Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation

Abstract

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form ${\left(r_3\left(t\right){\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+q\left(t\right)f\left(y\left(\sigma \left(t\right)\right)\right)=0,$ where $t\ge t_0.$The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.

1. Introduction

In recent years, the oscillation behavior of delay differential equations has been studied vigorously, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and the references cited therein. This is because of the fact that delay differential equations find various applications in some variational problems, in natural science and technology. Higher order delay differential equations are used in numerous applications in technology and natural science. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines, and therefore their qualitative properties are important, see [14].

Consider the oscillatory behavior of a fourth order differential equation of the form

$$L_y^{\prime }+q\left(t\right)f\left(y\left(\sigma \left(t\right)\right)\right)=0,$$

(1)

where $t\ge t_0$and$L_y=r_3\left(t\right){\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }.$Throughout this paper, we assume that the following assumptions hold:

(H₁) 

$r_i\in C\left([t_0,\infty ),\mathbb{R}\right),i=1,2,3$ are positive and satisfy

$${\int }_{t_0}^{\infty }\frac{1}{r_i\left(s\right)}ds=\infty ,$$

(2)

(H₂) 

$q\in C\left([t_0,\infty ),\mathbb{R}\right)$is nonnegative,

(H₃) 

$\sigma \in C^1\left([t_0,\infty ),\mathbb{R}\right),\sigma \left(t\right)\le t$ and${\displaystyle \underset{t\to \infty }{lim}}\sigma \left(t\right)=\infty$,

(H₄) 

$f\in C\left(\mathbb{R},\mathbb{R}\right),$there exists a constant $k>0$ such that $f\left(u\right)/u\ge k$, for $u\ne 0.$

We intend to a solution of Equation (1) a function $y\left(t\right):[t_y,\infty )\to \mathbb{R},t_y\ge t_0$ such that $r_1{y}^{\prime },$ $r_2{\left(r_1{y}^{\prime }\right)}^{\prime },$ $r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }$ are continuously differentiable for all $t\in [t_y,\infty )$ and $sup\{\left|y\left(t\right)\right|:t\ge T\}>0$ for any $T\ge t_y$.We assume that Equation (1) possesses such a solution. A solution y is said to be non-oscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. The equation it self is called oscillatory if all of its solutions are oscillatory.

Next, we quickly audit some significant oscillation criteria obtained for higher-order equations which can be viewed as an inspiration for this paper.

Dzurina et al. [8] studied the oscillation of solution of the fourth-order equation

$${\left(r_3\left(t\right){\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y\left(\tau \left(t\right)\right)=0$$

and prove that it is oscillatory if the auxiliary third order differential equation

$${\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+\frac{p\left(t\right)}{r_1\left(t\right)}=0,$$

is nonoscillatory, and they used the technique of comparison with first order delay equations.

Grace et al. [9] established some oscillation criteria of solutions for differential Equation (1) under the assumption (2), by using the only Riccati transformations, prove that it is oscillatory if

$${\int }_{t_0}^{\infty }\frac{1}{r_1\left(s\right)}\left({\int }_{t_1}^t\frac{1}{r_2\left(v\right)}{\int }_{t_1}^v\frac{1}{r_3\left(u\right)}{\int }_{t_1}^{u}q\left(\upsilon \right)d\upsilon dudv\right)ds=\infty .$$

Very recently, Cesarano et al. [7] proved that the solutions of Equation (1) is oscillatory if

$$\underset{t\to \infty }{lim}inf{\int }_{t_1}^t\frac{1}{r_1\left(s\right)}\left({\int }_{t_1}^t\frac{1}{r_2\left(u\right)}{\int }_u^{\infty }\frac{1}{r_3\left(v\right)}{\int }_v^{\infty }{\delta }_1\left(\upsilon \right)d\upsilon dvdu\right)ds>\frac{1}{4}$$

and

$$\underset{t\to \infty }{lim}inf\left({\int }_{t_1}^t\left({\int }_{t_1}^t\frac{dv}{r_2\left(v\right)}{\int }_{t_1}^v\frac{ds}{r_3\left(s\right)}\right)\frac{1}{r_1\left(s\right)}ds\right){\int }_{t_1}^{\infty }{\delta }_2\left(\upsilon \right)d\upsilon >\frac{1}{4}\mathrm{for}t\ge t_1,$$

where positive functions ${\delta }_1,{\delta }_2\in C^1\left(\left[t_0,\infty \right),{\mathbb{R}}^{+}\right).$

The purpose of this paper is to give new sufficient conditions for the oscillatory behavior of Equation (1) also, the results here complement of results in [7]. Firstly, we will provide some auxiliary lemmas that will help us to prove our oscillation criteria. Secondly, by employing a refinement of the Riccati transformations, we establish new Kamenev-type oscillation criteria of Equation (1). Finally, some examples are provided to illustrate the main results.

