Multiple Techniques for Studying Asymptotic Properties of a Class of Differential Equations with Variable Coefficients

Abstract

This manuscript is concerned with the oscillatory properties of 4th-order differential equations with variable coefficients. The main aim of this paper is the combination of the following three techniques used: the comparison method, Riccati technique and integral averaging technique. Two examples are given for applying the criteria.

1. Introduction

Differential equations of fourth-order have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems [1]. The p-Laplace equations have some significant applications in elasticity theory and continuum mechanics, see, for example, [2,3]. Symmetry plays an important role in determining the right way to study these equations [4]. The main aim of this paper is the combination of the following three techniques used:

(a) 

The comparison method.

(b) 

Riccati technique.

(c) 

Integral averaging technique.

We consider the following fourth-order delay differential equations with p-Laplacian like operators

$${\left(a\left(\zeta \right){\left|u^{‴}\left(\zeta \right)\right|}^{p-2}{u}^{‴}\left(\zeta \right)\right)}^{\prime }+q\left(\zeta \right)g\left(u\left(\eta \left(\zeta \right)\right)\right)=0,$$

(1)

where $\zeta \ge {\zeta }_0$. Throughout this work, we suppose that:

K1: 

$p>1$ is a real number.

K2: 

$a\in C^1\left([{\zeta }_0,\infty ),\mathbb{R}\right),$$a\left(\zeta \right)>0,a^{\prime }\left(\zeta \right)\ge 0$ and under the condition

$${\int }_{{\zeta }_0}^{\infty }\frac{1}{a^{1/\left(p-1\right)}\left(s\right)}\mathrm{d}s=\infty ,$$

(2)

K3: 

$q\in C\left([{\zeta }_0,\infty ),\mathbb{R}\right),$$q\left(\zeta \right)>0,$

K4: 

$\eta \in C\left([{\zeta }_0,\infty ),\mathbb{R}\right),\eta \left(\zeta \right)\le \zeta ,$$\underset{\zeta \to \infty }{lim}\eta \left(\zeta \right)=\infty ,$

K5: 

$g\in C\left(\mathbb{R},\mathbb{R}\right)$ such that $g\left(u\right)\ge m{\left|u\right|}^{p-2}u>0,$for $u\ne 0$ and m is a constant.

Definition 1.

The function $u\in C^3[{\zeta }_u,\infty ),{\zeta }_u\ge {\zeta }_0$ is called a solution of (1), if $a\left(\zeta \right){\left|u^{‴}\left(\zeta \right)\right|}^{p-2}{u}^{‴}\left(\zeta \right)\in C^1[{\zeta }_u,\infty ),$ and $u\left(\zeta \right)$ satisfies (1) on $[{\zeta }_u,\infty )$. Moreover, the equation (1) is oscillatory if all its solutions oscillate.

In the last few decades, there have been a constant interest to investigate the asymptotic property for oscillations of differential equation, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Furthermore, there are some results that study the oscillatory behavior of 4th-order equations with p-Laplacian, we refer the reader to [26,27].

Now the following results are presented.

Grace and Lalli [28], Karpuz et al. [29] and Zafer [30] studied the even-order equation

$$u^{\left(\gamma \right)}\left(\zeta \right)+q\left(\zeta \right)u\left(\eta \left(\zeta \right)\right)=0,$$

they used the Riccati substitution to find several oscillation criteria and established the following results, respectively:

$${\int }_{{\zeta }_0}^{\infty }\left(\delta \left(s\right)q\left(s\right)-\frac{\left(\gamma -1\right)!{\left({\delta }^{\prime }\left(s\right)\right)}^2}{2^{3-2\gamma }{\eta }^{\gamma -2}\left(s\right){\eta }^{\prime }\left(s\right)\delta \left(s\right)}\right)ds=\infty ,$$

(3)

where $\delta \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right).$

$$\underset{\zeta \to \infty }{liminf}{\int }_{\eta \left(\zeta \right)}^{\zeta }q\left(s\right){\eta }^{\gamma -2}\left(s\right)ds>\frac{\left(\gamma -1\right)2^{\left(\gamma -1\right)\left(\gamma -2\right)}}{\mathrm{e}}$$

