negative;

(D2)

$\omega ,$ ${\omega }^{\prime }$ and ${\omega }^{\left(n-2\right)}$ are positive, and ${\omega }^{\left(n-1\right)}$ is negative;

(D3)

${\left(-1\right)}^m{\omega }^{\left(m\right)}$ are positive, for $m=0,1,\dots ,n-1.$

Lemma 2. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. Then, eventually,

$${\left(-1\right)}^{s+1}\frac{{\mathrm{d}}^s}{\mathrm{d}{u}^s}\omega \le \left[a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right]·{\mathcal{E}}_{n-s-2},$$

(5)

for $s=0,1,\dots ,n-2$.

Proof. 

Since ${\left(a·{\omega }^{\left(n-1\right)}\right)}^{\prime }\le 0,$ ${\omega }^{\left(n-1\right)}\le 0$ and ${\omega }^{\left(n-2\right)}>0$ for all $u\ge u_1,$ where $u_1\ge u_0$, we conclude that

$${\omega }^{\left(n-2\right)}\left(u\right)\ge -{\int }_u^{\infty }\left[a\left(\upsilon \right){\omega }^{\left(n-1\right)}\left(\upsilon \right)\right]a^{-1}\left(\upsilon \right)\mathrm{d}\upsilon \ge -a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right){\mathcal{E}}_0\left(u\right).$$

Integrating this inequality $n-2$ times from u to ∞, we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\left(n-3\right)}\left(u\right)& \le & {\int }_u^{\infty }\left[a\left(\upsilon \right){\omega }^{\left(n-1\right)}\left(\upsilon \right)\right]{\mathcal{E}}_0\left(\upsilon \right)\mathrm{d}\upsilon \hfill \\ & \le & a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right){\int }_u^{\infty }{\mathcal{E}}_0\left(\upsilon \right)\mathrm{d}\upsilon \hfill \\ & =& a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right){\mathcal{E}}_1\left(u\right),\hfill \end{array}$$

and so on until we get

$${\left(-1\right)}^{s+1}{\omega }^{\left(s\right)}\le \left[a·{\omega }^{\left(n-1\right)}\right]·{\mathcal{E}}_{n-s-2},$$

for $s=0,1,\dots ,n-2$. The proof is complete. □

Lemma 3. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. Then,

$${\left(-1\right)}^s\frac{\mathrm{d}}{\mathrm{d}u}\left(\frac{1}{{\mathcal{E}}_{n-s-2}}·\frac{{\mathrm{d}}^s}{\mathrm{d}{u}^s}\omega \right)\ge 0,$$

(6)

for $s=0,1,\dots ,n-2$.

Proof. 

From Lemma 2, we have that (5) holds. Using (5) with $s=n-2$, we find ${\omega }^{\left(n-2\right)}\ge -\left[a·{\omega }^{\left(n-1\right)}\right]·{\mathcal{E}}_0$. Thus,

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(\frac{1}{{\mathcal{E}}_0}·{\omega }^{\left(n-2\right)}\right)=\frac{1}{{\mathcal{E}}_0^2}\left[{\mathcal{E}}_0·{\omega }^{\left(n-1\right)}+a^{-1}·{\omega }^{\left(n-2\right)}\right]\ge 0.$$

Furthermore, we find

$$-{\omega }^{\left(n-3\right)}\left(u\right)\ge {\int }_u^{\infty }\left[\frac{1}{{\mathcal{E}}_0\left(\upsilon \right)}{\omega }^{\left(n-2\right)}\left(\upsilon \right)\right]{\mathcal{E}}_0\left(\upsilon \right)\mathrm{d}\upsilon \ge \frac{1}{{\mathcal{E}}_0\left(u\right)}{\omega }^{\left(n-2\right)}\left(u\right){\mathcal{E}}_1\left(u\right),$$

and so, $-{\mathcal{E}}_0·{\omega }^{\left(n-3\right)}\ge {\mathcal{E}}_1·{\omega }^{\left(n-2\right)}$. This implies

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(\frac{1}{{\mathcal{E}}_1}·{\omega }^{\left(n-3\right)}\right)=\frac{1}{{\mathcal{E}}_1^2}\left[{\mathcal{E}}_1·{\omega }^{\left(n-2\right)}+{\mathcal{E}}_0·{\omega }^{\left(n-3\right)}\right]\le 0.$$

Proceeding in this manner, we arrive at

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(\frac{1}{{\mathcal{E}}_{n-2}}·\omega \right)\ge 0.$$

The proof is complete. □

Lemma 4. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. Then,

$$x\ge \left(1-p·\frac{\left[{\mathcal{E}}_{n-2}\circ h\right]}{{\mathcal{E}}_{n-2}}\right)·\omega ,$$

and

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)+\mathcal{Q}·\left[F\circ \omega \circ g\right]\le 0.$$

(7)

Proof. 

