Oscillation Results for Higher Order Differential Equations

Abstract

The objective of our research was to study asymptotic properties of the class of higher order differential equations with a p-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided.

1. Introduction

In this work, we are concerned with oscillations of higher-order differential equations with a p-Laplacian-like operator of the form

$${\left(r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-2}{y}^{\left(n-1\right)}\left(t\right)\right)}^{\prime }+q\left(t\right){\left|y\left(\tau \left(t\right)\right)\right|}^{p-2}y\left(\tau \left(t\right)\right)=0.$$

(1)

We assume that $p>1$ is a constant,$r\in C^1\left([t_0,\infty ),\mathbb{R}\right),$$r\left(t\right)>0,q,\tau \in C\left([t_0,\infty ),\mathbb{R}\right),$$q>0,\tau \left(t\right)\le t,$${lim}_{t\to \infty }\tau \left(t\right)=\infty$ and the condition

$$\eta \left(t_0\right)=\infty ,$$

(2)

where

$$\eta \left(t\right):={\int }_t^{\infty }\frac{ds}{r^{1/\left(p-1\right)}\left(s\right)}.$$

By a solution of (1) we mean a function y$\in C^{n-1}[T_y,\infty ),T_y\ge t_0,$which has the property $r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-2}{y}^{\left(n-1\right)}\left(t\right)\in C^1[T_y,\infty ),$and satisfies (1) on $[T_y,\infty )$. We consider only those solutions y of (1) which satisfy $sup\{\left|y\left(t\right)\right|:t\ge T\}>0,$for all $T>T_y.$A solution of (1) is called oscillatory if it has arbitrarily large number of zeros on $[T_y,\infty ),$ and otherwise it is called to be nonoscillatory; (1) is said to be oscillatory if all its solutions are oscillatory.

In recent decades, there has been a lot of research concerning the oscillation of solutions of various classes of differential equations; see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

It is interesting to study Equation (1) since the p-Laplace differential equations have applications in continuum mechanics [14,25]. In the following, we briefly review some important oscillation criteria obtained for higher-order equations, which can be seen as a motivation for this paper.

Elabbasy et al. [26] proved that the equation

$${\left(r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-2}{y}^{\left(n-1\right)}\left(t\right)\right)}^{\prime }+q\left(t\right)f\left(y\left(\tau \left(t\right)\right)\right)=0,$$

is oscillatory, under the conditions

$${\int }_{t_0}^{\infty }\frac{1}{r^{{}^{p-1}}\left(t\right)}dt=\infty ;$$

additionally,

$${\int }_{{\mathrm{𝓁}}_0}^{\infty }\left(\psi \left(s\right)-\frac{1}{p^p}{\varphi }^p\left(s\right)\frac{{\left(\left(n-1\right)!\right)}^{p-1}\rho \left(s\right)a\left(s\right)}{{\left(\left(p-1\right)\mu s^{n-1}\right)}^{p-1}}-\frac{\left(p-1\right)\rho \left(s\right)}{a^{1/\left(p-1\right)}\left(s\right){\eta }^p\left(s\right)}\right)ds=+\infty ,$$

for some constant $\mu \in \left(0,1\right)$and

$${\int }_{{\mathrm{𝓁}}_0}^{\infty }kq\left(s\right)\frac{\tau {\left(s\right)}^{p-1}}{s^{p-1}}ds=\infty .$$

Agarwal et al. [2] studied the oscillation of the higher-order nonlinear delay differential equation

$${\left[{\left|y^{\left(n-1\right)}\left(t\right)\right|}^{\alpha -1}{y}^{\left(n-1\right)}\left(t\right)\right]}^{\prime }+q\left(t\right){\left|y\left(\tau \left(t\right)\right)\right|}^{\alpha -1}y\left(\tau \left(t\right)\right)=0.$$

where $\alpha$ is a positive real number. In [27], Zhang et al. studied the asymptotic properties of the solutions of equation

$${\left[r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right]}^{\prime }+q\left(t\right)y^{\beta }\left(\tau \left(t\right)\right)=0t\ge t_0.$$

where $\alpha$ and $\beta$ are ratios of odd positive integers, $\beta \le \alpha$ and

$${\int }_{t_0}^{\infty }{r}^{-1/\alpha }\left(s\right)\mathrm{d}s<\infty .$$

(3)

In this work, by using the Riccati transformations, the integral averaging technique and comparison principles, we establish a new oscillation criterion for a class of higher-order neutral delay differential Equations (1). This theorem complements and improves results reported in [26]. An illustrative example is provided.

In the sequel, all occurring functional inequalities are assumed to hold eventually; that is, they are satisfied for all $t$large enough.