(4)

and

$$\underset{\zeta \to \infty }{liminf}{\int }_{\eta \left(\zeta \right)}^{\zeta }q\left(s\right){\eta }^{\gamma -2}\left(s\right)ds>\frac{\left(\gamma -1\right)!}{\mathrm{e}}.$$

(5)

Zhang et al. [31,32] studied the even-order equation

$${\left(a\left(\zeta \right){\left(u^{\left(\gamma -1\right)}\left(\zeta \right)\right)}^{\beta }\right)}^{\prime }+q\left(\zeta \right)u^{\beta }\left(\eta \left(\zeta \right)\right)=0,$$

(6)

where $\beta$ is a quotient of odd positive integers. They proved that it is oscillatory, if

$$\underset{\zeta \to \infty }{liminf}{\int }_{\zeta }^{\eta \left(\zeta \right)}\frac{q\left(s\right)}{a\left(\eta \left(s\right)\right)}{\left({\eta }^{\gamma -2}\left(s\right)\right)}^{\beta }ds>\frac{{\left(\left(\gamma -1\right)!\right)}^{\beta }}{e},$$

(7)

where $\gamma \ge 2$ is even and they used the compare with first order equations. If there exists a function $\delta \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ for all constants $M>0$ such that

$$\underset{\zeta \to \infty }{liminf}{\int }_{{\zeta }_0}^{\infty }\delta \left(s\right)\left(q\left(s\right)-\frac{a\left(s\right){\left(\theta M{\eta }^{\gamma -2}\left(s\right){\eta }^{\prime }\left(s\right)\right)}^{1-p}}{p^p}{\left(\frac{{\delta }^{\prime }\left(s\right)}{\delta \left(s\right)}-\frac{a\left(s\right)}{r\left(s\right)}\right)}^p\right)ds=\infty ,,$$

(8)

for some constant $\theta \in \left(0,1\right).$

Our aim in this work is to complement results in [28,29,30,31,32]. Two examples are given for applying the criteria.

2. Some Auxiliary Lemmas

Lemma 1.

[13] Fixing $V>0$ and $U\ge 0$, we have that

$$Ux-V{x}^{\left(\beta +1\right)/\beta }\le \frac{{\beta }^{\beta }}{{(\beta +1)}^{\beta +1}}\frac{U^{\beta +1}}{V^{\beta }}.$$

Lemma 2.

[14] For $i=0,1,\dots ,\gamma ,$ let $u^{(i)}\left(\zeta \right)>0,$ and $u^{\left(\gamma +1\right)}\left(\zeta \right)<0,$ then

$$\frac{u\left(\zeta \right)}{{\zeta }^{\gamma }/\gamma !}\ge \frac{u^{\prime }\left(\zeta \right)}{{\zeta }^{\gamma -1}/\left(\gamma -1\right)!}.$$

Lemma 3.

[16] Suppose that $u$ is an eventually positive solution of (1). Then, we distinguish the following situations:

$$\begin{array}{cc}\hfill \left({\mathbf{S}}_1\right)& u\left(\zeta \right)>0,u^{\prime }\left(\zeta \right)>0,u^{″}\left(\zeta \right)>0,u^{‴}\left(\zeta \right)>0,u^{\left(4\right)}\left(\zeta \right)<0,\\ \hfill \left({\mathbf{S}}_2\right)& u\left(\zeta \right)>0,u^{\prime }\left(\zeta \right)>0,u^{″}\left(\zeta \right)<0,u^{‴}\left(\zeta \right)>0,u^{\left(4\right)}\left(\zeta \right)<0,\end{array}$$

for $\zeta \ge {\zeta }_1,$ where ${\zeta }_1\ge {\zeta }_0$ is sufficiently large.

3. Main Results

Let the differential equation

$${\left[a\left(\zeta \right){\left(u^{\prime }\left(\zeta \right)\right)}^{\beta }\right]}^{\prime }+q\left(\zeta \right)u^{\beta }\left(g\left(\zeta \right)\right)=0\zeta \ge {\zeta }_{0,}$$

(9)

where a, $q\in C\left([{\zeta }_0,\infty ),{\mathbb{R}}^{+}\right)$, is nonoscillatory if and only if $\zeta \ge {\zeta }_0$, and a function $\varsigma \in C^1\left([\zeta ,\infty ),\mathbb{R}\right),$ satisfying the inequality

$${\varsigma }^{\prime }\left(\zeta \right)+\gamma a^{-1/\beta }\left(\zeta \right){\left(\varsigma \left(\zeta \right)\right)}^{\left(1+\beta \right)/\beta }+q\left(\zeta \right)\le 0\mathrm{on}[\zeta ,\infty ).$$

Definition 2.