From Lemma 3, we have that (6) holds. Using (6) with $s=0$, we obtain

$$\left[x\circ h\right]\le \left[\omega \circ h\right]\le \frac{\left[{\mathcal{E}}_{n-2}\circ h\right]}{{\mathcal{E}}_{n-2}}·\omega .$$

This implies

$$\begin{array}{ccc}\hfill x& \ge & \left(1-p·\frac{\left[\omega \circ h\right]}{\omega }\right)·\omega \hfill \\ & \ge & \left(1-p·\frac{\left[{\mathcal{E}}_{n-2}\circ h\right]}{{\mathcal{E}}_{n-2}}\right)·\omega .\hfill \end{array}$$

Thus, from (4) and (H3), we get

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)+q·\left[F\circ \left(1-\left[p\circ g\right]·\frac{\left[{\mathcal{E}}_{n-2}\circ h\circ g\right]}{\left[{\mathcal{E}}_{n-2}\circ g\right]}\right)\right]·\left[F\circ \omega \circ g\right]\le 0,$$

or

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)+\mathcal{Q}·\left[F\circ \omega \circ g\right]\le 0.$$

The proof is complete. □

Lemma 5. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. If there is a constant $\alpha >0$ such that, eventually,

$$\mathcal{Q}·{\mathcal{E}}_{n-2}^2\ge \alpha {\mathcal{E}}_{n-3},$$

(8)

then ω converges to zero as $u\to \infty$.

Proof. 

Since $\omega$ is positive and decreasing (from case $\left({\mathrm{D}}_3\right)$), we have that $\omega \left(u\right)\to L$ as $u\to \infty$, where $L\ge 0$.

Suppose that $L>0$. Then, there is $u_1\ge u_0$ such that $\left[\omega \circ g\right]\ge L$ for $u\ge u_1$. Hence, from Lemma 4, we get

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a\left(u\right)\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \left(u\right)\right)\le -\mathcal{Q}\left(u\right)F\left(L\right).$$

Integrating this inequality from $u_1$ to u and using (8), we find

$$\begin{array}{ccc}\hfill a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)& \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-F\left(L\right){\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon \hfill \\ & \le & -\alpha F\left(L\right){\int }_{u_1}^u\frac{{\mathcal{E}}_{n-3}\left(\upsilon \right)}{{\mathcal{E}}_{n-2}^2\left(\upsilon \right)}\mathrm{d}\upsilon \hfill \\ & =& -\alpha F\left(L\right)\left(\frac{1}{{\mathcal{E}}_{n-2}\left(u\right)}-\frac{1}{{\mathcal{E}}_{n-2}\left(u_1\right)}\right).\hfill \end{array}$$

(9)

We note that ${\mathcal{E}}_{n-2}\left(u\right)\to 0$ as $u\to \infty$. Then, for all $ϵ\in \left(0,1\right)$, we have that ${\mathcal{E}}_{n-2}^{-1}\left(u\right)-{\mathcal{E}}_{n-2}^{-1}\left(u_1\right)\ge ϵ{\mathcal{E}}_{n-2}^{-1}\left(u\right)$, eventually. Therefore, (9) becomes

$$a·{\omega }^{\left(n-1\right)}\le -ϵ\alpha F\left(L\right)\frac{1}{{\mathcal{E}}_{n-2}}.$$

(10)

From Lemma 2, we have that (5) holds. Combining (5) at $s=1$ and (10), we conclude that

$$\frac{{\omega }^{\prime }}{{\mathcal{E}}_{n-3}}\le a·{\omega }^{\left(n-1\right)}\le -ϵ\alpha F\left(L\right)\frac{1}{{\mathcal{E}}_{n-2}},$$

or

$${\omega }^{\prime }\le -ϵ\alpha F\left(L\right)\frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}}.$$

