A New Approach in the Study of Oscillation Criteria of Even-Order Neutral Differential Equations

Abstract

Based on the comparison with first-order delay equations, we establish a new oscillation criterion for a class of even-order neutral differential equations. Our new criterion improves a number of existing ones. An illustrative example is provided.

1. Introduction

In the last decade, many studies have been carried out on the oscillatory behavior of various types of functional 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] and the references cited therein. As a result of numerous applications in technology and natural science, the issue of oscillation of nonlinear neutral delay differential equation has caught the attention of many researchers, see [1,3,4,5,8,12,17,19,22,23,24]. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines, see [11].

In this paper, we are concerned with improving the oscillation criteria for the even-order neutral differential equation of the form

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0,$$

(1)

where $t\ge t_0,n\ge 4$ is an even natural number and $z\left(t\right):=x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)$. In this work, we assume that $\alpha$ is a quotient of odd positive integers, $r\in C[t_0,\infty ),$ $r\left(t\right)>0,$ $r^{\prime }\left(t\right)\ge 0,$ ${\int }_{t_0}^{\infty }{r}^{-1/\alpha }\left(s\right)\mathrm{d}s=\infty ,$ $p,q\in C[t_0,\infty ),$ $q\left(t\right)>0,$ $0\le p\left(t\right)<p_0<\infty ,$ $q\left(t\right)$ is not identically zero for large t, $\tau \in C^1[t_0,\infty ),\sigma \in C[t_0,\infty ),$${\tau }^{\prime }\left(t\right)>0,$$\tau \left(t\right)\le t$ and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty .$

By a solution of (1) we mean a function x$\in C^3[t_y,\infty ),t_y\ge t_0,$ which has the property $r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\in C^1[t_y,\infty ),$ and satisfies (1) on $[t_y,\infty )$. We consider only those solutions x of (1) which satisfy $sup\{\left|x\left(t\right)\right|:t\ge T\}>0,$ for all $T\ge t_y$. A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory.

A neutral delay differential equation is a differential equation in which the highest-order derivative of the unknown function appears both with and without delay.

In the following, we briefly review some important oscillation criteria obtained for higher-order neutral equations which can be seen as a motivation for this paper.

In 1998, based on establishing comparison theorems that compare the nth-order equation with only one first-order delay differential equations, Zafer [23] proved that the even-order differential equation

$$z^{\left(n\right)}\left(t\right)+q\left(t\right)x\left(\sigma \left(t\right)\right)=0$$

(2)

is oscillatory if

$$\underset{t\to \infty }{liminf}{\int }_{\sigma \left(t\right)}^{t}Q\left(s\right)\mathrm{d}s>\frac{\left(n-1\right)2^{\left(n-1\right)\left(n-2\right)}}{\mathrm{e}},$$

(3)

or

$$\underset{t\to \infty }{limsup}{\int }_{\sigma \left(t\right)}^{t}Q\left(s\right)\mathrm{d}s>\left(n-1\right)2^{\left(n-1\right)\left(n-2\right)},{\sigma }^{\prime }\left(t\right)\ge 0.$$

where $Q\left(t\right):={\sigma }^{n-1}\left(t\right)\left(1-p\left(\sigma \left(t\right)\right)\right)q\left(t\right)$. In a similar approach, Zhang and Yan [24] proved that (2) is oscillatory if either

$$lim\underset{t\to \infty }{inf}{\int }_{\sigma \left(t\right)}^{t}Q\left(s\right)\mathrm{d}s>\frac{\left(n-1\right)!}{\mathrm{e}},$$

(4)

or

$$lim\underset{t\to \infty }{sup}{\int }_{\sigma \left(t\right)}^{t}Q\left(s\right)\mathrm{d}s>\left(n-1\right)!,\sigma \left(t\right)\ge 0.$$

It’s easy to note that $\left(n-1\right)!<\left(n-1\right)2^{\left(n-1\right)\left(n-2\right)}$ for $n>3$, and hence results in [24] improved results of Zafer in [23].