Let

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

A kernel function $H_i\in C\left(D,\mathbb{R}\right)$ is said to belong to the function class ℑ, written by $H\in \Im$, if, for $i=1,2$,

(i) 

$H_i\left(\zeta ,s\right)=0$ for $\zeta \ge {\zeta }_0,H_i\left(\zeta ,s\right)>0,\left(\zeta ,s\right)\in D_0;$

(ii) 

$H_i\left(\zeta ,s\right)$ has a continuous and nonpositive partial derivative $\partial H_i/\partial s$ on $D_0$ and there exist functions $\delta ,\vartheta \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ and $h_i\in C\left(D_0,\mathbb{R}\right)$ such that

$$\frac{\partial }{\partial s}{H}_1\left(\zeta ,s\right)+\frac{{\delta }^{\prime }\left(s\right)}{\delta \left(s\right)}{H}_1\left(\zeta ,s\right)=h_1\left(\zeta ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\zeta ,s\right)$$

(10)

and

$$\frac{\partial }{\partial s}{H}_2\left(\zeta ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\zeta ,s\right)=h_2\left(\zeta ,s\right)\sqrt{H_2\left(\zeta ,s\right)}.$$

(11)

Theorem 1.

Let (2) holds. If the equations

$${\left(\frac{2a^{\frac{1}{p-1}}\left(\zeta \right)}{{\left(\theta {\zeta }^2\right)}^{p-1}}{\left(u^{\prime }\left(\zeta \right)\right)}^{p-1}\right)}^{\prime }+kq\left(\zeta \right){\left(\frac{{\eta }^3\left(\zeta \right)}{{\zeta }^3}\right)}^{p-1}{u}^{p-1}\left(\zeta \right)=0$$

(12)

and

$$u^{″}\left(\zeta \right)+u\left(\zeta \right){\int }_{\zeta }^{\infty }{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right){\left(\frac{\eta \left(\zeta \right)}{\zeta }\right)}^{p-1}\mathrm{d}s\right)}^{1/p-1}\mathrm{d}\varsigma =0$$

(13)

are oscillatory, then every solution of (1) is oscillatory.

Proof. 

Assume, for the sake of contradiction, that u is a positive solution of (1). Then, we let $u\left(\zeta \right)>0$ and $u\left(\eta \left(\zeta \right)\right)>0$. By Lemma 3, we have $\left({\mathbf{S}}_1\right)$ and $\left({\mathbf{S}}_2\right)$.

Let case $\left({\mathbf{S}}_1\right)$ holds. Using [25], [Lemma 2.2.3], we find

$$u^{\prime }\left(\zeta \right)\ge \frac{\theta }{2}{\zeta }^2{u}^{‴}\left(\zeta \right),$$

(14)

for every $\theta \in (0,1)$.

From Lemma 2, we get

$$\frac{u^{\prime }\left(\zeta \right)}{u\left(\zeta \right)}\le \frac{3}{\zeta }.$$

Integrating from $\eta \left(\zeta \right)$ to $\zeta$, we find

$$\frac{u\left(\eta \left(\zeta \right)\right)}{u\left(\zeta \right)}\ge \frac{{\eta }^3\left(\zeta \right)}{{\zeta }^3}.$$

(15)

Defining

$$\phi \left(\zeta \right):=\delta \left(\zeta \right)\left(\frac{a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}}{u^{p-1}\left(\zeta \right)}\right),\phi \left(\zeta \right)>0,$$

(16)

where$\delta \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ and

$$\begin{array}{c}{\phi }^{\prime }\left(\zeta \right)={\delta }^{\prime }\left(\zeta \right)\frac{a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}}{u^{p-1}\left(\zeta \right)}+\delta \left(\zeta \right)\frac{{\left(a{\left(u^{‴}\right)}^{p-1}\right)}^{\prime }\left(\zeta \right)}{u^{p-1}\left(\zeta \right)}\\ -\left(p-1\right)\delta \left(\zeta \right)\frac{u^{p-2}\left(\zeta \right)u^{\prime }\left(\zeta \right)a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}}{u^{2\left(p-1\right)}\left(\zeta \right)}.\end{array}$$