Integrating this inequality from $u_1$ to u, we find

$$\omega \left(u\right)\le \omega \left(u_1\right)-ϵ\alpha F\left(L\right)ln\frac{{\mathcal{E}}_{n-2}\left(u_1\right)}{{\mathcal{E}}_{n-2}\left(u\right)}.$$

Thus, $\omega \left(u\right)\to -\infty$ as $u\to \infty$, a contradiction. Then, $\omega \left(u\right)\to 0$ as $u\to \infty$. The proof is complete. □

3. Oscillation Criteria

In this section, we create a criterion that ensures that solutions to Equation (4) oscillate. The next theorem is a restatement of Theorem 2.1 in [28], when $\alpha =\beta$.

Theorem 1 

([28]). Assume that

$${\int }_{u_0}^{\infty }\left({\int }_u^{\infty }{\left(\mu -u\right)}^{n-3}\left(\frac{1}{a\left(\mu \right)}{\int }_{u_1}^{\mu }q\left(\upsilon \right)\mathrm{d}\upsilon \right)\mathrm{d}v\right)\mathrm{d}u=\infty$$

and

$$\underset{u\to \infty }{limsup}{\int }_{u_0}^u\left({\mathcal{E}}_0\left(\upsilon \right)q\left(\upsilon \right)\left(1-p\left(g\left(\upsilon \right)\right)\right)\left(\frac{{\eta }_0{g}^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)-\frac{1}{4a\left(\upsilon \right){\mathcal{E}}_0\left(\upsilon \right)}\right)\mathrm{d}\upsilon =\infty ,$$

(11)

for some constant ${\eta }_0\in \left(0,1\right)$. If the DDE

$$\frac{\mathrm{d}}{\mathrm{d}u}\psi +q·\frac{{\eta }_1\left(1-\left[p\circ g\right]\right)·g^{n-1}\left(u\right)}{\left(n-1\right)!\left[a\circ g]\right]}·\left[\psi \circ g\right]=0$$

(12)

is oscillatory for some constant ${\eta }_1\in \left(0,1\right)$, then every solution of (3), with $\alpha =\beta ,$ is either oscillatory or converges to zero as $u\to \infty$.

Lemma 6. 

Assume that ${lim}_{x\to 0}\frac{x}{F\left(x\right)}=K<\infty$, and (8) holds. If

$$\begin{array}{ccc}& & \underset{u\to \infty }{limsup}\left[{\mathcal{E}}_{n-2}\left(g\left(u\right)\right){\int }_{u_0}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon +{\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon \right.\hfill \\ & & \left.+F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right){\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \right]>K,\hfill \end{array}$$

(13)

then ω satisfies case $\left({\mathrm{D}}_2\right)$ in Lemma 1.

Proof. 

Assume the contrary that $\omega$ satisfies case $\left({\mathrm{D}}_1\right)$ or $\left({\mathrm{D}}_3\right)$.

Assume first that $\omega$ satisfies case $\left({\mathrm{D}}_3\right)$ for $u\ge u_1\ge u_0$. From Lemma 4, we obtain that (7) holds. Integrating (7) from $u_1$ to u, we find

$$\begin{array}{ccc}\hfill -a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)& \ge & -a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)+{\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & \ge & {\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon .\hfill \end{array}$$

(14)

Next, it follows from Lemma 2 that (5) holds. From (5) with $s=0$, we note that the function

$$\omega +\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)·{\mathcal{E}}_{n-2}$$

is positive for $u\ge u_1$. Then, from (5) with $s=1$,

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}}{\mathrm{d}u}\left(\omega +a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-2}\right)& =& {\omega }^{\prime }-a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}+{\left(a·{\omega }^{\left(n-1\right)}\right)}^{\prime }·{\mathcal{E}}_{n-2}\hfill \\ & \le & {\left(a·{\omega }^{\left(n-1\right)}\right)}^{\prime }·{\mathcal{E}}_{n-2},\hfill \end{array}$$

which, with (7), gives

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(\omega +a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-2}\right)\le -{\mathcal{E}}_{n-2}·\mathcal{Q}·\left[F\circ \omega \circ g\right]\le 0.$$