Notation 1.

For convenience, we denote

$$\begin{array}{ccc}\hfill \eta \left(s\right)& :& ={\int }_{t_1}^t\frac{ds}{r_3\left(s\right)},\theta \left(s\right):={\int }_{t_2}^t\frac{\eta \left(s\right)ds}{r_2\left(s\right)},\hfill \\ \hfill \pi \left(t\right)& :& =\frac{\rho \left(t\right)}{r_2\left(t\right)}{\int }_t^{\infty }\left[\frac{1}{r_3\left(t\right)}{\int }_s^{\infty }kq\left(\nu \right)\left(\frac{{\sigma }^3\left(\nu \right)}{{\nu }^3}\right)d\nu \right]ds\hfill \end{array}$$

and

$$\delta \left(t\right):=\frac{1}{\eta \left(t\right)\theta \left(\sigma \left(t\right)\right)}\left({\int }_{t_3}^{\sigma \left(t\right)}\frac{\theta \left(s\right)ds}{r_1\left(s\right)}\right)\left({\int }_{t_2}^{\sigma \left(t\right)}\frac{\eta \left(s\right)ds}{r_2\left(s\right)}\right).$$

2. Some Auxiliary Lemmas

We shall employ the following lemmas:

Lemma 1

([2]). Let $\alpha \ge 1$bea ratio of two odd numbers $C>0$ and D are constants. Then

$$Dz-C{z}^{\left(\alpha +1\right)/\alpha }\le \frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{D^{\alpha +1}}{C^{\alpha }}.$$

Lemma 2

([15]). If the function y satisfies $y^{\left(i\right)}\left(t\right)>0,$ $i=0,1,\dots ,n,$ and $y^{\left(n+1\right)}\left(t\right)<0$for $t\ge t_0,$ then

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

Lemma 3

([7]). Assume that Equation (2) is satisfied and let $y\left(t\right)$ be an eventually positive solution of Equation (1). Then, there exist two possible cases for $t\ge t_1$ large enough:

$$\begin{array}{cc}\hfill \left({\mathbf{N}}_1\right)& y^{\prime }\left(t\right)>0,{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)>0,{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)>0,{\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)<0,\\ \hfill \left({\mathbf{N}}_2\right)& y^{\prime }\left(t\right)>0,{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)<0,{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)>0,{\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)<0.\end{array}$$

Lemma 4.

Assume that $y$is an eventually positive solution of Equation (1). If $\left({\mathbf{N}}_1\right)$ holds and there exists a function $\vartheta \in C^1\left(\left[t_0,\infty \right),{\mathbb{R}}^{+}\right)$ such that

$$\omega \left(t\right):=\vartheta \left(t\right)\left(\frac{r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}\right)$$

(3)

then

$${\omega }^{\prime }\left(t\right)\le -k\vartheta \left(t\right)q\left(t\right)\delta \left(t\right)+\frac{r_3\left(t\right){\left({\vartheta }^{\prime }\left(t\right)\right)}^2}{2\vartheta \left(t\right)},$$

(4)

also, if $\left({\mathbf{N}}_2\right)$ holds and there exists a function $\rho \in C^1\left(\left[t_0,\infty \right),{\mathbb{R}}^{+}\right)$ such that

$$\psi \left(t\right):=\rho \left(t\right)\left(\frac{\left(r_1{y}^{\prime }\right)\left(t\right)}{y\left(t\right)}\right),$$

(5)

then

$${\psi }^{\prime }\left(t\right)\le -\pi \left(t\right)+\frac{r_1\left(t\right){\left({\rho }^{\prime }\left(t\right)\right)}^2}{4\rho \left(t\right)}.$$

(6)

Proof. 