2. Main Results

In this section, we establish some oscillation criteria for Equation (1). For convenience, we denote that $F_{+}\left(t\right):=max\left\{0,F\left(t\right)\right\},$

$$B\left(t\right):=\frac{1}{\left(n-4\right)!}{\int }_t^{\infty }{\left(\theta -t\right)}^{n-4}{\left(\frac{{\int }_{\theta }^{\infty }q\left(s\right){\left(\frac{\tau \left(s\right)}{s}\right)}^{p-1}ds}{r\left(\theta \right)}\right)}^{1/\left(p-1\right)}d\theta$$

and

$$D\left(s\right):=\frac{r\left(s\right)\delta \left(s\right){\left|h\left(t,s\right)\right|}^p}{p^p{\left[H\left(t,s\right)A\left(s\right)\mu \frac{s^{n-2}}{\left(n-2\right)!}\right]}^{p-1}}.$$

We begin with the following lemmas.

Lemma 1

(Agarwal [1]). Let $y\left(t\right)\in C^m\left[t_0,\infty \right)$ be of constant sign and $y^{\left(m\right)}\left(t\right)\ne 0$ on $\left[t_0,\infty \right)$ which satisfies $y\left(t\right)y^{\left(m\right)}\left(t\right)\le 0.$Then,

$\left(\mathbf{I}\right)$ There exists a $t_1\ge t_0$ such that the functions $y^{\left(i\right)}\left(t\right),i=1,2,\dots ,m-1$ are of constant sign on $\left[t_0,\infty \right);$

$\left(\mathbf{II}\right)$ There exists a number $k\in \left\{1,3,5,\dots ,m-1\right\}$ when m is even, $k\in \{0,2,4,\dots ,m-1\}$ when m is odd, such that, for$t\ge t_1$,

$$y\left(t\right)y^{\left(i\right)}\left(t\right)>0,$$

for all $i=0,1,\dots ,k$ and

$${\left(-1\right)}^{m+i+1}y\left(t\right)y^{\left(i\right)}\left(t\right)>0,$$

for all $i=k+1,\dots ,m.$

Lemma 2

(Kiguradze [15]). If the function y satisfies $y^{\left(j\right)}>0$for all $j=0,1,\dots ,m,$ and $y^{\left(m+1\right)}<0,$ then

$$\frac{m!}{t^m}y\left(t\right)-\frac{\left(m-1\right)!}{t^{m-1}}{y}^{\prime }\left(t\right)\ge 0.$$

Lemma 3

(Bazighifan [7]). Let $h\in C^m\left(\left[t_0,\infty \right),\left(0,\infty \right)\right).$ Suppose that $h^{\left(m\right)}\left(t\right)$ is of a fixed sign, on $\left[t_0,\infty \right)$, $h^{\left(m\right)}\left(t\right)$ not identically zero, and that there exists a $t_1\ge t_0$ such that, for all $t\ge t_1,$

$$h^{\left(m-1\right)}\left(t\right)h^{\left(m\right)}\left(t\right)\le 0.$$

If we have $\underset{t\to \infty }{\lim }h\left(t\right)\ne 0,$ then there exists $t_{\lambda }\ge t_0$ such that

$$h\left(t\right)\ge \frac{\lambda }{\left(m-1\right)!}{t}^{m-1}\left|h^{\left(m-1\right)}\left(t\right)\right|,$$

for every $\lambda \in \left(0,1\right)$ and $t\ge t_{\lambda }$.

Lemma 4.

Let $n\ge 4$be even, and assume that y is an eventually positive solution of Equation (1). If (2) holds, then there exists two possible cases for$t\ge t_1,$where$t_1\ge t_0$is sufficiently large:

$$\begin{array}{cc}\left(C_1\right)\hfill & y^{\prime }\left(t\right)>0,y^{″}\left(t\right)>0,y^{\left(n-1\right)}\left(t\right)>0,y^{\left(n\right)}\left(t\right)<0,\\ \left(C_2\right)\hfill & y^{\left(j\right)}\left(t\right)>0,y^{(j+1)}\left(t\right)<0foralloddinteger\hfill \\ \hfill & j\in \{1,2,...,n-3\},y^{\left(n-1\right)}\left(t\right)>0,y^{\left(n\right)}\left(t\right)<0.\end{array}$$

Proof. 