Integrating this inequality from u to ∞, we obtain

$$\omega \left(u\right)+a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right){\mathcal{E}}_{n-2}\left(u\right)\ge {\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon .$$

(15)

Combining (14) and (15), we find

$$\begin{array}{ccc}\hfill \omega \left(u\right)& \ge & -a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right){\mathcal{E}}_{n-2}\left(u\right)+{\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & \ge & {\mathcal{E}}_{n-2}\left(u\right){\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon +{\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon ,\hfill \end{array}$$

and hence,

$$\begin{array}{ccc}\hfill \omega \left(g\left(u\right)\right)& \ge & {\mathcal{E}}_{n-2}\left(g\left(u\right)\right){\int }_{u_1}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon +{\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & & +{\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon .\hfill \end{array}$$

(16)

Using Lemma 3, we obtain that $\omega /{\mathcal{E}}_{n-2}$ is increasing, and so

$$F\left(\omega \left(g\left(\upsilon \right)\right)\right)\ge F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)F\left(\omega \left(g\left(u\right)\right)\right),\mathrm{for}u\le \upsilon .$$

(17)

Thus, (16) reduces to

$$\begin{array}{ccc}\hfill \frac{\omega \left(g\left(u\right)\right)}{F\left(\omega \left(g\left(u\right)\right)\right)}& \ge & {\mathcal{E}}_{n-2}\left(g\left(u\right)\right){\int }_{u_1}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon +{\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon \hfill \\ & & +F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right){\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon ,\hfill \end{array}$$

(18)

which contradicts (13).

Suppose that $\omega$ satisfies case $\left({\mathrm{D}}_1\right)$ for $u\ge u_1\ge u_0$. From (13), we can demonstrate that

$${\int }_{u_1}^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon =\infty ,$$

by following the same procedure as in the proof of Theorem 1 in [10]. Further, it follows from (H1) that

$${\int }_{u_1}^{\infty }\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon =\infty .$$

Since $\omega$ is positive and increasing, we get that $x\ge \left(1-p\right)\omega$, and there is $u_2\ge u_1$ with $\omega \left(g\left(u\right)\right)\ge M$ for $u\ge u_2$. Integrating (4) from $u_1$ to u, we find

$$\begin{array}{ccc}\hfill a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)& \ge & {\int }_{u_1}^{\infty }q\left(\upsilon \right)F\left(x\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & \ge & {\int }_{u_1}^{\infty }q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\omega \left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & \ge & F\left(M\right){\int }_{u_1}^{\infty }q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon .\hfill \end{array}$$

(19)

From the fact that $\frac{\left[{\mathcal{E}}_{n-2}\circ h\right]}{{\mathcal{E}}_{n-2}}\ge 1$, we find

$$q\left(u\right)F\left(1-p\left(g\left(u\right)\right)\right)\ge \mathcal{Q}\left(u\right),$$

which with (19) gives

$$a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)\ge F\left(M\right){\int }_{u_1}^{\infty }\mathcal{Q}\left(\upsilon \right)\mathrm{d}\upsilon ,$$

a contradiction. Therefore, $\omega$ satisfies case $\left({\mathrm{D}}_2\right)$, eventually. The proof is complete. □

Theorem 2. 

Assume that $F\left(x\right)=x$. If conditions (11) and (13) are satisfied, then Equation (4) becomes oscillatory.

Proof. 

Assuming that there is a non-oscillatory solution to (4) necessarily means that there is a solution x of (4), in which $x\in {\mathcal{P}}_s$. From Lemma 1, the derivatives of the function $\omega$ have three possibilities. It follows from Lemma 6 that $\omega$ satisfies case $\left({\mathrm{D}}_2\right)$, eventually. Following the same approach in Theorem 1, we can prove that if $\omega$ fulfills case $\left({\mathrm{D}}_2\right)$, then we get a conflict with condition (11). The proof is complete. □

In addition to the above, we also present the following theorem, which provides a criterion for the oscillation of (4) based on the comparison principle. So, we need to review the following lemma.