Let y be an eventually positive solution of Equation (1) on $\left[t_0,\infty \right)$. It follows from Lemma 3 that there exist two possible cases $\left({\mathbf{N}}_1\right)$ and $\left({\mathbf{N}}_2\right)$.

Assume that Case $\left({\mathbf{N}}_1\right)$ holds. From ${\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)>0$ and ${\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)<0$for $t\ge t_1$, we obtain

$$\begin{array}{ccc}\hfill \left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)& =& \left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t_1\right)+{\int }_{t_1}^t\frac{r_3\left(s\right){\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(s\right)}{r_3\left(s\right)}ds\hfill \\ & \ge & r_3\left(t\right)\left({\int }_{t_1}^t\frac{ds}{r_3\left(s\right)}\right){\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right).\hfill \end{array}$$

(7)

Thus for $t\ge t_1$, we have

$$\begin{array}{ccc}\hfill {\left(\frac{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}{\eta \left(t\right)}\right)}^{\prime }& =& \frac{{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)\eta \left(t\right)-\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right){\eta }^{\prime }\left(t\right)}{{\eta }^2\left(t\right)}\hfill \\ & =& \frac{{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)}{\eta \left(t\right)}-\frac{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right){\eta }^{\prime }\left(t\right)}{{\eta }^2\left(t\right)}\hfill \\ & \le & \frac{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}{{\eta }^2\left(t\right)}\left[\frac{\eta \left(t\right)}{r_3\left(t\right){\int }_{t_1}^t\frac{ds}{r_3\left(s\right)}}-{\eta }^{\prime }\left(t\right)\right]\hfill \\ & \le & 0.\hfill \end{array}$$

(8)

Therefore, $\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)/\eta \left(t\right)$ is a nonincreasing function for $t_2\ge t_1,t\ge t_2$. Then, we get

$$\begin{array}{ccc}\hfill \left(r_1{y}^{\prime }\right)\left(t\right)& =& \left(r_1{y}^{\prime }\right)\left(t_2\right)+{\int }_{t_2}^t\frac{r_2\left(s\right){\left(r_1{y}^{\prime }\right)}^{\prime }\left(s\right)\eta \left(s\right)}{r_2\left(s\right)\eta \left(s\right)}ds\hfill \\ & \ge & \frac{r_2\left(t\right)}{\eta \left(t\right)}\left({\int }_{t_2}^t\frac{\eta \left(s\right)ds}{r_2\left(s\right)}\right){\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right).\hfill \end{array}$$

(9)

Thus, from Equation (9), we obtain

$$\begin{array}{ccc}\hfill {\left(\frac{\left(r_1{y}^{\prime }\right)\left(t\right)}{\theta \left(t\right)}\right)}^{\prime }& =& \frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)\theta \left(t\right)-\left(r_1{y}^{\prime }\right)\left(t\right){\theta }^{\prime }\left(t\right)}{{\theta }^2\left(t\right)}\hfill \\ & =& \frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)}{\theta \left(t\right)}-\frac{\left(r_1{y}^{\prime }\right)\left(t\right){\theta }^{\prime }\left(t\right)}{{\theta }^2\left(t\right)}\hfill \\ & \le & \frac{\left(r_1{y}^{\prime }\right)\left(t\right)}{{\theta }^2\left(t\right)}\left[\frac{\theta \left(t\right)}{\frac{r_2\left(t\right)}{\eta \left(t\right)}{\int }_{t_2}^t\frac{\eta \left(s\right)ds}{r_2\left(s\right)}}-{\theta }^{\prime }\left(t\right)\right]\hfill \\ & \le & 0.\hfill \end{array}$$

(10)

Therefore, $\left(\left(r_1{y}^{\prime }\right)\left(t\right)\right)/\theta \left(t\right)$ is a nonincreasing function for $t_3\ge t_2,t\ge t_3$. So, we get

$$\begin{array}{ccc}\hfill y\left(t\right)& =& y\left(t_3\right)+{\int }_{t_3}^t\frac{\left(r_1{y}^{\prime }\right)\left(s\right)\theta \left(s\right)}{r_1\left(s\right)\theta \left(s\right)}ds\hfill \\ & \ge & \frac{r_1\left(t\right)}{\theta \left(t\right)}\left({\int }_{t_3}^t\frac{\theta \left(s\right)ds}{r_1\left(s\right)}\right)y^{\prime }\left(t\right).\hfill \end{array}$$

(11)