Let y be an eventually positive solution of Equation (1). By virtue of (1), we get

$${\left(r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-2}{y}^{\left(n-1\right)}\left(t\right)\right)}^{\prime }<0.$$

(4)

From ([11] Lemma 4), we have that $y^{\left(n-1\right)}\left(t\right)>0$ eventually. Then, we can write (4) in the from

$${\left(r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}\right)}^{\prime }<0,$$

which gives

$$r^{\prime }\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}+r\left(t\right)\left(p-1\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-2}{y}^{\left(n\right)}\left(t\right)<0.$$

Thus, $y^{\left(n\right)}\left(t\right)<0$ eventually. Thus, by Lemma 1, we have two possible cases $\left(C_1\right)$and$\left(C_2\right)$. This completes the proof. □

Lemma 5.

Let y be an eventually positive solution of Equation (1) and assume that Case$\left(C_1\right)$ holds. If

$$\omega \left(t\right):=\delta \left(t\right)\left(\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}}{y^{p-1}\left(t\right)}\right),$$

(5)

where $\delta \in C^1\left(\left[t_0,\infty \right),\left(0,\infty \right)\right),$then

$${\omega }^{\prime }\left(t\right)\le \frac{{\delta }_{+}^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)-\delta \left(t\right)q\left(t\right){\left(\frac{{\tau }^{n-1}\left(t\right)}{t^{n-1}}\right)}^{p-1}-\frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!{\left(\delta \left(t\right)r\left(t\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(t\right).$$

(6)

Proof. 

Let y be an eventually positive solution of Equation (1) and assume that Case $\left(C_1\right)$ holds. From the definition of $\omega$, we see that $\omega \left(t\right)>0$for$t\ge t_1,$and

$$\begin{array}{cc}{\omega }^{\prime }\left(t\right)\le \hfill & {\delta }^{\prime }\left(t\right)\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}}{y^{p-1}\left(t\right)}+\delta \left(t\right)\frac{{\left(r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}\right)}^{\prime }}{y^{p-1}\left(t\right)}\hfill \\ & -\delta \left(t\right)\frac{\left(p-1\right)y^{\prime }\left(t\right)r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}}{y^p\left(t\right)}.\hfill \end{array}$$

Using Lemma 3 with $m=n-1,h\left(t\right)=y^{\prime }\left(t\right)$, we get

$$y^{\prime }\left(t\right)\ge \frac{\mu }{\left(n-2\right)!}{t}^{n-2}{y}^{\left(n-1\right)}\left(t\right),$$

(7)

for every constant $\mu \in (0,1)$.From (5) and (7), we obtain

$$\begin{array}{ccc}{\omega }^{\prime }\left(t\right)\hfill & \le \hfill & {\delta }^{\prime }\left(t\right)\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}}{y^{p-1}\left(t\right)}+\delta \left(t\right)\frac{{\left(r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}\right)}^{\prime }}{y^{p-1}\left(t\right)}\hfill \\ \hfill & \hfill & -\delta \left(t\right)\frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!}\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^p}{y^p\left(t\right)}.\hfill \end{array}$$

(8)

By Lemma 2, we have

$$\frac{y\left(t\right)}{y^{\prime }\left(t\right)}\ge \frac{t}{n-1}.$$

Integrating this inequality from $\tau \left(t\right)$ to t, we obtain

$$\frac{y\left(\tau \left(t\right)\right)}{y\left(t\right)}\ge \frac{{\tau }^{n-1}\left(t\right)}{t^{n-1}}.$$

(9)

Combining (1) and (8), we get

$$\begin{array}{ccc}{\omega }^{\prime }\left(t\right)\hfill & \le \hfill & {\delta }^{\prime }\left(t\right)\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^{p-1}}{y^{p-1}\left(t\right)}-\delta \left(t\right)\frac{q\left(t\right)\left(y^{\left(p-1\right)}\left(\tau \left(t\right)\right)\right)}{y^{p-1}\left(t\right)}\hfill \\ \hfill & \hfill & -\delta \left(t\right)\frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!}\frac{r\left(t\right){\left|\left(y^{\left(n-1\right)}\left(t\right)\right)\right|}^p}{y^p\left(t\right)}.\hfill \end{array}$$

(10)

From (9) and (10), we obtain

$${\omega }^{\prime }\left(t\right)\le \frac{{\delta }_{+}^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)-\delta \left(t\right)q\left(t\right){\left(\frac{{\tau }^{n-1}\left(t\right)}{t^{n-1}}\right)}^{p-1}-\frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!{\left(\delta \left(t\right)r\left(t\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(t\right).$$

(11)

It follows from (11) that

$$\delta \left(t\right)q\left(t\right){\left(\frac{{\tau }^{n-1}\left(t\right)}{t^{n-1}}\right)}^{p-1}\le \frac{{\delta }_{+}^{\prime }\left(t\right)}{\delta \left(t\right)}\omega \left(t\right)-{\omega }^{\prime }\left(t\right)-\frac{\left(p-1\right)\mu t^{n-2}}{\left(n-2\right)!{\left(\delta \left(t\right)r\left(t\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(t\right).$$

This completes the proof. □

Lemma 6.