For nonlinear equation, Xing et al. [22] proved that (1) is oscillatory if

$${\left({\sigma }^{-1}\left(t\right)\right)}^{\prime }\ge {\sigma }_0>0,{\tau }^{\prime }\left(t\right)\ge {\tau }_0>0,{\tau }^{-1}\left(\sigma \left(t\right)\right)<t$$

and

$$lim\underset{t\to \infty }{inf}{\int }_{{\tau }^{-1}\left(\sigma \left(t\right)\right)}^t\frac{\widehat{q}\left(s\right)}{r\left(s\right)}{\left(s^{n-1}\right)}^{\alpha }\mathrm{d}s>\left(\frac{1}{{\sigma }_0}+\frac{p_0^{\alpha }}{{\sigma }_0{\tau }_0}\right)\frac{{\left(\left(n-1\right)!\right)}^{\alpha }}{\mathrm{e}},$$

(5)

where $\widehat{q}\left(t\right):=min\left\{q\left({\sigma }^{-1}\left(t\right)\right),q\left({\sigma }^{-1}\left(\tau \left(t\right)\right)\right)\right\}$.

If we apply the previous results to the equation

$${\left(x\left(t\right)+\frac{7}{8}x\left(\frac{1}{\mathrm{e}}t\right)\right)}^{\left(4\right)}+\frac{q_0}{t^4}x\left(\frac{1}{{\mathrm{e}}^2}t\right)=0,t\ge 1,$$

(6)

then we get that (6) is oscillatory if

The condition	(3)	(4)	(5)
The criterion	q0 > 113, 981.3	q0 > 561.9	q0 > 3008.5

Hence, Xing et al. [22] improved the results in [23,24].

By establishing a new comparison theorem that compare the higher-order Equation (1) with a couple of first-order delay differential equations, we improve the results in [22,23,24]. An example is presented to illustrate our main results.

In order to discuss our main results, we need the following lemmas:

Lemma 1

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

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

Lemma 2

([2] Lemma 2.2.3). Let $x\in C^n\left(\left[t_0,\infty \right),\left(0,\infty \right)\right).$ Assume that $x^{\left(n\right)}\left(t\right)$ is of fixed sign and not identically zero on $\left[t_0,\infty \right)$ and that there exists a $t_1\ge t_0$ such that $x^{\left(n-1\right)}\left(t\right)x^{\left(n\right)}\left(t\right)\le 0$ for all $t\ge t_1$. If ${lim}_{t\to \infty }x\left(t\right)\ne 0,$ then for every $\mu \in \left(0,1\right)$ there exists $t_{\mu }\ge t_1$ such that

$$x\left(t\right)\ge \frac{\mu }{\left(n-1\right)!}{t}^{n-1}\left|x^{\left(n-1\right)}\left(t\right)\right|fort\ge t_{\mu }.$$

Lemma 3

([3] Lemmas 1 and 2). Assume that $u,v\ge 0$ and β is a positive real number. Then

$${\left(u+v\right)}^{\beta }\le 2^{\beta -1}\left(u^{\beta }+v^{\beta }\right),for\beta \ge 1$$

and

$${\left(u+v\right)}^{\beta }\le u^{\beta }+v^{\beta },for\beta \le 1.$$

2. Main Results

Here, we define the next notation:

$$P_k\left(t\right)=\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left(1-\frac{{\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}^{k-1}}{{\left({\tau }^{-1}\left(t\right)\right)}^{k-1}p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right),\mathrm{for}k=2,n,$$

$$R_0\left(t\right)={\left(\frac{1}{r\left(t\right)}{\int }_t^{\infty }q\left(s\right)P_2^{\alpha }\left(\sigma \left(s\right)\right)\mathrm{d}s\right)}^{1/\alpha }$$

and

$$R_m\left(t\right)={\int }_t^{\infty }{R}_{m-1}\left(s\right)\mathrm{d}s,m=1,2,\dots ,n-3.$$