From Equation (1), we have

$${\left(r_3\left(t\right){\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }=-kq\left(t\right)y\left(\sigma \left(t\right)\right).$$

(12)

Since, $\left(\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)\right)/\eta \left(t\right)$ is a nonincreasing, we get

$$\frac{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(\sigma \left(t\right)\right)}{\eta \left(\sigma \left(t\right)\right)}\ge \frac{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}{\eta \left(t\right)},\sigma \left(t\right)\le t,$$

i.e.,

$$\frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(\sigma \left(t\right)\right)}{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)}\ge \frac{r_2\left(t\right)\eta \left(\sigma \left(t\right)\right)}{r_2\left(\sigma \left(t\right)\right)\eta \left(t\right)}.$$

(13)

Thus, from Equations (9), (11) and (13), we have

$$\begin{array}{ccc}\hfill \frac{y\left(\sigma \left(t\right)\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}& =& \frac{1}{r_2\left(t\right)}\frac{y\left(\sigma \left(t\right)\right)}{y^{\prime }\left(\sigma \left(t\right)\right)}\frac{y^{\prime }\left(\sigma \left(t\right)\right)}{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(\sigma \left(t\right)\right)}\frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(\sigma \left(t\right)\right)}{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)}\hfill \\ & \ge & \frac{1}{r_2\left(t\right)}\frac{r_1\left(\sigma \left(t\right)\right)}{\theta \left(\sigma \left(t\right)\right)}\left({\int }_{t_3}^{\sigma \left(t\right)}\frac{\theta \left(s\right)ds}{r_1\left(s\right)}\right)\left(\frac{r_2}{r_1\eta }\right)\hfill \\ & & \left(\sigma \left(t\right)\right)\left({\int }_{t_2}^{\sigma \left(t\right)}\frac{\eta \left(s\right)ds}{r_2\left(s\right)}\right)\frac{r_2\left(t\right)\eta \left(\sigma \left(t\right)\right)}{r_2\left(\sigma \left(t\right)\right)\eta \left(t\right)}.\hfill \end{array}$$

Thus,

$$\frac{y\left(\sigma \left(t\right)\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}\ge \frac{1}{\eta \left(t\right)\theta \left(\sigma \left(t\right)\right)}\left({\int }_{t_3}^{\sigma \left(t\right)}\frac{\theta \left(s\right)ds}{r_1\left(s\right)}\right)\left({\int }_{t_2}^{\sigma \left(t\right)}\frac{\eta \left(s\right)ds}{r_2\left(s\right)}\right).$$

(14)

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

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& =& {\vartheta }^{\prime }\left(t\right)\left[\frac{r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}\right]+\vartheta \left(t\right)\frac{{\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}\hfill \\ & & -\vartheta \left(t\right)\frac{r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right){\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)}{{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^2\left(t\right)}.\hfill \end{array}$$

Using Equations (12), (14) and (3), we obtain

$${\omega }^{\prime }\left(t\right)\le \frac{{\vartheta }^{\prime }\left(t\right)}{\vartheta \left(t\right)}\omega \left(t\right)-k\vartheta \left(t\right)q\left(t\right)\frac{y\left(\sigma \left(t\right)\right)}{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)\left(t\right)}-\frac{1}{r_3\left(t\right)\vartheta \left(t\right)}{\omega }^2\left(t\right),$$

which yields

$${\omega }^{\prime }\left(t\right)\le -k\vartheta \left(t\right)q\left(t\right)\delta \left(t\right)+\frac{{\vartheta }^{\prime }\left(t\right)}{\vartheta \left(t\right)}\omega \left(t\right)-\frac{1}{r_3\left(t\right)\vartheta \left(t\right)}{\omega }^2\left(t\right).$$

(15)

Using Lemma 1 with $C=1/\left(r_3\left(t\right)\vartheta \left(t\right)\right),D={\vartheta }^{\prime }\left(t\right)/\vartheta \left(t\right)$ and $z=\omega \left(t\right)$, we get

$$\frac{{\vartheta }^{\prime }\left(t\right)}{\vartheta \left(t\right)}\omega \left(t\right)-\frac{1}{r_3\left(t\right)\vartheta \left(t\right)}{\omega }^2\left(t\right)\le \frac{r_3\left(t\right){\left({\vartheta }^{\prime }\left(t\right)\right)}^2}{2\vartheta \left(t\right)}.$$

(16)

From Equations (15) and (16), we get

$${\omega }^{\prime }\left(t\right)\le -k\vartheta \left(t\right)q\left(t\right)\delta \left(t\right)+\frac{r_3\left(t\right){\left({\vartheta }^{\prime }\left(t\right)\right)}^2}{2\vartheta \left(t\right)}.$$

Thus, Equation (4) holds.