Combining (14) and (16), we obtain

$$\begin{array}{c}{\phi }^{\prime }\left(\zeta \right)\le \frac{{\delta }_{+}^{\prime }\left(\zeta \right)}{\delta \left(\zeta \right)}\phi \left(\zeta \right)+\delta \left(\zeta \right)\frac{{\left(a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}\right)}^{\prime }}{u^{p-1}\left(\zeta \right)}\\ -\left(p-1\right)\delta \left(\zeta \right)\frac{\theta }{2}{\zeta }^2\frac{a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^p}{u^p\left(\zeta \right)}\\ \le \frac{{\delta }^{\prime }\left(\zeta \right)}{\delta \left(\zeta \right)}\phi \left(\zeta \right)+\delta \left(\zeta \right)\frac{{\left(a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{\beta }\right)}^{\prime }}{u^{\beta }\left(\zeta \right)}\\ -\frac{\left(p-1\right)\theta {\zeta }^2}{2{\left(\delta \left(\zeta \right)a\left(\zeta \right)\right)}^{\frac{1}{p-1}}}{\phi }^{\frac{p}{p-1}}\left(\zeta \right).\end{array}$$

(17)

From (1) and (17), we find

$${\phi }^{\prime }\left(\zeta \right)\le \frac{{\delta }^{\prime }\left(\zeta \right)}{\delta \left(\zeta \right)}\phi \left(\zeta \right)-m\delta \left(\zeta \right)\frac{q\left(\zeta \right)u^{p-1}\left(\eta \left(\zeta \right)\right)}{u^{p-1}\left(\zeta \right)}-\frac{\left(p-1\right)\theta {\zeta }^2}{2{\left(\delta \left(\zeta \right)a\left(\zeta \right)\right)}^{\frac{1}{p-1}}}{\phi }^{\frac{p}{p-1}}\left(\zeta \right).$$

From (15), we have

$${\phi }^{\prime }\left(\zeta \right)\le \frac{{\delta }^{\prime }\left(\zeta \right)}{\delta \left(\zeta \right)}\phi \left(\zeta \right)-m\delta \left(\zeta \right)q\left(\zeta \right){\left(\frac{{\eta }^3\left(\zeta \right)}{{\zeta }^3}\right)}^{p-1}-\frac{\left(p-1\right)\theta {\zeta }^2}{2{\left(\delta \left(\zeta \right)a\left(\zeta \right)\right)}^{\frac{1}{p-1}}}{\phi }^{\frac{p}{p-1}}\left(\zeta \right).$$

(18)

Let $\delta \left(\zeta \right)=m=1$ in (18), we have

$${\phi }^{\prime }\left(\zeta \right)+\frac{\left(p-1\right)\theta {\zeta }^2}{2a^{\frac{1}{p-1}}\left(\zeta \right)}{\phi }^{\frac{p}{p-1}}\left(\zeta \right)+q\left(\zeta \right){\left(\frac{{\eta }^3\left(\zeta \right)}{{\zeta }^3}\right)}^{p-1}\le 0.$$

Hence, the equation (12) is nonoscillatory $,$ which is a contradiction.

Let case $\left({\mathbf{S}}_2\right)$ holds. By Lemma 2, we find

$$\frac{u^{\prime }\left(\zeta \right)}{u\left(\zeta \right)}\le \frac{1}{\zeta }.$$

Integrating again from $\eta \left(\zeta \right)$ to $\zeta$, we find

$$\frac{u\left(\eta \left(\zeta \right)\right)}{u\left(\zeta \right)}\ge \frac{\eta \left(\zeta \right)}{\zeta }.$$

(19)

Defining

$$\psi \left(\zeta \right):=\vartheta \left(\zeta \right)\frac{u^{\prime }\left(\zeta \right)}{u\left(\zeta \right)}>0,$$

where $\vartheta \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ and

$${\psi }^{\prime }\left(\zeta \right)=\frac{{\vartheta }^{\prime }\left(\zeta \right)}{\vartheta \left(\zeta \right)}\psi \left(\zeta \right)+\vartheta \left(\zeta \right)\frac{u^{″}\left(\zeta \right)}{u\left(\zeta \right)}-\frac{1}{\vartheta \left(\zeta \right)}\psi {\left(\zeta \right)}^2.$$