Lemma 4

([20] Lemma 1.2). Assume that $x$ is an eventually positive solution of (1). Then, there exist two possible cases:

$$\begin{array}{cc}\hfill \left({\mathbf{I}}_1\right)& z\left(t\right)>0,z^{\prime }\left(t\right)>0,z^{″}\left(t\right)>0,z^{\left(n-1\right)}\left(t\right)>0,z^{\left(n\right)}\left(t\right)<0,\\ \hfill \left({\mathbf{I}}_2\right)& z\left(t\right)>0,z^{\left(j\right)}\left(t\right)>0,z^{(j+1)}\left(t\right)<0foralloddinteger\\ \hfill & j\in \{1,3,\dots ,n-3\},z^{(n-1)}\left(t\right)>0,z^{\left(n\right)}\left(t\right)<0,\end{array}$$

for $t\ge t_1,$ where $t_1\ge t_0$ is sufficiently large.

Theorem 1.

Let

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

(7)

Assume that there exist positive functions $\eta ,\zeta \in C^1\left(\left[t_0,\infty \right),\mathbb{R}\right)$ satisfying

$$\eta \left(t\right)\le \sigma \left(t\right),\eta \left(t\right)<\tau \left(t\right),\zeta \left(t\right)\le \sigma \left(t\right),\zeta \left(t\right)<\tau \left(t\right),{\zeta }^{\prime }\left(t\right)\ge 0and\underset{t\to \infty }{lim}\eta \left(t\right)=\underset{t\to \infty }{lim}\zeta \left(t\right)=\infty .$$

(8)

If there exists a $\mu \in \left(0,1\right)$ such that the differential equations

$${\psi }^{\prime }\left(t\right)+{\left(\frac{\mu {\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)}^{n-1}}{\left(n-1\right)!r^{1/\alpha }\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)}\right)}^{\alpha }q\left(t\right)P_n^{\alpha }\left(\sigma \left(t\right)\right)\psi \left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)=0$$

(9)

and

$${\varphi }^{\prime }\left(t\right)+{\tau }^{-1}\left(\zeta \left(t\right)\right)R_{n-3}\left(t\right)\varphi \left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right)=0$$

(10)

are oscillatory, then Equation (1) is oscillatory.

Proof. 

Let x be a non-oscillatory solution of (1) on $\left[t_0,\infty \right)$. Without loss of generality, we can assume that $x$ is eventually positive. It follows from Lemma 4 that there exist two possible cases $\left({\mathbf{I}}_1\right)$ and $\left({\mathbf{I}}_2\right)$.

Assume that Case $\left({\mathbf{I}}_1\right)$ holds. From the definition of $z\left(t\right)$, we see that

$$x\left(t\right)=\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left(z\left({\tau }^{-1}\left(t\right)\right)-x\left({\tau }^{-1}\left(t\right)\right)\right).$$

By repeating the same process, we find that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \frac{z\left({\tau }^{-1}\left(t\right)\right)}{p\left({\tau }^{-1}\left(t\right)\right)}-\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left(\frac{z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}-\frac{x\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right)\hfill \\ & \ge & \frac{z\left({\tau }^{-1}\left(t\right)\right)}{p\left({\tau }^{-1}\left(t\right)\right)}-\frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\frac{z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}.\hfill \end{array}$$

(11)

Using Lemma 1, we get $z\left(t\right)\ge \frac{1}{\left(n-1\right)}t{z}^{\prime }\left(t\right)$ and hence the function $t^{1-n}z\left(t\right)$ is nonincreasing, which with the fact that $\tau \left(t\right)\le t$ gives

$${\left({\tau }^{-1}\left(t\right)\right)}^{n-1}z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)\le {\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}^{n-1}z\left({\tau }^{-1}\left(t\right)\right).$$

(12)

Combining Equations (11) and (12), we conclude that

$$\begin{array}{ccc}\hfill x\left(t\right)& \ge & \frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left(1-\frac{{\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}^{n-1}}{{\left({\tau }^{-1}\left(t\right)\right)}^{n-1}p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right)z\left({\tau }^{-1}\left(t\right)\right)\hfill \\ & =& P_n\left(t\right)z\left({\tau }^{-1}\left(t\right)\right).\hfill \end{array}$$

