Oscillation Results of Emden–Fowler-Type Differential Equations

Abstract

In this article, we obtain oscillation conditions for second-order differential equation with neutral term. Our results extend, improve, and simplify some known results for neutral delay differential equations. Several effective and illustrative implementations are provided.

1. Introduction

The neutral differential equations are of great importance in life, due to their contribution to many applications in science and technology. We find this type of equations in applications of engineering in-flight vibration, medicine represented in the heartbeat, physics, see [1,2]. For this, many researchers have been interested in studying the oscillations of Emden-Fowler neutral differential equations, see [3].

This manuscript focuses on the oscillatory properties of solutions to second-order differential equation with neutral term:

$${\left(\varsigma \left(y\right){\left[{\left({\xi }_1\left(y\right)+{\alpha }_1\left(y\right){\xi }_1\left({\alpha }_2\left(y\right)\right)\right)}^{\prime }\right]}^{r_1}\right)}^{\prime }+f\left(y,{\xi }_1\left({\beta }_{ı}\left(y\right)\right)\right) =0,$$

(1)

where $y\ge y_0$ and ${\xi }_2\left(y\right)={\xi }_1\left(y\right)+{\alpha }_1\left(y\right){\xi }_1\left({\alpha }_2\left(y\right)\right).$ Our novel outcomes are obtained by considering the following suppositions:

$$\begin{array}{c}\left\{\begin{array}{cc}{\alpha }_2,{\beta }_{ı}\in C\left(\left[y_0,\infty \right),\mathbb{R}\right),{\beta }_{ı}^{\prime }\left(y\right)>0,{\alpha }_2\left(y\right)\le y,\underset{y\to \infty }{lim}{\alpha }_2\left(y\right)=\infty ,\underset{y\to \infty }{lim}{\beta }_{ı}\left(y\right)=\infty ,\hfill & \\ f\left(y,{\xi }_1\right)\in C\left(\left[y_0,\infty \right)\times \mathbb{R},\mathbb{R}\right),\zeta \in C\left(\left[y_0,\infty \right),\mathbb{R}\right),{\alpha }_1\left(y\right)\in \left[0,1\right),\hfill & \\ {\xi }_{1}f\left(y,{\xi }_1\right)>0\mathrm{for}\mathrm{all}{\xi }_1\ne 0,\left|f\left(y,{\xi }_1\right)\right|\ge \sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\left|{\xi }_1\right|}^{r_2},j\ge 1,ı=1,2,..,j,\hfill & \\ \varsigma \mathrm{and}{\alpha }_1\mathrm{are}\mathrm{positive}\mathrm{functions};r_1,r_2\mathrm{are}\mathrm{quotient}\mathrm{of}\mathrm{odd}\mathrm{positive}\mathrm{integers}.\hfill \end{array}\right.\hfill \end{array}$$

(2)

The following relations are satisfied:

$$\varpi \left(y_0\right) :={\int }_{y_0}^{\infty }{\varsigma }^{-1/r_1}\left(v\right)\mathrm{d}v<\infty ,$$

(3)

and

$${\alpha }_1\left(y\right) <\varpi \left(y\right)/\varpi \left({\alpha }_2\left(y\right)\right).$$

(4)

In the context of oscillation theory, it has been the object of many researchers who have investigated this notion for Emden–Fowler neutral differential and difference equations; the reader can refer to Reference [4,5,6,7,8,9,10,11,12,13,14,15,16].

The authors in Reference [5,10,14,15] considered equation of the form

$${\left(\varsigma \left(y\right){\left[{\left({\xi }_1\left(y\right)+{\alpha }_1\left(y\right){\xi }_1\left({\alpha }_2\left(y\right)\right)\right)}^{\prime }\right]}^{r_1}\right)}^{\prime }+\zeta \left(y\right){\xi }_1^{r_2}\left(\beta \left(y\right)\right) =0.$$

(5)

The authors used Riccati technique that differs from the one we used in this article. Moreover, they also studied the equation under the condition $\varpi \left(y_0\right)=\infty$ which is different from our condition $\varpi \left(y_0\right) <\infty$.