(20)

Integrating (1) from $\zeta$ to x and using $u^{\prime }\left(\zeta \right)>0$, we have

$$a\left(x\right){\left(u^{‴}\left(x\right)\right)}^{p-1}-a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}=-{\int }_{\zeta }^{x}q\left(s\right)g\left(u\left(\eta \left(s\right)\right)\right)ds.$$

From (19), we get

$$a\left(x\right){\left(u^{‴}\left(x\right)\right)}^{p-1}-a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}\le -k{y}^{p-1}\left(\zeta \right){\int }_{\zeta }^{x}q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}ds.$$

Letting $x\to \infty$, we have

$$a\left(\zeta \right){\left(u^{‴}\left(\zeta \right)\right)}^{p-1}\ge k{y}^{p-1}\left(\zeta \right){\int }_{\zeta }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s$$

and so

$$u^{\prime \prime \prime }\left(\zeta \right)\ge u\left(\zeta \right){\left(\frac{m}{a\left(\zeta \right)}{\int }_{\zeta }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s\right)}^{1/\left(p-1\right)}.$$

Integrating again from $\zeta$ to ∞, we get

$$u^{\prime \prime }\left(\zeta \right)+u\left(\zeta \right){\int }_{\zeta }^{\infty }{\left(\frac{m}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s\right)}^{1/\left(p-1\right)}\mathrm{d}\varsigma \le 0.$$

(21)

Combining (20) and (21), we find

$${\psi }^{\prime }\left(\zeta \right)\le \frac{{\vartheta }^{\prime }\left(\zeta \right)}{\vartheta \left(\zeta \right)}\psi \left(\zeta \right)-\vartheta \left(\zeta \right){\int }_{\zeta }^{\infty }{\left(\frac{m}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s\right)}^{1/\left(p-1\right)}\mathrm{d}\varsigma -\frac{1}{\vartheta \left(\zeta \right)}\psi {\left(\zeta \right)}^2.$$

(22)

If $\vartheta \left(\zeta \right)=m=1$ in (22), we get

$${\psi }^{\prime }\left(\zeta \right)+{\psi }^2\left(\zeta \right)+{\int }_{\zeta }^{\infty }{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s\right)}^{1/\left(p-1\right)}\mathrm{d}\varsigma \le 0.$$

Thus, the Equation (13) is nonoscillatory, which is a contradiction. The proof of the theorem is complete. ☐

Next, we obtain the following Hille and Nehari type oscillation criteria for (1) with $p=2.$

Theorem 2.

Let $p=2,m=1.$ Assume that

$${\int }_{{\zeta }_0}^{\infty }\frac{\theta {\zeta }^2}{2a\left(\zeta \right)}\mathrm{d}\zeta =\infty$$

and

$$\underset{\zeta \to \infty }{liminf}\left({\int }_{{\zeta }_0}^{\zeta }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right){\int }_{\zeta }^{\infty }q\left(s\right)\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)\mathrm{d}s>\frac{1}{4},$$

(23)

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

$$\underset{\zeta \to \infty }{liminf}\zeta {\int }_{{\zeta }_0}^{\zeta }{\int }_v^{\infty }\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\left(\frac{\eta \left(s\right)}{s}\right)\mathrm{d}s\right)\mathrm{d}\varsigma \mathrm{d}v>\frac{1}{4},$$

(24)

then all solutions of (1) is oscillatory.

In this theorem, we use the integral averaging technique:

Theorem 3.

Let (2) holds. If there exist positive functions $\delta ,\vartheta \in C^1\left(\left[{\zeta }_0,\infty \right),\mathbb{R}\right)$ such that

$$\underset{\zeta \to \infty }{limsup}\frac{1}{H_1\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_1\left(\zeta ,s\right)m\delta \left(s\right)q\left(s\right){\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)}^{p-1}-\pi \left(s\right)\right)\mathrm{d}s=\infty$$

(25)

and

$$\underset{\zeta \to \infty }{limsup}\frac{1}{H_2\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_2\left(\zeta ,s\right)\vartheta \left(s\right)\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\zeta ,s\right)}{4}\right)\mathrm{d}s=\infty ,$$

(26)

where

$$\pi \left(s\right)=\frac{h_1^p\left(\zeta ,s\right)H_1^{p-1}\left(\zeta ,s\right)}{p^p}\frac{2^{p-1}\delta \left(s\right)a\left(s\right)}{{\left(\theta s^2\right)}^{p-1}},$$

for all $\theta \in \left(0,1\right),$ and

$$\varpi \left(s\right)={\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right){\left(\frac{\eta \left(s\right)}{s}\right)}^{p-1}\mathrm{d}s\right)}^{1/\left(p-1\right)}\mathrm{d}\varsigma ,$$

then (1) is oscillatory.