Assume that Case $\left({\mathbf{N}}_2\right)$ holds. From the definition of $\psi \left(t\right)$, we see that $\psi \left(t\right)>0$for$t\ge t_1,$and

$${\psi }^{\prime }\left(t\right)={\rho }^{\prime }\left(t\right)\frac{\left(r_1{y}^{\prime }\right)\left(t\right)}{y\left(t\right)}+\rho \left(t\right)\frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)}{y\left(t\right)}-\rho \left(t\right)\frac{\left(r_1{\left(y^{\prime }\right)}^2\right)\left(t\right)}{y^2\left(t\right)}.$$

(17)

Hence by Equation (5), we get

$${\psi }^{\prime }\left(t\right)=\frac{{\rho }^{\prime }\left(t\right)}{\rho \left(t\right)}\psi \left(t\right)+\rho \left(t\right)\frac{{\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)}{y\left(t\right)}-\frac{{\psi }^2\left(t\right)}{r_1\left(t\right)\rho \left(t\right)}.$$

(18)

From Lemma 2, we have that

$$y\left(t\right)\ge \frac{t}{3}{y}^{\prime }\left(t\right),\mathrm{where}n=3.$$

(19)

Integrating Equation (19) from $\sigma \left(t\right)$ to $t,$ we get

$$\frac{y\left(\sigma \left(t\right)\right)}{y\left(t\right)}\ge \frac{{\sigma }^3\left(t\right)}{t^3}.$$

(20)

Integrating Equation (1) from t to $u,$ we find

$$\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)\left(u\right)-\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)\left(t\right)+{\int }_t^{u}kq\left(s\right)y\left(\sigma \left(s\right)\right)ds\le 0.$$

From $y^{\prime }\left(t\right)>0$ and Equation (20), we have

$$\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)\left(u\right)-\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)\left(t\right)+y\left(t\right){\int }_t^{u}kq\left(s\right)\left(\frac{{\sigma }^3\left(s\right)}{s^3}\right)ds\le 0.$$

Letting $u\to \infty$, we arrive at the inequality

$$-\left(r_3{\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\right)\left(t\right)+y\left(t\right){\int }_t^{\infty }kq\left(s\right)\left(\frac{{\sigma }^3\left(s\right)}{s^3}\right)ds\le 0.$$

Thus,

$${\left(r_2{\left(r_1{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)\ge y\left(t\right)\left[\frac{1}{r_3\left(s\right)}{\int }_t^{\infty }kq\left(s\right)\left(\frac{{\sigma }^3\left(s\right)}{s^3}\right)ds\right].$$

(21)

Integrating Equation (21) from t to ∞ we obtain

$${\left(r_1{y}^{\prime }\right)}^{\prime }\left(t\right)+y\left(t\right)\frac{1}{r_2\left(t\right)}{\int }_t^{\infty }\left[\frac{1}{r_3\left(s\right)}{\int }_s^{\infty }kq\left(\nu \right)\left(\frac{{\sigma }^3\left(\nu \right)}{{\nu }^3}\right)d\nu \right]ds\le 0.$$

(22)

Hence, by Equation (22) in Equation (18), we find

$${\psi }^{\prime }\left(t\right)\le -\pi \left(t\right)+\frac{{\rho }^{\prime }\left(t\right)}{\rho \left(t\right)}\psi \left(t\right)-\frac{{\psi }^2\left(t\right)}{r_1\left(t\right)\rho \left(t\right)}.$$

(23)

Thus, we have

$${\psi }^{\prime }\left(t\right)\le -\pi \left(t\right)+\frac{r_1\left(t\right){\left({\rho }^{\prime }\left(t\right)\right)}^2}{4\rho \left(t\right)}.$$

Thus, Equation (6) holds. This completes the proof. □

3. Kamenev-Type Criteria

In the theorem following, we establish new Kamenev-type oscillation criteria for Equation (1).

Theorem 1.