Lemma 7 

(Lemma 2.2.3, [33]). Let $\Im \in C^s\left(\left[u_0,\infty \right)\right)$, $\Im \left(u\right)>0$, ${lim}_{u\to \infty }\Im \left(u\right)\ne 0$, ${\Im }^{\left(s\right)}\left(u\right)$ be of constant sign eventually, and ${\Im }^{\left(s\right)}\ne 0$ on a subray of $\left[u_0,\infty \right)$. If ${\Im }^{\left(s-1\right)}\left(u\right){\Im }^{\left(s\right)}\left(u\right)<0$ for $u\ge u_1$, then there is a $u_{\mu }\ge u_1$ such that

$$\Im \left(u\right)\ge \frac{\mu }{\left(s-1\right)!}{u}^{s-1}\left|{\Im }^{\left(s-1\right)}\left(u\right)\right|,$$

for $0<\mu <1$ and $u\in \left[u_{\mu },\infty \right)$.

Theorem 3. 

Assume that ${lim}_{x\to 0}\frac{x}{F\left(x\right)}=K<\infty$, and let (8) and (13) hold. The oscillation of the DDE

$${\psi }^{\prime }\left(u\right)+F\left(\psi \left(g\left(u\right)\right)\right)\frac{1}{a\left(u\right)}{\int }_{u_1}^{u}q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\frac{\mu g^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)\mathrm{d}\upsilon =0,$$

(20)

for some $\mu \in \left(0,1\right)$, ensures the oscillation of Equation (4).

Proof. 

Assuming that there is a non-oscillatory solution to (4) necessarily means that there is a solution x of (4), in which $x\in {\mathcal{P}}_s$. From Lemma 1, the derivatives of the function $\omega$ have three possibilities. It follows from Lemma 6 that $\omega$ satisfies case $\left({\mathrm{D}}_2\right)$, eventually.

Using Lemma 7 with $\Im =\omega$ and $s=n-1$, we obtain

$$\omega \left(u\right)\ge \frac{\mu }{\left(n-2\right)!}{u}^{n-2}{\omega }^{\left(n-2\right)}\left(u\right),$$

(21)

eventually. Since $\omega$ is positive and increasing, we get that $x\ge \left(1-p\right)\omega$, and so, (4) becomes

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)+q·\left[F\circ \left(1-\left[p\circ g\right]\right)\right]·\left[F\circ \omega \circ g\right]\le 0,$$

which, with (21), yields

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(a·\frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{u}^{n-1}}\omega \right)+q·\left[F\circ \left(1-\left[p\circ g\right]\right)\right]·\left[F\circ \frac{\mu g^{n-2}}{\left(n-2\right)!}\right]·\left[F\circ {\omega }^{\left(n-2\right)}\circ g\right]\le 0.$$

Integrating this inequality from $u_1$ to u, we find

$$\begin{array}{ccc}\hfill a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)& \le & -{\int }_{u_1}^{u}q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\frac{\mu g^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)F\left({\omega }^{\left(n-2\right)}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & \le & -F\left({\omega }^{\left(n-2\right)}\left(g\left(u\right)\right)\right){\int }_{u_1}^{u}q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\frac{\mu g^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)\mathrm{d}\upsilon .\hfill \end{array}$$

If we set $\psi ={\omega }^{\left(n-2\right)}>0$, then we have that $\psi$ is a positive solution of

$${\psi }^{\prime }\left(u\right)+F\left(\psi \left(g\left(u\right)\right)\right)\frac{1}{a\left(u\right)}{\int }_{u_1}^{u}q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\frac{\mu g^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)\mathrm{d}\upsilon \le 0.$$

From Theorem 1 in [34], Equation (20) also has a positive solution, which is a contradiction. The proof is complete. □

Corollary 1. 

Assume that ${lim}_{x\to 0}\frac{x}{F\left(x\right)}=K<\infty$, $F\left(x\right)/x\ge 1$ for $\left|x\right|\in \left(0,1\right]$, and let (8) and (13) hold. The fulfillment of the following condition ensures the oscillation of Equation (4):

$$\underset{u\to \infty }{liminf}{\int }_{g\left(u\right)}^u\frac{1}{a\left(s\right)}\left({\int }_{u_1}^{s}q\left(\upsilon \right)F\left(1-p\left(g\left(\upsilon \right)\right)\right)F\left(\frac{s{g}^{n-2}\left(\upsilon \right)}{\left(n-2\right)!}\right)\mathrm{d}\upsilon \right)\mathrm{d}s>\frac{1}{\mathrm{e}}.$$

(22)

Proof. 