Liu et al. [17] proved that the Equation (5) is oscillates or ${lim}_{y\to \infty }{\xi }_1\left(y\right)=0$ if

$${\int }_{y_0}^{\infty }\left({\eta }_1\left(s\right)\zeta \left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\right)}^{r_2}-\frac{2^{r_2}{\left({\eta }_1^{\prime }\left(s\right)\right)}^2{\varsigma }^{r_2/r_1}\left({\beta }_{ı}\left(s\right)\right)}{8r_2{M}^{\frac{r_1-r_2}{r_1}}{\beta }_{ı}^{r_2-1}\left(s\right){\beta }_{ı}^{\prime }\left(s\right){\eta }_1\left(s\right)}\right)\mathrm{d}s=\infty ,$$

(6)

and

$${\int }_{y_0}^{\infty }{\left(\frac{1}{\varsigma \left(y\right)\phi \left(y\right)}{\int }_{y_0}^y\phi \left(s\right)\zeta \left(s\right)\mathrm{d}s\right)}^{1/r_1}\mathrm{d}y=\infty ,$$

(7)

where ${\eta }_1,\phi \in C\left(\left[y_0,\infty \right),\left[0,\infty \right)\right)$ and ${\eta }_1>0$ and

$${\alpha }_1^{\prime }\left(y\right)\ge 0,{\alpha }_2^{\prime }\left(y\right)\ge 0\mathrm{and}\underset{y\to \infty }{lim}{\alpha }_1\left(y\right)=C.$$

(8)

On the other hand, in Reference [18], the authors obtained oscillation criteria for equation

$${\left(\varsigma \left(y\right){\left|{\xi }_2^{\prime }\left(y\right)\right|}^{r_1-1}{\xi }_2^{\prime }\left(y\right)\right)}^{\prime }+\zeta \left(y\right){\left|{\xi }_1\left({\beta }_{ı}\left(y\right)\right)\right|}^{r_2-1}{\xi }_1\left({\beta }_{ı}\left(y\right)\right) =0,$$

(9)

where

$${\alpha }_1^{\prime }\left(y\right) \ge 0,{\varsigma }^{\prime }\left(y\right) \ge 0,{\alpha }_2^{\prime }\left(y\right) \ge 0\mathrm{and}{\beta }_{ı}\left(y\right) \le {\alpha }_2\left(y\right),$$

(10)

and used the comparing technique to ensure that the equation is oscillatory if

$${\int }_{y_0}^{\infty }\left({\eta }_1\left(s\right)\zeta \left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\right)}^{r_2}-\frac{{\left({\eta }_1^{\prime }\left(s\right)\right)}^{{\epsilon }_1+1}\varsigma \left({\epsilon }_1\left(s\right)\right)}{{\left({\epsilon }_1+1\right)}^{{\epsilon }_1+1}{\left(m{\eta }_1\left(s\right){\beta }_{ı}^{\prime }\left(s\right)\right)}^{{\epsilon }_1}}\right)\mathrm{d}s=\infty ,$$

(11)

and

$${\int }_{y_0}^{\infty }\left({\varpi }^{{\epsilon }_2}\left(s\right)\zeta \left(s\right){\left(1-{\alpha }_1\left(s\right)\right)}^{r_2}-\frac{K{\varsigma }^{1-\frac{{\epsilon }_2+1}{r_1}}\left(s\right)}{\varpi \left(s\right)}\right)\mathrm{d}s=\infty ,$$

(12)

where ${\eta }_1\in C\left(\left[y_0,\infty \right),\left(0,\infty \right)\right)$ and ${\epsilon }_1=min\left\{r_1,r_2\right\}$, ${\epsilon }_2=max\left\{r_1,r_2\right\}$,

$${\epsilon }_1\left(y\right)=\left\{\begin{array}{cc}\beta \left(y\right)& \mathrm{if}{r}_2\ge r_1\\ y& \mathrm{if}{r}_2<r_1\end{array}\right.\mathrm{and}m=\left\{\begin{array}{cc}1& r_1=r_2\\ 0<m\le 1& r_1\ne r_2\end{array}\right..$$

Saker [13] explained that the Equation (5) is oscillatory if

$${\int }_y^{\infty }{\epsilon }_2\left(s\right)\zeta \left(s\right){\left(\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\right)\frac{{\int }_y^{{\beta }_{ı}\left(y\right)}{\varsigma }^{-1/r_1}\left(x\right)\mathrm{d}x}{{\int }_y^y{\varsigma }^{-1/r_1}\left(x\right)\mathrm{d}x}\right)}^{r_2}\mathrm{d}s=\infty ,$$

(13)

and

$${\int }_y^{\infty }{\left(\frac{1}{\varsigma \left(s\right)}{\int }_y^s{\varpi }^{r_2}\left(x\right)\zeta \left(x\right){\left(1-{\alpha }_1\left(x\right)\right)}^{r_2}\mathrm{d}x\right)}^{1/r_1}\mathrm{d}s=\infty ,$$

(14)

where

$${\epsilon }_2\left(y\right)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}{r}_2=r_1\\ c_2{\left({\int }_y^y{\varsigma }^{-1/r_1}\left(s\right)\mathrm{d}s\right)}^{r_2-r_1}\hfill & \mathrm{if}{r}_2<r_1\\ c_1\hfill & \mathrm{if}{r}_2>r_1\end{array}\right.,$$

and $c_1,$$c_2$ are any positive constants.