Proof. 

Proceeding as in the proof of Theorem 1. Assume that $\left({\mathbf{S}}_1\right)$ holds. From Theorem 1, we get that (18) holds. Multiplying (18) by $H_1\left(\zeta ,s\right)$ and integrating the resulting inequality from ${\zeta }_1$ to $\zeta$, we find that

$$\begin{array}{ccc}\hfill {\int }_{{\zeta }_1}^{\zeta }{H}_1\left(\zeta ,s\right)m\delta \left(s\right)q\left(s\right){\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)}^{p-1}\mathrm{d}s& \le & \phi \left({\zeta }_1\right)H_1\left(\zeta ,{\zeta }_1\right)+{\int }_{{\zeta }_1}^{\zeta }\left(\frac{\partial }{\partial s}{H}_1\left(\zeta ,s\right)+\frac{{\delta }^{\prime }\left(s\right)}{\delta \left(s\right)}{H}_1\left(\zeta ,s\right)\right)\phi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\zeta }_1}^{\zeta }\frac{\left(p-1\right)\theta s^2}{2{\left(\delta \left(s\right)a\left(s\right)\right)}^{\frac{1}{p-1}}}{H}_1\left(\zeta ,s\right){\phi }^{\frac{p}{p-1}}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

From (10), we get

$$\begin{array}{ccc}\hfill {\int }_{{\zeta }_1}^{\zeta }{H}_1\left(\zeta ,s\right)m\delta \left(s\right)q\left(s\right){\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)}^{p-1}\mathrm{d}s& \le & \phi \left({\zeta }_1\right)H_1\left(\zeta ,{\zeta }_1\right)+{\int }_{{\zeta }_1}^{\zeta }{h}_1\left(\zeta ,s\right)H_1^{\left(p-1\right)/p}\left(\zeta ,s\right)\phi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\zeta }_1}^{\zeta }\frac{\left(p-1\right)\theta s^2}{2{\left(\delta \left(s\right)a\left(s\right)\right)}^{\frac{1}{p-1}}}{H}_1\left(\zeta ,s\right){\phi }^{\frac{p}{p-1}}\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(27)

Using Lemma 1 with $V=\left(p-1\right)\theta s^2/\left(2{\left(\delta \left(s\right)a\left(s\right)\right)}^{\frac{1}{p-1}}\right)H_1\left(\zeta ,s\right),U=h_1\left(\zeta ,s\right)H_1^{\left(p-1\right)/p}\left(\zeta ,s\right)$ and $u=\phi \left(s\right)$, we get

$$\begin{array}{c}{h}_1\left(\zeta ,s\right)H_1^{\left(p-1\right)/p}\left(\zeta ,s\right)\phi \left(s\right)-\frac{\left(p-1\right)\theta s^2}{2{\left(\delta \left(s\right)a\left(s\right)\right)}^{\frac{1}{p-1}}}{H}_1\left(\zeta ,s\right){\phi }^{\frac{p}{p-1}}\left(s\right)\hfill \\ \le \frac{h_1^p\left(\zeta ,s\right)H_1^{p-1}\left(\zeta ,s\right)}{p^p}\frac{2^{p-1}\delta \left(s\right)a\left(s\right)}{{\left(\theta s^2\right)}^{p-1}},\hfill \end{array}$$

which, with (27) gives

$$\frac{1}{H_1\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_1\left(\zeta ,s\right)m\delta \left(s\right)q\left(s\right){\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)}^{p-1}-\pi \left(s\right)\right)\mathrm{d}s\le \phi \left({\zeta }_1\right).$$

This contradicts (25).