Let y be an eventually positive solution of Equation (1) and assume that Case$\left(C_2\right)$ holds. If

$$\psi \left(t\right):=\sigma \left(t\right)\frac{y^{\prime }\left(t\right)}{y\left(t\right)},$$

(12)

where $\sigma \in C^1\left(\left[t_0,\infty \right),\left(0,\infty \right)\right),$then

$$\sigma \left(t\right)B\left(t\right)\le -{\psi }^{\prime }\left(t\right)+\frac{{\sigma }^{\prime }\left(t\right)}{\sigma \left(t\right)}\psi \left(t\right)-\frac{1}{\sigma \left(t\right)}{\psi }^2\left(t\right).$$

(13)

Proof. 

Let y be an eventually positive solution of Equation (1) and assume that Case $\left(C_2\right)$ holds. Using Lemma 2, we obtain

$$y\left(t\right)\ge t{y}^{\prime }\left(t\right).$$

Thus we find that $y/t$is nonincreasing, and hence

$$y\left(\tau \left(t\right)\right)\ge y\left(t\right)\frac{\tau \left(t\right)}{t}.$$

(14)

Since $y>0$, (1) becomes

$${\left(r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}\right)}^{\prime }+q\left(t\right)y^{p-1}\left(\tau \left(t\right)\right)=0.$$

Integrating that equation from t to ∞, we see that

$$\underset{t\to \infty }{lim}\left(r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}\right)-r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}+{\int }_t^{\infty }q\left(s\right)y^{p-2}\left(\tau \left(s\right)\right)=0.$$

(15)

Since the function $r{\left(y^{\left(n-1\right)}\right)}^{p-1}$ is positive $\left[r>0\mathrm{and}{y}^{\left(n-1\right)}>0\right]$ and nonincreasing $\left({\left(r{\left(y^{\left(n-1\right)}\right)}^{p-1}\right)}^{\prime }<0\right)$, there exists a $t_2\ge t_0$ such that $r{\left(y^{\left(n-1\right)}\right)}^{p-1}$ is bounded above for all $t\ge t_2$, and so $\underset{t\to \infty }{lim}\left(r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}\right)=c\ge 0.$ Then, from (15), we obtain

$$-r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}+{\int }_t^{\infty }q\left(s\right)y^{p-2}\left(\tau \left(s\right)\right)\le -c\le 0.$$

From (14), we obtain

$$-r\left(t\right){\left(y^{\left(n-1\right)}\left(t\right)\right)}^{p-1}+{\int }_t^{\infty }q\left(s\right)y{\left(s\right)}^{p-1}\frac{\tau {\left(s\right)}^{p-1}}{s^{p-1}}ds\le 0.$$

It follows from $y^{\prime }\left(t\right)>0$that

$$-y^{\left(n-1\right)}\left(t\right)+\frac{y\left(t\right)}{r^{1/\left(p-1\right)}\left(t\right)}{\left({\int }_t^{\infty }q\left(s\right){\left(\frac{\tau \left(s\right)}{s}\right)}^{p-1}ds\right)}^{1/\left(p-1\right)}\le 0.$$

Integrating the above inequality from t to $\infty$for a total of $\left(n-3\right)$ times, we get

$$y^{″}\left(t\right)+\frac{{\int }_t^{\infty }{\left(\theta -t\right)}^{n-4}{\left(\frac{{\int }_{\theta }^{\infty }q\left(s\right){\left(\frac{\tau \left(s\right)}{s}\right)}^{p-1}ds}{r\left(\theta \right)}\right)}^{1/\left(p-1\right)}d\theta }{\left(n-4\right)!}y\left(t\right)\le 0.$$

(16)

From the definition of $\psi \left(t\right)$, we see that $\psi \left(t\right)>0$for$t\ge t_1,$and

$${\psi }^{\prime }\left(t\right)={\sigma }^{\prime }\left(t\right)\frac{y^{\prime }\left(t\right)}{y\left(t\right)}+\sigma \left(t\right)\frac{y^{″}\left(t\right)y\left(t\right)-{\left(y^{\prime }\left(t\right)\right)}^2}{y^2\left(t\right)}.$$

(17)

It follows from (16) and (17) that

$$\sigma \left(t\right)B\left(t\right)\le -{\psi }^{\prime }\left(t\right)+\frac{{\sigma }^{\prime }\left(t\right)}{\sigma \left(t\right)}\psi \left(t\right)-\frac{1}{\sigma \left(t\right)}{\psi }^2\left(t\right).$$

This completes the proof. □

Definition 1.