In Reference [8,9,19,20,21,22], the authors used the comparing technique with first-order to ensure that Equation (5) is oscillatory and also explained that the equation

$${\xi }_2^{\prime }\left(y\right)+\tilde{X}\left(y\right){\xi }_2^{r_2/r_1}\left({\beta }_{ı}\left(y\right)\right) =0,$$

(15)

is oscillatory if and only if

$${\int }_{y_0}^y\tilde{X}\left(s\right)\mathrm{d}s=\infty ,$$

(16)

and

$$\underset{y\to \infty }{lim}{\int }_{{\beta }_{ı}\left(y\right)}^y\tilde{X}\left(s\right)\mathrm{d}s=\infty ,$$

where

$$X\left(y\right) =\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(y\right)\right)\frac{\varpi \left({\alpha }_2\left({\beta }_{ı}\left(y\right)\right)\right)}{\varpi \left({\beta }_{ı}\left(y\right)\right)}\right)}^{r_2},$$

and

$$\tilde{X}\left(y\right) ={\left(\frac{1}{\varsigma \left(y\right)}{\int }_{y_1}^{y}X\left(s\right)\mathrm{d}s\right)}^{1/r_1}.$$

From the above, we can observe the following:

-

The conditions that [17] obtained guarantee that all solutions of (5) are either oscillatory or tends to zero;

-

To apply the results in Reference [13,17,18], we must ensure that ${\eta }_1\left(y\right)$ and ${\alpha }_2\left(y\right)$ are nondecreasing.

The purpose of this paper is to extend, improve, and simplify results in [13,17,18] and establish new oscillation criteria for (1). Our results improved results in Reference [17] so that the condition guarantee that all solution of (1) is oscillatory, and without imposing restrictions on the derivatives of ${\eta }_1\left(y\right)$ and ${\alpha }_2\left(y\right)$. In addition, we simplify the results in Reference [13,18].

2. Oscillation Criteria

Theorem 1.

If

$${\int }_{y_0}^{\infty }{\left(\frac{1}{\varsigma \left(x\right)}{\int }_{y_0}^x{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s\right)}^{1/r_1}\mathrm{d}x=\infty ,$$

(17)

where

$$X\left(y\right)=\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(y\right)\right)\frac{\varpi \left({\alpha }_2\left({\beta }_{ı}\left(y\right)\right)\right)}{\varpi \left({\beta }_{ı}\left(y\right)\right)}\right)}^{r_2},$$

then Equation (1) is oscillatory.

Proof. 

Let ${\xi }_1$ be a nonoscillatory solution of Equation (1); then, ${\xi }_1\left(y\right) >0,{\xi }_1\left({\alpha }_2\left(y\right)\right) >0$ and ${\xi }_1\left({\beta }_{ı}\left(y\right)\right) >0$ for all $y\ge y_1$. Since ${\xi }_1\left(y\right) >0$, we see that ${\xi }_2\left(y\right) >0$ and

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }=-f\left(y,{\xi }_1\left({\beta }_{ı}\left(y\right)\right)\right)\le -\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\xi }_1^{r_2}\left({\beta }_{ı}\left(y\right)\right)<0.$$

(18)

Then, $\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}$ is strictly decreasing on $\left[y_0,\infty \right)$ and of one sign.