From Theorem 2.1.1 in [1], criterion (22) ensures the oscillation of Equation (20). □

4. Criterion of an Iterative Nature

In this section, we create a criterion of an iterative nature that ensures that the solutions to Equation (4) oscillate, when $F\left(x\right)=x$.

Lemma 8. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. If (8) holds, then the function $\omega /{\mathcal{E}}_{n-2}^{\alpha }$ is decreasing and also converges to zero as $u\to \infty$.

Proof. 

From Lemma 4, we have that (7) holds for $u\ge u_1\ge u_0$. Integrating (7) from $u_1$ to u and using (8), we get

$$\begin{array}{ccc}\hfill a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)& \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-{\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)\omega \left(g\left(\upsilon \right)\right)\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\alpha \omega \left(u\right){\int }_{u_1}^u\frac{{\mathcal{E}}_{n-3}\left(\upsilon \right)}{{\mathcal{E}}_{n-2}^2\left(\upsilon \right)}\mathrm{d}\upsilon \hfill \\ & =& a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\alpha \omega \left(u\right)\left(\frac{1}{{\mathcal{E}}_{n-2}\left(u\right)}-\frac{1}{{\mathcal{E}}_{n-2}\left(u_1\right)}\right).\hfill \end{array}$$

(23)

It follows from Lemma 5 that $\omega \left(u\right)\to 0$ as $u\to \infty$. Thus, there is a $u_2\ge u_1$ such that $a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)+\frac{\alpha \omega \left(u\right)}{{\mathcal{E}}_{n-2}\left(u_1\right)}\le 0$ for $u\ge u_2$. Hence, (23) becomes

$$a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)\le -\frac{\alpha }{{\mathcal{E}}_{n-2}\left(u\right)}\omega \left(u\right).$$

(24)

Using Lemma 2 with $s=1$, we get ${\omega }^{\prime }\le a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}$, which with (24) yields

$$\frac{{\omega }^{\prime }\left(u\right)}{{\mathcal{E}}_{n-3}\left(u\right)}\le a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)\le -\frac{\alpha }{{\mathcal{E}}_{n-2}\left(u\right)}\omega \left(u\right).$$

(25)

Hence,

$$\frac{\mathrm{d}}{\mathrm{d}u}\left(\frac{\omega }{{\mathcal{E}}_{n-2}^{\alpha }}\right)=\frac{1}{{\mathcal{E}}_{n-2}^{\alpha +1}}\left[{\mathcal{E}}_{n-2}{\omega }^{\prime }+\alpha {\mathcal{E}}_{n-3}\omega \right]\le 0.$$

Now, we have that $\omega /{\mathcal{E}}_{n-2}^{\alpha }$ is positive and decreasing. Then, $\omega /{\mathcal{E}}_{n-2}^{\alpha }\to \ell$ as $u\to \infty$, where $\ell \ge 0$.

Suppose that $\ell >0$. Hence,

$$\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}\ge \ell .$$

(26)

Next, we define

$$\varphi :=\frac{a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-2}+\omega }{{\mathcal{E}}_{n-2}^{\alpha }},$$

and so

$$\begin{array}{ccc}\hfill {\varphi }^{\prime }& =& \frac{{\left(a·{\omega }^{\left(n-1\right)}\right)}^{\prime }}{{\mathcal{E}}_{n-2}^{\alpha -1}}-\frac{1}{{\mathcal{E}}_{n-2}^{\alpha }}\left(a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}-{\omega }^{\prime }\right)+\alpha \frac{a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha }}+\alpha \frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha +1}}·\omega \hfill \\ & \le & -\frac{1}{{\mathcal{E}}_{n-2}^{\alpha -1}}·\mathcal{Q}·\left[\omega \circ g\right]+\alpha \frac{a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha }}+\alpha \frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha +1}}·\omega .\hfill \end{array}$$

Thus, from (8), we get

$$\begin{array}{ccc}\hfill {\varphi }^{\prime }& \le & -\alpha \frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha +1}}·\left[\omega \circ g\right]+\alpha \frac{a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha }}+\alpha \frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha +1}}·\omega \hfill \\ & \le & \alpha \frac{a·{\omega }^{\left(n-1\right)}·{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha }}.\hfill \end{array}$$