Let

$$D=\{\left(t,s\right)\in {\mathbb{R}}^2:t\ge s\ge t_0\}\mathrm{and}{D}_0=\{\left(t,s\right)\in {\mathbb{R}}^2:t>s\ge t_0\}.$$

We say that a function $H\in C\left(D,\mathbb{R}\right)$ belongs to the class ℜ if

$\left({\mathbf{i}}_1\right)H\left(t,t\right)=0$for $t\ge t_0,H\left(t,s\right)>0,\left(t,s\right)\in D_0.$

$\left({\mathbf{i}}_2\right)H$has a nonpositive continuous partial derivative $\partial H/\partial s$ on $D_0$ with respect to the second variable.

Theorem 1.

Let $n\ge 4$be even. Assume that there exist functions $H,H_{*}\in \Re ,$$\delta ,A,\sigma ,A_{*}\in C^1\left(\left[t_0,\infty \right),\left(0,\infty \right)\right)$ and $h,h_{*}\in C\left(D_0,\mathbb{R}\right)$ such that

$$-\frac{\partial }{\partial s}\left(H\left(t,s\right)A\left(s\right)\right)=H\left(t,s\right)A\left(s\right)\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}+h\left(t,s\right).$$

(18)

and

$$-\frac{\partial }{\partial s}\left(H_{*}\left(t,s\right)A_{*}\left(s\right)\right)=H_{*}\left(t,s\right)A_{*}\left(s\right)\frac{{\sigma }^{\prime }\left(t\right)}{\sigma \left(t\right)}+h_{*}\left(t,s\right).$$

(19)

If

$$\underset{t\to \infty }{limsup}\frac{1}{H\left(t,t_0\right)}{\int }_{t_0}^t\left[H\left(t,s\right)A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}-D\left(s\right)\right]ds=\infty ,$$

(20)

for some constant $\mu \in \left(0,1\right)$and

$$\underset{t\to \infty }{limsup}\frac{1}{H_{*}\left(t,t_0\right)}{\int }_{t_0}^t\left(H_{*}\left(t,s\right)A_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)-\frac{\sigma \left(s\right){\left|h_{*}\left(t,s\right)\right|}^2}{4H_{*}\left(t,s\right)A_{*}\left(s\right)}\right)ds=\infty ,$$

(21)

then every solution of (1) is oscillatory.

Proof. 

Let y be a nonoscillatory solution of Equation (1) on the interval $\left[t_0,\infty \right)$. Without loss of generality, we can assume that $y$is an eventually positive. By Lemma 4, there exist two possible cases for$t\ge t_1,$where$t_1\ge t_0$is sufficiently large.

Assume that $\left(C_1\right)$ holds. From Lemma 5, we get that (6) holds. Multiplying (6) by $H\left(t,s\right)A\left(s\right)$ and integrating the resulting inequality from $t_1$to t, we have

$$\begin{array}{cc}{\int }_{t_1}^{t}H\left(t,s\right)\hfill & A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}ds\hfill \\ & \le -{\int }_{t_1}^{t}H\left(t,s\right)A\left(s\right){\omega }^{\prime }\left(s\right)ds+{\int }_{t_1}^{t}H\left(t,s\right)A\left(s\right)\frac{{\delta }^{\prime }\left(s\right)}{\delta \left(s\right)}\omega \left(s\right)ds\hfill \\ & -{\int }_{t_1}^{t}H\left(t,s\right)A\left(s\right)\frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)!{\left(\delta \left(s\right)r\left(s\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(s\right)ds\hfill \end{array}$$

Thus

$$\begin{array}{cc}{\int }_{t_1}^t\hfill & H\left(t,s\right)A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}ds\hfill \\ & \le H\left(t,t_1\right)A\left(t_1\right)\omega \left(t_1\right)-{\int }_{t_1}^t\left(-\frac{\partial }{\partial s}\left(H\left(t,s\right)A\left(s\right)\right)-H\left(t,s\right)A\left(s\right)\frac{{\delta }^{\prime }\left(t\right)}{\delta \left(t\right)}\right)\omega \left(s\right)ds\hfill \\ & -{\int }_{t_1}^{t}H\left(t,s\right)A\left(s\right)\frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)!{\left(\delta \left(s\right)r\left(s\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(s\right)ds\hfill \end{array}$$