Let ${\xi }_2^{\prime }\left(y\right)<0$ for $y\ge y_1$. Hence,

$$\begin{array}{ccc}\hfill {\xi }_2\left(y\right)& \ge & -{\int }_y^{\infty }\frac{1}{{\varsigma }^{1/r_1}\left(s\right)}\left({\varsigma }^{1/r_1}\left(s\right){\xi }_2^{\prime }\left(s\right)\right)\mathrm{d}s\hfill \\ & \ge & -{\varsigma }^{1/r_1}\left(y\right){\xi }_2^{\prime }\left(y\right)\varpi \left(y\right).\hfill \end{array}$$

Since

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }<0,$$

we find

$$\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\le \varsigma \left(y_1\right){\left[{\xi }_2^{\prime }\left(y_1\right)\right]}^{r_1}:=-K<0,$$

for all $y\ge y_1$; thus,

$${\xi }_2\left(y\right)\ge K^{1/r_1}\varpi \left(y\right)\mathrm{for}\mathrm{all}y\ge y_1.$$

(19)

From (19), we see

$$\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac{{\xi }_2\left(y\right)}{\varpi \left(y\right)}\right)\ge 0.$$

We can also note that

$${\xi }_1\left(y\right) ={\xi }_2\left(y\right)-{\alpha }_1\left(y\right){\xi }_1\left({\alpha }_2\left(y\right)\right),$$

and

$${\xi }_1\left(y\right) \ge {\xi }_2\left(y\right)-{\alpha }_1\left(y\right){\xi }_2\left({\alpha }_2\left(y\right)\right).$$

Then,

$${\xi }_1\left(y\right) \ge {\xi }_2\left(y\right)\left(1-{\alpha }_1\left(y\right)\frac{\varpi \left({\alpha }_2\left(y\right)\right)}{\varpi \left(y\right)}\right).$$

From (18), we have

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }\le -\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\xi }_2^{r_2}\left({\beta }_{ı}\left(y\right)\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(y\right)\right)\frac{\varpi \left({\alpha }_2\left({\beta }_{ı}\left(y\right)\right)\right)}{\varpi \left({\beta }_{ı}\left(y\right)\right)}\right)}^{r_2}.$$

(20)

Combining (19) with (20) yields

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }\le -K^{r_2/r_1}{\varpi }^{r_2}\left({\beta }_{ı}\left(y\right)\right)X\left(y\right).$$

(21)

By integrating (21) from $y_1$ to y, we find

$$\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\le -K^{r_2/r_1}{\int }_{y_1}^y{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s.$$

(22)

Integrating (22) from $y_1$ to y, we obtain

$${\xi }_2\left(y\right)\le {\xi }_2\left(y_1\right)-K^{r_2/r_1^2}{\int }_{y_1}^y{\left(\frac{1}{\varsigma \left(x\right)}{\int }_{y_1}^x{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s\right)}^{1/r_1}\mathrm{d}x.$$

(23)

From (17) and (23), we find that

$${\xi }_2\left(y\right) \to -\infty \mathrm{as}y\to \infty ,$$

while this is a contradiction.

Now, let ${\xi }_2^{\prime }\left(y\right)>0$. Then, ${\xi }_2\left(y\right)>{\xi }_2\left({\alpha }_2\left(y\right)\right)>{\xi }_1\left({\alpha }_2\left(y\right)\right)$; hence,

$${\xi }_1\left(y\right) = \left(1-{\alpha }_1\left(y\right)\right){\xi }_2\left(y\right).$$

From (18), we find

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }\le -\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(y\right)\right)\right)}^{r_2}{\xi }_2^{r_2}\left({\beta }_{ı}\left(y\right)\right).$$

(24)

Since ${\varpi }^{\prime }\left(y\right)<0$, we see that

$$\varpi \left({\alpha }_2\left({\beta }_{ı}\left(y\right)\right)\right)\ge \varpi \left({\beta }_{ı}\left(y\right)\right)\mathrm{for}y\ge y_2\ge y_1.$$

Then, and from (24), we get

$${\left(\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\right)}^{\prime }\le -\sum _{ı=1}^j{\zeta }_{ı}\left(y\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(y\right)\right)\right)}^{r_2}{\xi }_2^{r_2}\left({\beta }_{ı}\left(y\right)\right).$$

(25)