Let (2). Assume that there exists a positive functions $\vartheta ,\rho ,\eta ,\theta \in C^1\left(\left[t_0,\infty \right)\right)$ and an integer $n\in \mathbb{N}.$If

$$\underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(s\right)q\left(s\right)\delta \left(s\right)-\frac{r_3\left(s\right){\left({\vartheta }^{\prime }\left(s\right)\right)}^2}{2\vartheta \left(s\right)}\right)\mathrm{d}s=\infty$$

(24)

and

$$\underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(\pi \left(s\right)-\frac{r_1\left(s\right){\left({\rho }^{\prime }\left(s\right)\right)}^2}{4\rho \left(s\right)}\right)\mathrm{d}s=\infty ,$$

(25)

then every solution of Equation (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 $y\left(t\right)$is eventually positive. Using Lemma 3, we have two cases $\left({\mathbf{N}}_1\right)$ and $\left({\mathbf{N}}_2\right)$.

For case $\left({\mathbf{N}}_1\right)$, from Lemma 4, we get that Equation (4) holds. Thus, we have

$$-{\int }_{t_0}^t{\left(t-s\right)}^n{\omega }^{\prime }\left(s\right)\mathrm{d}s\ge {\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(s\right)q\left(s\right)\delta \left(s\right)-\frac{r_3\left(s\right){\left({\vartheta }^{\prime }\left(s\right)\right)}^2}{2\vartheta \left(s\right)}\right)\mathrm{d}s.$$

Since

$${\int }_{t_0}^t{\left(t-s\right)}^n{\omega }^{\prime }\left(s\right)\mathrm{d}s=n{\int }_{t_0}^t{\left(t-s\right)}^{n-1}\omega \left(s\right)\mathrm{d}s-{\left(t-t_0\right)}^n\omega \left(t_0\right).$$

(26)

Thus, we get

$$\begin{array}{cc}& {\left(\frac{t-t_0}{t}\right)}^n\omega \left(t_0\right)-\frac{n}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^{n-1}\omega \left(s\right)\mathrm{d}s\hfill \\ \ge & \frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(s\right)q\left(s\right)\delta \left(s\right)-\frac{r_3\left(s\right){\left({\vartheta }^{\prime }\left(s\right)\right)}^2}{2\vartheta \left(s\right)}\right)\mathrm{d}s.\hfill \end{array}$$

Hence,

$$\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(s\right)q\left(s\right)\delta \left(s\right)-\frac{r_3\left(s\right){\left({\vartheta }^{\prime }\left(s\right)\right)}^2}{2\vartheta \left(s\right)}\right)\mathrm{d}s\le {\left(\frac{t-t_0}{t}\right)}^n\omega \left(t_0\right),$$

and so

$$\underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(s\right)q\left(s\right)\delta \left(s\right)-\frac{r_3\left(s\right){\left({\vartheta }^{\prime }\left(s\right)\right)}^2}{2\vartheta \left(s\right)}\right)\mathrm{d}s\le \omega \left(t_0\right),$$

which contradicts Equation (24).

For case $\left({\mathbf{N}}_2\right)$, from Lemma 4, we get that Equation (6) holds. Thus, we obtain

$$-{\int }_{t_0}^t{\left(t-s\right)}^n{\psi }^{\prime }\left(s\right)\mathrm{d}s\ge {\int }_{t_0}^t{\left(t-s\right)}^n\left(\pi \left(t\right)-\frac{r_1\left(t\right){\left({\rho }^{\prime }\left(t\right)\right)}^2}{4\rho \left(t\right)}\right)\mathrm{d}s.$$

(27)

From Equations (26) and (27), we get

$$\begin{array}{cc}& {\left(\frac{t-t_0}{t}\right)}^n\psi \left(t_0\right)-\frac{n}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^{n-1}\psi \left(s\right)\mathrm{d}s\hfill \\ \ge & \frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(\pi \left(s\right)-\frac{r_1\left(s\right){\left({\rho }^{\prime }\left(s\right)\right)}^2}{4\rho \left(s\right)}\right)\mathrm{d}s,\hfill \end{array}$$

which yields

$$\underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(\pi \left(s\right)-\frac{r_1\left(s\right){\left({\rho }^{\prime }\left(s\right)\right)}^2}{4\rho \left(s\right)}\right)\mathrm{d}s\le \psi \left(t_0\right),$$

which contradicts Equation (25).