It follows from (25) and (26) that

$$\begin{array}{ccc}\hfill {\varphi }^{\prime }& \le & -{\alpha }^2\frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}^{\alpha +1}}·\omega \left(u\right)\hfill \\ & \le & -\ell {\alpha }^2\frac{{\mathcal{E}}_{n-3}}{{\mathcal{E}}_{n-2}}<0.\hfill \end{array}$$

Integrating this inequality from $u_1$ to u, we find

$$\varphi \left(u_1\right)\ge \ell {\alpha }^{2}ln\frac{{\mathcal{E}}_{n-2}\left(u_1\right)}{{\mathcal{E}}_{n-2}\left(u\right)},$$

which leads to a contradiction. Then, $\ell =0.$ The proof is complete. □

Lemma 9. 

Assume that (8) holds. If

$$\begin{array}{ccc}& & \underset{u\to \infty }{limsup}\left[{\mathcal{E}}_{n-2}\left(g\left(u\right)\right)F\left({\mathcal{E}}_{n-2}^{-\alpha }\left(g\left(u\right)\right)\right){\int }_{u_1}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \right.\hfill \\ & & +F\left({\mathcal{E}}_{n-2}^{-\alpha }\left(g\left(u\right)\right)\right){\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & & \left.+F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right){\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \right]>K,\hfill \end{array}$$

(27)

then ω satisfies case $\left({\mathrm{D}}_2\right)$ in Lemma 1.

Proof. 

Proceeding as in the proof of Lemma 6, we get that (16) and (17) hold. Using Lemma 8, we get

$$\omega \left(g\left(\upsilon \right)\right)\ge \frac{{\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(g\left(u\right)\right)}\omega \left(g\left(u\right)\right)\mathrm{for}\upsilon \le u.$$

(28)

From (17) and (28), (16) becomes

$$\begin{array}{ccc}\hfill \frac{\omega \left(g\left(u\right)\right)}{F\left(\omega \left(g\left(u\right)\right)\right)}& \ge & {\mathcal{E}}_{n-2}\left(g\left(u\right)\right)F\left({\mathcal{E}}_{n-2}^{-\alpha }\left(g\left(u\right)\right)\right){\int }_{u_1}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & & +F\left({\mathcal{E}}_{n-2}^{-\alpha }\left(g\left(u\right)\right)\right){\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & & +F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right){\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon ,\hfill \end{array}$$

which contradicts (27). □

Theorem 4. 

Assume that (8), (22) and (27) hold. Then, Equation (4) is oscillatory.

It is also possible to continue to improve the monotonic property of the function $\omega /{\mathcal{E}}_{n-2}$ and then use it in the oscillation criteria.

Notation 3. 

Since ${\mathcal{E}}_{n-2}\left(u\right)$ is decreasing, there is a $ϵ>1$ such that

$$\frac{\left[{\mathcal{E}}_{n-2}\circ g\right]}{{\mathcal{E}}_{n-2}}\ge ϵ.$$

Let ${\alpha }_0\in \left(0,1\right)$, we define ${\alpha }_0=\alpha$

$${\alpha }_{k+1}:={\alpha }_0\frac{{ϵ}^{{\alpha }_k}}{1-{\alpha }_k},$$

for $k=0,1,\dots$.

Lemma 10. 

Suppose that $x\in {\mathcal{P}}_s$ and ω satisfies case $\left({\mathrm{D}}_3\right)$ in Lemma 1. Suppose also that there is $m\in \mathbb{N}$ such that ${\alpha }_k\in \left(0,1\right)$ and ${\alpha }_{k-1}<{\alpha }_k$ for $k=0,1,\dots ,m$. If (8) holds, then the function $\omega /{\mathcal{E}}_{n-2}^m$ is decreasing, and also converges to zero as $u\to \infty$.

Proof. 

From Lemma 8, we have that $\omega /{\mathcal{E}}_{n-2}^{\alpha }$ is decreasing and also converges to zero as $u\to \infty$. We will prove the required when $m=1$.