Integrating (25) from $y_1$ to y, we obtain

$$\begin{array}{ccc}\hfill \varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}& \le & \varsigma \left(y_2\right){\left[{\xi }_2^{\prime }\left(y_2\right)\right]}^{r_1}-{\int }_{y_2}^y\sum _{ı=1}^j{\zeta }_{ı}\left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\right)}^{r_2}{\xi }_2^{r_2}\left({\beta }_{ı}\left(s\right)\right)\mathrm{d}s\hfill \\ & \le & \varsigma \left(y_2\right){\left[{\xi }_2^{\prime }\left(y_2\right)\right]}^{r_1}-{\xi }_2^{r_2}\left({\beta }_{ı}\left(y_2\right)\right){\int }_{y_2}^y\sum _{ı=1}^j{\zeta }_{ı}\left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\right)}^{r_2}\mathrm{d}s\hfill \\ & \le & \varsigma \left(y_2\right){\left[{\xi }_2^{\prime }\left(y_2\right)\right]}^{r_1}-{\xi }_2^{r_2}\left({\beta }_{ı}\left(y_2\right)\right){\int }_{y_2}^y\sum _{ı=1}^j{\zeta }_{ı}\left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\frac{\varpi \left({\alpha }_2\left({\beta }_{ı}\left(y\right)\right)\right)}{\varpi \left({\beta }_{ı}\left(y\right)\right)}\right)}^{r_2}\mathrm{d}s\hfill \\ & \le & \varsigma \left(y_2\right){\left[{\xi }_2^{\prime }\left(y_2\right)\right]}^{r_1}-{\xi }_2^{r_2}\left({\beta }_{ı}\left(y_2\right)\right){\int }_{y_2}^{y}X\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(26)

Since ${\varpi }^{\prime }\left(y\right)<0$, we obtain

$${\int }_{y_2}^y{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s\le {\varpi }^{r_2}\left({\beta }_{ı}\left(y_2\right)\right){\int }_{y_2}^{y}X\left(s\right)\mathrm{d}s.$$

(27)

From (17), we see that ${\int }_{y_2}^y{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s$ must be unbounded. Hence, from (27), we obtain

$${\int }_{y_2}^{y}X\left(s\right)\mathrm{d}s\to \infty \mathrm{as}y\to \infty .$$

(28)

From (26), we find

$${\xi }_2^{\prime }\left(y\right) \to -\infty \mathrm{as}y\to \infty ,$$

while this is a contradiction to ${\xi }_2^{\prime }\left(y\right)>0$. Thus, the theorem is proved. □

Theorem 2.

If

$${\xi }_2^{\prime }\left(y\right)+\tilde{X}\left(y\right){\xi }_2^{r_2/r_1}\left({\beta }_{ı}\left(y\right)\right) =0$$

(29)

is oscillatory and

$${\int }_{y_0}^{\infty }X\left(s\right)\mathrm{d}s=\infty .$$

(30)

Then, Equation (1) is oscillatory.

Proof. 

Let ${\xi }_1$ be a nonoscillatory solution of Equation (1); then, ${\xi }_1\left(y\right)>0$ and ${\xi }_2^{\prime }\left(y\right)$ is of one sign eventually.

Let ${\xi }_2^{\prime }\left(y\right)<0$ for all $y\ge y_1$. Integrating (20) from $y_1$ to y, we see that

$$\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\le \varsigma \left(y_1\right){\left[{\xi }_2^{\prime }\left(y_1\right)\right]}^{r_1}-{\int }_{y_1}^y{\xi }_2^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s.$$

Since ${\beta }_{ı}^{\prime }\left(y\right)>0$, we find

$$\varsigma \left(y\right){\left[{\xi }_2^{\prime }\left(y\right)\right]}^{r_1}\le -{\xi }_2^{r_2}\left({\beta }_{ı}\left(y\right)\right){\int }_{y_1}^{y}X\left(s\right)\mathrm{d}s,$$

so

$${\xi }_2^{\prime }\left(y\right)\le -{\xi }_2^{r_2/r_1}\left({\beta }_{ı}\left(y\right)\right){\left(\frac{1}{\varsigma \left(y\right)}{\int }_{y_1}^{y}X\left(s\right)\mathrm{d}s\right)}^{1/r_1}.$$

Hence, we have that ${\xi }_2\left(y\right) >0$ for equation

$${\xi }_2^{\prime }\left(y\right) + \tilde{X}\left(y\right){\xi }_2^{r_2/r_1}\left({\beta }_{ı}\left(y\right)\right) \le 0.$$

(31)

From Reference Lemma 1 in [20], Equation (29) has a positive solution. This indicates that is no positive solution of (1), while this is a contradiction.

Let ${\xi }_2^{\prime }\left(y\right)>0$ for all $y\ge y_1$. Thus, we find (30) leads to (28), and then the remainder of the proof is the same as the proof of Theorem 1. Thus, the theorem is proved. □

Next, we establish new oscillation conditions for Equation (1) according to the results obtained by Reference [8,20].

Corollary 1.