(13)

From Equations (1) and (13), we obtain

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)P_n^{\alpha }\left(\sigma \left(t\right)\right)z^{\alpha }\left({\tau }^{-1}\left(\sigma \left(t\right)\right)\right)\le 0.$$

Since $\eta \left(t\right)\le \sigma \left(t\right)$ and $z^{\prime }\left(t\right)>0$, we get

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)P_n^{\alpha }\left(\sigma \left(t\right)\right)z^{\alpha }\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right).$$

(14)

Now, by using Lemma 2, we have

$$z\left(t\right)\ge \frac{\mu }{\left(n-1\right)!}{t}^{n-1}{z}^{\left(n-1\right)}\left(t\right).$$

(15)

for some $\mu \in \left(0,1\right)$. It follows from (14) and (15) that, for all $\mu \in \left(0,1\right),$

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+{\left(\frac{\mu {\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)}^{n-1}}{\left(n-1\right)!}\right)}^{\alpha }q\left(t\right)P_n^{\alpha }\left(\sigma \left(t\right)\right){\left(z^{\left(n-1\right)}\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)\right)}^{\alpha }\le 0.$$

Thus, if we set $\psi \left(t\right)=r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }$, then we see that $\psi$ is a positive solution of the first-order delay differential inequality

$${\psi }^{\prime }\left(t\right)+{\left(\frac{\mu {\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)}^{n-1}}{\left(n-1\right)!r^{1/\alpha }\left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)}\right)}^{\alpha }q\left(t\right)P_n^{\alpha }\left(\sigma \left(t\right)\right)\psi \left({\tau }^{-1}\left(\eta \left(t\right)\right)\right)\le 0.$$

It is well known (see [21] (Theorem 1)) that the corresponding Equation (9) also has a positive solution, which is a contradiction.

Assume that Case $\left({\mathbf{I}}_2\right)$ holds. Using Lemma 1, we get that

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

(16)

and thus the function $t^{-1}z\left(t\right)$ is nonincreasing, eventually. Since ${\tau }^{-1}\left(t\right)\le {\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)$, we obtain

$${\tau }^{-1}\left(t\right)z\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)\le {\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)z\left({\tau }^{-1}\left(t\right)\right).$$

(17)

Combining (11) and (17), we find

$$\begin{array}{ccc}\hfill x\left(t\right)& \ge & \frac{1}{p\left({\tau }^{-1}\left(t\right)\right)}\left(1-\frac{\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}{\left({\tau }^{-1}\left(t\right)\right)p\left({\tau }^{-1}\left({\tau }^{-1}\left(t\right)\right)\right)}\right)z\left({\tau }^{-1}\left(t\right)\right)\hfill \\ & =& P_2\left(t\right)z\left({\tau }^{-1}\left(t\right)\right),\hfill \end{array}$$

which with (1) yields

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+q\left(t\right)P_2^{\alpha }\left(\sigma \left(t\right)\right)z^{\alpha }\left({\tau }^{-1}\left(\sigma \left(t\right)\right)\right)\le 0.$$

Since $\zeta \left(t\right)\le \sigma \left(t\right)$ and $z^{\prime }\left(t\right)>0$, we have that

$${\left(r\left(t\right){\left(z^{\left(n-1\right)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)P_2^{\alpha }\left(\sigma \left(t\right)\right)z^{\alpha }\left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right).$$

(18)

Integrating the (18) from t to ∞, we obtain

$$z^{\left(n-1\right)}\left(t\right)\ge R_0\left(t\right)z\left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right).$$

Integrating this inequality from t to ∞ a total of $n-3$ times, we obtain

$$z^{″}\left(t\right)+R_{n-3}\left(t\right)z\left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right)\le 0.$$

(19)

Thus, if we set $\varphi \left(t\right):=z^{\prime }\left(t\right)$ and using (16), then we conclude that $\varphi$ is a positive solution of

$${\varphi }^{\prime }\left(t\right)+{\tau }^{-1}\left(\zeta \left(t\right)\right)R_{n-3}\left(t\right)\varphi \left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right)\le 0.$$

(20)

It is well known (see [21] (Theorem 1)) that the corresponding Equation (10) also has a positive solution, which is a contradiction. The proof is complete. ☐

Corollary 1.