This implies

$$\begin{array}{cc}{\int }_{t_1}^{t}H(t,s\hfill & )A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}ds\hfill \\ & \le H\left(t,t_1\right)A\left(t_1\right)\omega \left(t_1\right)+{\int }_{t_1}^t\left|h\left(t,s\right)\right|\omega \left(s\right)d\left(s\right)\hfill \\ & -{\int }_{t_1}^{t}H\left(t,s\right)A\left(s\right)\frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)!{\left(\delta \left(s\right)r\left(s\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\left(s\right)ds.\hfill \end{array}$$

(22)

Using the inequality

$$\beta U{V}^{\beta -1}-U^{\beta }\le \left(\beta -1\right)V^{\beta },\beta >1,U\ge 0\mathrm{and}V\ge 0,$$

(23)

with $\beta =p/\left(p-1\right),$

$$U={\left(\left(p-1\right)H\left(t,s\right)A\left(s\right)\frac{\mu s^{n-2}}{\left(n-2\right)!}\right)}^{\left(p-1\right)/p}\frac{\omega \left(s\right)}{{\left(\delta \left(s\right)r\left(s\right)\right)}^{1/p}}$$

and

$$V={\left(\frac{p-1}{p}\right)}^{p-1}{\left|h\left(t,s\right)\right|}^{p-1}{\left(\frac{\delta \left(s\right)r\left(s\right)}{{\left(\left(p-1\right)H\left(t,s\right)A\left(s\right)\frac{\mu s^{n-2}}{\left(n-2\right)!}\right)}^{p-1}}\right)}^{\left(p-1\right)/p},$$

we get

$$\begin{array}{cc}\left|h\left(t,s\right)\right|\omega \left(s\right)-H\left(t,s\right)A\left(s\right)\hfill & \frac{\left(p-1\right)\mu s^{n-2}}{\left(n-2\right)!{\left(\delta \left(s\right)r\left(s\right)\right)}^{1/\left(p-1\right)}}{\omega }^{p/\left(p-1\right)}\hfill \\ & \le \frac{\delta \left(s\right)r\left(s\right)}{{\left(H\left(t,s\right)A\left(s\right)\frac{\mu s^{n-2}}{\left(n-2\right)!}\right)}^{p-1}}{\left(\frac{\left|h\left(t,s\right)\right|}{p}\right)}^p,\hfill \end{array}$$

which with (23) gives

$$\begin{array}{ccc}{\int }_{t_1}^t\left(H\left(t,s\right)A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}-D\left(s\right)\right)ds\hfill & \le & H\left(t,t_1\right)A\left(t_1\right)\omega \left(t_1\right)\hfill \\ & \le & H\left(t,t_0\right)A\left(t_1\right)\omega \left(t_1\right).\hfill \end{array}$$

Then

$$\begin{array}{cc}\frac{1}{H\left(t,t_0\right)}{\int }_{t_0}^t\hfill & \left(H\left(t,s\right)A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}-D\left(s\right)\right)ds\hfill \\ & \le A\left(t_1\right)\omega \left(t_1\right)+{\int }_{t_0}^{t_1}A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}ds\hfill \\ & <\infty ,\hfill \end{array}$$

for some $\mu \in \left(0,1\right)$, which contradicts (20).

Assume that Case $\left(C_2\right)$ holds. From Lemma 6, we get that (13) holds. Multiplying (13) by $H_{*}\left(t,s\right)A_{*}\left(s\right)$, and integrating the resulting inequality from $t_1$to t, we have

$$\begin{array}{ccc}{\int }_{t_1}^t{H}_{*}\left(t,s\right)A_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)ds\hfill & \le & -{\int }_{t_1}^t{H}_{*}\left(t,s\right)A_{*}\left(s\right){\psi }^{\prime }\left(s\right)ds+{\int }_{t_1}^t{H}_{*}\left(t,s\right)A_{*}\left(s\right)\frac{{\sigma }^{\prime }\left(s\right)}{\sigma \left(s\right)}\psi \left(s\right)ds\hfill \\ & & -{\int }_{t_1}^t\frac{H_{*}\left(t,s\right)A_{*}\left(s\right)}{\sigma \left(s\right)}{\psi }^2\left(s\right)ds\hfill \\ & =& H_{*}\left(t,t_1\right)A_{*}\left(t_1\right)\psi \left(t_1\right)-{\int }_{t_1}^t\frac{H_{*}\left(t,s\right)A_{*}\left(s\right)}{\sigma \left(s\right)}{\psi }^2\left(s\right)ds\hfill \\ & & -{\int }_{t_1}^t\left(-\frac{\partial }{\partial s}\left(H_{*}\left(t,s\right)A_{*}\left(s\right)\right)-H_{*}\left(t,s\right)A_{*}\left(s\right)\frac{{\sigma }^{\prime }\left(t\right)}{\sigma \left(t\right)}\right)\psi \left(s\right)ds.\hfill \end{array}$$