Assume that $\left({\mathbf{S}}_2\right)$ holds. From Theorem 1, (22) holds. Multiplying (22) by $H_2\left(\zeta ,s\right)$and integrating the resulting inequality from ${\zeta }_1$ to $\zeta$, we get

$$\begin{array}{ccc}\hfill {\int }_{{\zeta }_1}^{\zeta }{H}_2\left(\zeta ,s\right)\vartheta \left(s\right)\varpi \left(s\right)\mathrm{d}s& \le & \psi \left({\zeta }_1\right)H_2\left(\zeta ,{\zeta }_1\right)\hfill \\ & & +{\int }_{{\zeta }_1}^{\zeta }\left(\frac{\partial }{\partial s}{H}_2\left(\zeta ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\zeta ,s\right)\right)\psi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\zeta }_1}^{\zeta }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\zeta ,s\right){\psi }^2\left(s\right)\mathrm{d}s.\hfill \end{array}$$

Thus, from (11), we get

$$\begin{array}{ccc}\hfill {\int }_{{\zeta }_1}^{\zeta }{H}_2\left(\zeta ,s\right)\vartheta \left(s\right)\varpi \left(s\right)\mathrm{d}s& \le & \psi \left({\zeta }_1\right)H_2\left(\zeta ,{\zeta }_1\right)+{\int }_{{\zeta }_1}^{\zeta }{h}_2\left(\zeta ,s\right)\sqrt{H_2\left(\zeta ,s\right)}\psi \left(s\right)\mathrm{d}s\hfill \\ & & -{\int }_{{\zeta }_1}^{\zeta }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\zeta ,s\right){\psi }^2\left(s\right)\mathrm{d}s\hfill \\ & \le & \psi \left({\zeta }_1\right)H_2\left(\zeta ,{\zeta }_1\right)+{\int }_{{\zeta }_1}^{\zeta }\frac{\vartheta \left(s\right)h_2^2\left(\zeta ,s\right)}{4}\mathrm{d}s\hfill \end{array}$$

and so

$$\frac{1}{H_2\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_2\left(\zeta ,s\right)\vartheta \left(s\right)\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\zeta ,s\right)}{4}\right)\mathrm{d}s\le \psi \left({\zeta }_1\right),$$

which contradicts (26). The proof of the theorem is complete.

Example 1.

Consider the equation

$$u^{\left(4\right)}\left(\zeta \right)+\frac{q_0}{{\zeta }^4}u\left(\frac{9\zeta }{10}\right)=0,\zeta \ge 1,q_0>0.$$

(28)

Let $p=2,a\left(\zeta \right)=1,q\left(\zeta \right)=q_0/{\zeta }^4$ and $\eta \left(\zeta \right)=9\zeta /10$. If we set $m=1,H_1\left(\zeta ,s\right)={\left(\zeta -s\right)}^2$ and $\delta \left(s\right)=s^3,$ then $h_1\left(\zeta ,s\right)=\left(\zeta -s\right)\left(5-3\zeta s^{-1}\right),$ and conditions (23) becomes

$$\begin{array}{c}\underset{\zeta \to \infty }{limsup}\frac{1}{H_1\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_1\left(\zeta ,s\right)m\delta \left(s\right)q\left(s\right){\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)}^{p-1}-\pi \left(s\right)\right)\mathrm{d}s\hfill \\ =\underset{\zeta \to \infty }{limsup}\frac{1}{{\left(\zeta -1\right)}^2}{\int }_{{\zeta }_1}^{\zeta }\left(\frac{729q_0{\zeta }^2{s}^{-1}}{1000}+\frac{729q_{0}s}{1000}-\frac{729q_0\zeta }{500}-\frac{s\left(25+9{\zeta }^2{s}^{-2}-30\zeta s^{-1}\right)}{2\theta }\right)\mathrm{d}s\hfill \\ =\infty ,\hfill \end{array}$$

if $q_0>500/\left(81\theta \right)$ for some $\theta \in \left(0,1\right)$, letting $\theta =81/82,$ then $q_0>6.25.$