Assume that (7) holds and there exist positive functions $\eta ,\zeta$ such that (8) holds. If

$$\underset{t\to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\eta \left(t\right)\right)}^t{\left(\frac{{\left({\tau }^{-1}\left(\eta \left(s\right)\right)\right)}^{n-1}}{r^{1/\alpha }\left({\tau }^{-1}\left(\eta \left(s\right)\right)\right)}\right)}^{\alpha }q\left(s\right)P_n^{\alpha }\left(\sigma \left(s\right)\right)\mathrm{d}s>\frac{{\left(\left(n-1\right)!\right)}^{\alpha }}{\mathrm{e}}$$

(21)

and

$$\underset{t\to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\zeta \left(t\right)\right)}^t{\tau }^{-1}\left(\zeta \left(s\right)\right)R_{n-3}\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}},$$

(22)

then (1) is oscillatory.

Proof. 

It is well-known (see, e.g., [14] (Theorem 2)) that Condition (21) and (22) imply oscillation of (9) and (10), respectively. ☐

Example 1.

Consider the equation

$${\left(x\left(t\right)+p_{0}x\left(\delta t\right)\right)}^{\left(n\right)}+\frac{q_0}{t^n}x\left(\lambda t\right)=0,$$

(23)

where $t\ge 1,q_0>0,$ $\delta \in \left(p_0^{-1/\left(n-1\right)},1\right)$ and $\lambda \in \left(0,\delta \right).$ We note that $r\left(t\right)=1,$ $p\left(t\right)=p_0,$ $\tau \left(t\right)=\delta t,\sigma \left(t\right)=\lambda t$ and $q\left(t\right)=q_0/t^n$. Thus, if we choose $\eta \left(t\right)=\zeta \left(t\right)=\lambda t$, then it’s easy to see that (7) and (8) are satisfied. Moreover, we have

$$\begin{array}{ccc}\hfill P_k\left(t\right)& =& \frac{1}{p_0}\left(1-\frac{{\delta }^{1-k}}{p_0}\right),fork=2,n,\hfill \\ \hfill R_0\left(t\right)& =& \frac{q_0}{p_0}\left(1-\frac{1}{\delta p_0}\right)\frac{t^{1-n}}{\left(n-1\right)},\hfill \end{array}$$

and

$$R_{n-3}\left(t\right)=\frac{1}{\left(n-3\right)!}\frac{q_0}{p_0}\left(1-\frac{1}{\delta p_0}\right)\frac{1}{\left(n-2\right)\left(n-1\right)t^2}.$$

Hence, Condition (21) and (22) become

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

(24)

and

$$q_0\frac{1}{p_0}\frac{\lambda }{\delta }\left(1-\frac{1}{\delta p_0}\right)ln\frac{\delta }{\lambda }>\frac{\left(n-1\right)!}{\mathrm{e}},$$

(25)

respectively. It’s easy to see that (24) implies (25).

Therefore, by Corollary 1, we conclude that (23) is oscillatory if (24) holds.

Remark 1.

For Equation (23), in particular case that $n=4,$ $p_0=16,$ $\delta =1/2$ and $\lambda =1/3$, Condition (24) yields $q_0>587.93$. Whereas, the criterion obtained from the results of [22] is $q_0>4850.4$. Hence, our results improve the results in [22].

3. Conclusions

In this paper, our method is based on presenting a new comparison theorem that compare the higher-order Equation (1) with a couple of first-order equations. There are numerous results concerning the oscillation criteria of first order Equations (9) and (10) (see, e.g., [14,25,26,27]), 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 obtain oscillation criteria of (1) if $z\left(t\right):=x\left(t\right)-p\left(t\right)x\left(\tau \left(t\right)\right)$ in the future work.