Theorem 1 is proved. □

Example 1.

Consider the equation

$$y^{\left(4\right)}\left(t\right)+\frac{q_0}{t^4}y\left(t\right)=0,t\ge 1,$$

(28)

where $q_0>0$. We note that $r_1=r_2=r_3=1,n=4,\sigma \left(t\right)=t$ and $q\left(t\right)=q_0/t^4$. Hence, it is easy to see that

$$\eta \left(t\right)=t,\theta \left(t\right)=\frac{1}{2}{t}^2$$

and

$$\begin{array}{ccc}\hfill \delta \left(t\right)& =& \frac{1}{\eta \left(t\right)\theta \left(\sigma \left(t\right)\right)}\left({\int }_{t_3}^{\sigma \left(t\right)}\frac{\theta \left(s\right)\mathrm{d}s}{r_1\left(s\right)}\right)\left({\int }_{t_2}^{\sigma \left(t\right)}\frac{\eta \left(s\right)\mathrm{d}s}{r_2\left(s\right)}\right)\hfill \\ & =& \left(\frac{2}{t^3}\right)\left(\frac{t^3}{6}\right)\left(\frac{t^2}{2}\right)=\frac{t^2}{6}.\hfill \end{array}$$

Now, if we set $\vartheta \left(t\right)=\rho \left(t\right)=t$ and $k=1$, then we have

$$\begin{array}{cc}& \underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(k\vartheta \left(t\right)q\left(t\right)\delta \left(t\right)-\frac{r_3\left(t\right){\left({\vartheta }^{\prime }\left(t\right)\right)}^2}{2\vartheta \left(t\right)}\right)\mathrm{d}s\hfill \\ =& \underset{t\to \infty }{limsup}\frac{1}{t^2}{\int }_{t_0}^t{\left(t-s\right)}^2\left(\frac{q_0}{6s}-\frac{1}{2s}\right)ds\hfill \\ =& \underset{t\to \infty }{limsup}\left(\frac{q_0}{6}-\frac{1}{2}\right)\frac{1}{t^2}{\int }_{t_0}^t{\left(t-s\right)}^2\frac{1}{s}ds\hfill \end{array}$$

and

$$\begin{array}{cc}& \underset{t\to \infty }{limsup}\frac{1}{t^n}{\int }_{t_0}^t{\left(t-s\right)}^n\left(\pi \left(t\right)-\frac{r_1\left(t\right){\left({\rho }^{\prime }\left(t\right)\right)}^2}{4\rho \left(t\right)}\right)\mathrm{d}s\hfill \\ =& \underset{t\to \infty }{limsup}\frac{1}{t^2}{\int }_{t_0}^t{\left(t-s\right)}^2\left(\frac{q_0}{6s}-\frac{1}{4s}\right)ds.\hfill \end{array}$$

So, the conditions become

$$q_0>3$$

(29)

and

$$q_0>1.5.$$

(30)

Thus, by using Theorem 1, Equation (28) is oscillatory if $q_0>3.$

Example 2.

For $t\ge 1,$consider a differential equation

$${\left(t{\left(t{\left(t{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+ty\left(\lambda t\right)=0,$$

(31)

where $\lambda \in \left(0,1\right)$ is a constant. Let $r_1=r_2=r_3=t,q\left(t\right)=t,\sigma \left(t\right)=t.$Moreover, we have

$${\int }_{t_0}^{\infty }\frac{ds}{s}=\infty .$$

If we now set $\vartheta \left(t\right)=\rho \left(t\right)=k=1,$it is easy to see that all conditions of Theorem 1 are satisfied. Hence, every solution of Equation (31) is oscillatory.

4. Conclusions

This article is concerned with oscillatory behavior of a class of fourth-order delay differential equations. Using a Riccati transformation a new Kamenev-type asymptotic criterion for (1) is presented. In future work, we will present a new comparison theorem that compares the higher-order Equation (1) with first-order equations. There are numerous results concerning the oscillation criteria of first order equations, which include various forms of criteria as Hille/Nehari, Philos, etc. This allows us to obtain also various criteria for the oscillation of (1). Further, we can try to get some oscillation criteria of (1) if $z\left(t\right):=y\left(t\right)-p\left(t\right)y\left(\tau \left(t\right)\right)$.