Then

$$\begin{array}{ccc}{\int }_{t_1}^t{H}_{*}\left(t,s\right)A_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)ds\hfill & \le & H_{*}\left(t,t_1\right)A_{*}\left(t_1\right)\psi \left(t_1\right)+{\int }_{t_1}^t\left|h_{*}\left(t,s\right)\right|\psi \left(s\right)d\left(s\right)\hfill \\ & & -{\int }_{t_1}^t\frac{H_{*}\left(t,s\right)A_{*}\left(s\right)}{\sigma \left(s\right)}{\psi }^2\left(s\right)ds.\hfill \end{array}$$

Hence we have

$$\begin{array}{ccc}{\int }_{t_1}^t(H_{*}\left(t,s\right)A_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)-\frac{\sigma \left(s\right){\left|h_{*}\left(t,s\right)\right|}^2}{4H_{*}\left(t,s\right)A_{*}})ds\hfill & \le & H_{*}\left(t,t_1\right)A_{*}\left(t_1\right)\psi \left(t_1\right)\hfill \\ & \le & H_{*}\left(t,t_0\right)A_{*}\left(t_1\right)\psi \left(t_1\right).\hfill \end{array}$$

This implies

$$\begin{array}{cc}\frac{1}{H_{*}\left(t,t_0\right)}{\int }_{t_0}^t(H_{*}(t,s)A_{*}\hfill & (s)\sigma \left(s\right)B\left(s\right)-\frac{\sigma \left(s\right){\left|h_{*}\left(t,s\right)\right|}^2}{4H_{*}\left(t,s\right)A_{*}})ds\hfill \\ & \le A_{*}\left(t_1\right)\psi \left(t_1\right)+{\int }_{t_0}^t{A}_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)ds<\infty \hfill \end{array}$$

which contradicts (21). Therefore, every solution of (1) is oscillatory. □

In the next theorem, we establish new oscillation results for Equation (1) by using the comparison technique with the first-order differential inequality:

Theorem 2.

Let $n\ge 2$be even and $r^{\prime }\left(t\right)>0$. Assume that for some constant $\lambda \in \left(0,1\right)$, the differential equation

$${\phi }^{\prime }\left(t\right)+\frac{q\left(t\right)}{r\left(\tau \left(t\right)\right)}{\left(\frac{\lambda {\tau }^{{}^{n-1}}\left(t\right)}{\left(n-1\right)!}\right)}^{p-1}\phi \left(\tau \left(t\right)\right)=0$$

(24)

is oscillatory. Then every solution of (1) is oscillatory.

Proof. 

Let (1) have a nonoscillatory solution y. Without loss of generality, we can assume that $y\left(t\right)>0$ for$t\ge t_1,$where$t_1\ge t_0$is sufficiently large. Since $r^{\prime }\left(t\right)>0$, we have

$$y^{\prime }\left(t\right)>0,y^{\left(n-1\right)}\left(t\right)>0\mathrm{and}{y}^{\left(n\right)}\left(t\right)<0.$$

(25)

From Lemma 3, we get

$$y\left(t\right)\ge \frac{\lambda t^{n-1}}{\left(n-1\right)!r^{1/p-1}\left(t\right)}{r}^{1/p-1}\left(t\right)y^{\left(n-1\right)}\left(t\right),$$

(26)

for every $\lambda \in \left(0,1\right)$. Thus, if we set

$$\phi \left(t\right)=r\left(t\right){\left[y^{\left(n-1\right)}\left(t\right)\right]}^{p-1}>0,$$

then we see that $\phi$ is a positive solution of the inequality

$${\phi }^{\prime }\left(t\right)+\frac{q\left(t\right)}{r\left(\tau \left(t\right)\right)}{\left(\frac{\lambda {\tau }^{{}^{n-1}}\left(t\right)}{\left(n-1\right)!}\right)}^{p-1}\phi \left(\tau \left(t\right)\right)\le 0.$$

(27)

From [22] (Theorem 1), we conclude that the corresponding Equation (24) also has a positive solution, which is a contradiction.

Theorem 2 is proved. □

Corollary 1.

Assume that (2) holds and let $n\ge 2$be even. If

$$\underset{t\to \infty }{lim}inf{\int }_{\tau \left(t\right)}^t\frac{q\left(s\right)}{r\left(\tau \left(s\right)\right)}{\left({\tau }^{{}^{n-1}}\left(s\right)\right)}^{p-1}ds>\frac{{\left(\left(n-1\right)!\right)}^{p-1}}{e},$$

(28)

then every solution of (1) is oscillatory.