Also, set $H_2\left(\zeta ,s\right)={\left(\zeta -s\right)}^2$ and $\vartheta \left(s\right)=s,$ then $h_2\left(\zeta ,s\right)=\left(\zeta -s\right)\left(3-\zeta s^{-1}\right),\varpi \left(s\right)=3q_0/\left(20{\zeta }^2\right)$ and conditions (24) becomes

$$\begin{array}{c}\underset{\zeta \to \infty }{limsup}\frac{1}{H_2\left(\zeta ,{\zeta }_1\right)}{\int }_{{\zeta }_1}^{\zeta }\left(H_2\left(\zeta ,s\right)\vartheta \left(s\right)\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\zeta ,s\right)}{4}\right)\mathrm{d}s\hfill \\ =\underset{\zeta \to \infty }{limsup}\frac{1}{{\left(\zeta -1\right)}^2}{\int }_{{\zeta }_1}^{\zeta }\left(\frac{3q_0{\zeta }^2{s}^{-1}}{20}+\frac{3q_{0}s}{20}-\frac{3q_0\zeta }{10}-\frac{s\left(9-6\zeta s^{-1}+{\zeta }^2{s}^{-2}\right)}{4}\right)\mathrm{d}s\hfill \\ =\infty ,\hfill \end{array}$$

if $q_0>5/3$, From Theorem 3, all solutions of (28) are oscillatory, if $q_0>6.25$.

Remark 1.

By comparing our results with previous results

1. By applying condition (3) in [28], we get

$$q_0>1728,$$

2. By applying condition (4) in [29], we get

$$q_0>919.6,$$

3. By applying condition (5) in [30], we get

$$q_0>28.73,$$

4. By applying condition (7) in [31], we get

$$q_0>28.73,$$

5. The condition (8) in [32] cannot be applied to Equation (28) due to the arbitrariness in the choice of θ. Therefore, our result complement results [28,29,30,31,32].

Example 2.

Let the equation

$$u^{\left(4\right)}\left(\zeta \right)+\frac{q_0}{{\zeta }^4}u\left(\frac{1}{2}\zeta \right)=0,\zeta \ge 1,q_0>0.$$

(29)

Let$a\left(\zeta \right)=1,q\left(\zeta \right)=q_0/{\zeta }^4$ and $\eta \left(\zeta \right)=\zeta /2$. If we set $m=1,$ then condition (23) becomes

$$\begin{array}{ccc}\hfill \underset{\zeta \to \infty }{liminf}\left({\int }_{{\zeta }_0}^{\zeta }\frac{\theta s^2}{2a\left(s\right)}\mathrm{d}s\right){\int }_{\zeta }^{\infty }q\left(s\right)\left(\frac{{\eta }^3\left(s\right)}{s^3}\right)\mathrm{d}s& =& \underset{\zeta \to \infty }{liminf}\left(\frac{{\zeta }^3}{3}\right){\int }_{\zeta }^{\infty }\frac{q_0}{8s^4}ds\hfill \\ & =& \frac{q_0}{72}>\frac{1}{4}\hfill \end{array}$$

and condition (24) becomes

$$\begin{array}{ccc}\hfill \underset{\zeta \to \infty }{liminf}\zeta {\int }_{{\zeta }_0}^{\zeta }{\int }_v^{\infty }\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\left(\frac{\eta \left(s\right)}{s}\right)\mathrm{d}s\right)\mathrm{d}\varsigma \mathrm{d}v& =& \underset{\zeta \to \infty }{liminf}\zeta \left(\frac{q_0}{12\zeta }\right)\hfill \\ & =& \frac{q_0}{12}>\frac{1}{4}.\hfill \end{array}$$

Hence, by Theorem 2, all solution equation (29) is oscillatory if $q_0>18$.

Remark 2.

We point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type:

$${\left(a\left(\zeta \right){\left|u^{\prime \prime \prime }\left(\zeta \right)\right|}^{p-2}{u}^{‴}\left(\zeta \right)\right)}^{\prime }+\sum _{i=1}^m{q}_i\left(\zeta \right){\left|u\left({\eta }_i\left(\zeta \right)\right)\right|}^{p-2}u\left({\eta }_i\left(\zeta \right)\right)=0,\mathrm{where}\zeta \ge {\zeta }_0,m\ge 1,$$

under the condition

$${\int }_{{\zeta }_0}^{\infty }\frac{1}{a^{1/\left(p-1\right)}\left(s\right)}\mathrm{d}s<\infty .$$

4. Conclusions

In this article, we studied some oscillation conditions for 4th-order differential equations by the comparison method, Riccati technique and integral averaging technique.

Further, in the future work we study Equation (1) under the condition ${\int }_{{\zeta }_0}^{\infty }\frac{1}{a^{1/\left(p-1\right)}\left(s\right)}\mathrm{d}s<\infty .$