From Lemma 4, we have that (7) holds for $u\ge u_1\ge u_0$. Integrating (7) from $u_1$ to u and using the fact that $\omega /{\mathcal{E}}_{n-2}^{\alpha }$ is decreasing, we get

$$\begin{array}{ccc}\hfill a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)& \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-{\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)\omega \left(g\left(\upsilon \right)\right)\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-{\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right)\frac{{\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(\upsilon \right)}\omega \left(\upsilon \right)\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}{\int }_{u_1}^u\mathcal{Q}\left(\upsilon \right){\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}{\int }_{u_1}^u\frac{\alpha {\mathcal{E}}_{n-3}\left(\upsilon \right)}{{\mathcal{E}}_{n-2}^{2-\alpha }\left(\upsilon \right)}\frac{{\mathcal{E}}_{n-2}^{\alpha }\left(g\left(\upsilon \right)\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(\upsilon \right)}\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\alpha {ϵ}^{\alpha }\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}{\int }_{u_1}^u\frac{{\mathcal{E}}_{n-3}\left(\upsilon \right)}{{\mathcal{E}}_{n-2}^{2-\alpha }\left(\upsilon \right)}\mathrm{d}\upsilon \hfill \\ & \le & a\left(u_1\right){\omega }^{\left(n-1\right)}\left(u_1\right)-\frac{\alpha {ϵ}^{\alpha }}{1-\alpha }\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}\left(\frac{1}{{\mathcal{E}}_{n-2}^{1-\alpha }\left(u\right)}-\frac{1}{{\mathcal{E}}_{n-2}^{1-\alpha }\left(u_1\right)}\right).\hfill \end{array}$$

(29)

It follows from Lemma 5 that $\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}^{\alpha }\left(u\right)}\to 0$ as $u\to \infty$. Hence, (29) becomes

$$a\left(u\right){\omega }^{\left(n-1\right)}\left(u\right)\le -\frac{\alpha {ϵ}^{\alpha }}{1-\alpha }\frac{\omega \left(u\right)}{{\mathcal{E}}_{n-2}\left(u\right)}.$$

The remainder of the proof has not been considered because it is identical to the proof of Lemma 8. □

Theorem 5. 

Assume that (8) and (22) hold. If there is $m\in \mathbb{N}$ such that ${\alpha }_k\in \left(0,1\right)$ and ${\alpha }_{k-1}<{\alpha }_k$ for $k=0,1,\dots ,m$, and

$$\begin{array}{ccc}& & \underset{u\to \infty }{limsup}\left[{\mathcal{E}}_{n-2}\left(g\left(u\right)\right)F\left({\mathcal{E}}_{n-2}^{-{\alpha }_m}\left(g\left(u\right)\right)\right){\int }_{u_1}^{g\left(u\right)}\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{{\alpha }_m}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \right.\hfill \\ & & +F\left({\mathcal{E}}_{n-2}^{-{\alpha }_m}\left(g\left(u\right)\right)\right){\int }_{g\left(u\right)}^u{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}^{{\alpha }_m}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \hfill \\ & & \left.+F\left({\mathcal{E}}_{n-2}^{-1}\left(g\left(u\right)\right)\right){\int }_u^{\infty }{\mathcal{E}}_{n-2}\left(\upsilon \right)\mathcal{Q}\left(\upsilon \right)F\left({\mathcal{E}}_{n-2}\left(g\left(\upsilon \right)\right)\right)\mathrm{d}\upsilon \right]>K,\hfill \end{array}$$

then Equation (4) is oscillatory.

5. Conclusions

Our aim in this article was to extend the approach taken in [14] to neutral equations and also to the non-canonical case. The study of the non-canonical case contains more analytical difficulties than the canonical case due to the possibility of the existence of positive decreasing solutions.

We deduced some asymptotic and monotonic properties of the positive solutions whose corresponding function is in class ${\mathcal{P}}_s$. Then, we created new oscillation parameters depending on the inferred characteristics. In addition, we iteratively derived these properties, so that it allows them to be applied more than once in the case of failure at the beginning. The results obtained in this article are characterized by the fact that they do not require the existence of unknown functions, unlike the results in [26] that require this. In addition, our results do not need the conditions $h\circ g=g\circ h$ and h is nondecreasing, which are necessary conditions for the results in [28].

Extending our results in this study to the nonlinear case of the investigated equation would be very interesting. This is due to many analytical difficulties that must be addressed to obtain improved monotonic properties in the nonlinear case.