Next, we give the following example to illustrate our main results.

Example 1.

Consider the equation

$$y^{\left(4\right)}\left(t\right)+\frac{\gamma }{t^4}y\left(\frac{9}{10}t\right)=0,t\ge 1,$$

(29)

where $\gamma >0$ is a constant. We note that $n=4,r\left(t\right)=1,p=2,\tau \left(t\right)=9t/10$ and$q\left(t\right)=\gamma /t^4$. If we set $H\left(t,s\right)=H_{*}\left(t,s\right)={\left(t-s\right)}^2,$$A\left(s\right)=A_{*}\left(s\right)=1,$$\delta \left(s\right)=t^3,$$\sigma \left(s\right)=t,h\left(t,s\right)=\left(t-s\right)\left(5-3t{s}^{-1}\right)$ and $h_{*}\left(t,s\right)=\left(t-s\right)\left(3-t{s}^{-1}\right)$ then we get

$$\eta \left(s\right)={\int }_{t_0}^{\infty }\frac{1}{r^{1/\left(p-1\right)}\left(s\right)}ds=\infty$$

and

$$\begin{array}{ccc}B\left(t\right)\hfill & =& \frac{1}{\left(n-4\right)!}{\int }_t^{\infty }{\left(\theta -t\right)}^{n-4}{\left(\frac{{\int }_{\theta }^{\infty }q\left(s\right){\left(\frac{\tau \left(s\right)}{s}\right)}^{p-1}ds}{r\left(\theta \right)}\right)}^{1/\left(p-1\right)}d\theta \hfill \\ & =& 3\gamma /\left(20t^2\right).\hfill \end{array}$$

Hence conditions (20) and (21) become

$$\begin{array}{c}\underset{t\to \infty }{limsup}\frac{1}{H\left(t,t_0\right)}{\int }_{t_0}^t\left(H\left(t,s\right)A\left(s\right)\delta \left(s\right)q\left(s\right){\left(\frac{{\tau }^{n-1}\left(s\right)}{s^{n-1}}\right)}^{p-1}-D\left(s\right)\right)ds\hfill \\ =\underset{t\to \infty }{limsup}\frac{1}{{\left(t-1\right)}^2}{\int }_1^t\left[\frac{729\gamma }{1000}{t}^2{s}^{-1}+\frac{729\gamma }{1000}s-\frac{729\gamma }{500}t-\frac{s}{2\mu }\left(25+9t^2{s}^{-2}-30t{s}^{-1}\right)\right]ds\hfill \\ =\infty \left(\mathrm{if}\gamma >500/81\right)\hfill \end{array}$$

and

$$\begin{array}{c}\underset{t\to \infty }{limsup}\frac{1}{H_{*}\left(t,t_0\right)}{\int }_{t_0}^t\left(H_{*}\left(t,s\right)A_{*}\left(s\right)\sigma \left(s\right)B\left(s\right)-\frac{\sigma \left(s\right){\left|h_{*}\left(t,s\right)\right|}^2}{4H_{*}\left(t,s\right)A_{*}\left(s\right)}\right)ds\hfill \\ =\underset{t\to \infty }{limsup}\frac{1}{{\left(t-1\right)}^2}{\int }_1^t\left[\frac{3\gamma }{20}{t}^2{s}^{-1}+\frac{3\gamma }{20}s-\frac{3\gamma }{10}t-\frac{s}{4}\left(9-630t{s}^{-1}+t^2{s}^{-2}\right)\right]ds\hfill \\ =\infty \left(\mathrm{if}\gamma >5/3\right).\hfill \end{array}$$

Thus, by Theorem 1, every solution of Equation (29) is oscillatory if $\gamma >500/81$.

3. Conclusions

In this work, we have discussed the oscillation of the higher-order differential equation with a p-Laplacian-like operator and we proved that Equation (1) is oscillatory by using the following methods:

• The Riccati transformation technique.
• Comparison principles.
• The Integral averaging technique.

Additionally, in future work we could try to get some oscillation criteria of Equation (1) under the condition ${\int }_{t_0}^{\infty }\frac{1}{r^{1/\left(p-1\right)}\left(t\right)}dt<\infty$. Thus, we would discuss the following two cases:

$$\begin{array}{cc}\left(C_1\right)\hfill & y\left(t\right)>0,y^{\left(n-1\right)}\left(t\right)>0,y^{\left(n\right)}\left(t\right)<0,\hfill \\ \left(C_2\right)\hfill & y\left(t\right)>0,y^{\left(n-2\right)}\left(t\right)>0,y^{\left(n-1\right)}\left(t\right)<0.\hfill \end{array}$$