Let $0<r_2<r_1$ and

$${\int }_{y_0}^y{\left(\frac{1}{\varsigma \left(s\right)}{\int }_{y_1}^{y}X\left(z\right)\mathrm{d}z\right)}^{1/r_1}\mathrm{d}s=\infty ;$$

then, Equation (1) is oscillatory.

Corollary 2.

Let $r_2>r_1$ and

$${\int }_{y_0}^{\infty }\sum _{ı=1}^j{\zeta }_{ı}\left(s\right){\left(1-{\alpha }_1\left({\beta }_{ı}\left(s\right)\right)\frac{\varpi \left({\alpha }_2\left({\beta }_{ı}\left(s\right)\right)\right)}{\varpi \left({\beta }_{ı}\left(s\right)\right)}\right)}^{r_2}\mathrm{d}s=\infty .$$

If

$$\underset{y\to \infty }{limsup}\frac{r_2{\theta }^{\prime }\left({\beta }_{ı}\left(y\right)\right){\beta }_{ı}^{\prime }\left(y\right)}{r_1{\theta }^{\prime }\left(y\right)}<1,$$

and

$$\underset{y\to \infty }{liminf}\left(\frac{1}{{\theta }^{\prime }\left(y\right)}{e}^{-\theta \left(y\right)}\tilde{X}\left(y\right)\right)>0,$$

(32)

where $\theta \left(y\right)\in C^1\left(\left[y_0,\infty \right),\mathbb{R}\right),$${\theta }^{\prime }\left(y\right)>0$ and ${lim}_{y\to \infty }\theta \left(y\right)=\infty .$ Then, Equation (1) is oscillatory.

3. Applications

Through this section, we will provide some examples and applications that support the validity of the results and conditions that we got in our article.

Example 1.

Consider the equation

$${\left(y^2{\left({\xi }_1\left(y\right)+\frac{\mu }{y}{\xi }_1\left(y-{\epsilon }_1\right)\right)}^{\prime }\right)}^{\prime }+a\left(y-{\epsilon }_2\right){\xi }_1\left(y-{\epsilon }_2\right)=0,y\ge y_0,$$

(33)

where $\mu ,{\epsilon }_1,a$, and ${\epsilon }_2$ are positive real numbers, and $y_0=max\left\{\mu +{\epsilon }_1,{\epsilon }_2+{\epsilon }_1\right\}.$

We have that $r_1=r_2=1,$ $\varsigma \left(y\right)=y^2,$ ${\alpha }_1\left(y\right)=\mu /y,$ ${\alpha }_2\left(y\right)=y-{\epsilon }_1,$ $\zeta \left(y\right)=a\left(y-{\epsilon }_2\right)$, and ${\beta }_{ı}\left(y\right)=y-{\epsilon }_2$. This gives that $\varpi \left(y\right)=y^{-1}$ and

$$X\left(y\right)=a\left(y-{\epsilon }_2\right)\left(1-\frac{\mu }{y-{\epsilon }_2-{\epsilon }_1}\right).$$

Since $y>\mu +{\epsilon }_1$, we have that $\mu /y<\left(y-{\epsilon }_1\right)/y$, and hence ${\alpha }_1\left(y\right)<\varpi \left(y\right)/\varpi \left({\alpha }_2\left(y\right)\right)$. Note that,

$${\int }_{y_0}^{\infty }\left(\frac{1}{x^2}{\int }_{y_0}^x\left(1-\frac{\mu }{y-{\epsilon }_2-{\epsilon }_1}\right)\mathrm{d}s\right)\mathrm{d}x=\infty .$$

By Theorem 1, Equation (33) is oscillatory.

Example 2.

Let the equation

$${\left(e^{-y}{\left({\left({\xi }_1\left(y\right)+e^{-y}{\xi }_1\left({\alpha }_2\left(y\right)\right)\right)}^{\prime }\right)}^{11/3}\right)}^{\prime }+\frac{1}{y+1}{\left({\xi }_1\left(y-2\right)\right)}^{7/3}+\frac{1}{2+y}{\left({\xi }_1\left(y-3\right)\right)}^{7/3}=0.$$

(34)

Let, $r_1=11/3,r_2=7/3,\varsigma \left(y\right)=e^{-y},{\alpha }_1\left(y\right)=e^{-y},$${\beta }_{ı}\left(y\right)=y-\left(j+1\right)$ with index $j=1,2$ and

$$\varpi \left(y_0\right)={\int }_{y_0}^{\infty }{\varsigma }^{-1/r_1}\left(v\right)\mathrm{d}v=\frac{11}{3}\left(e^{3y/11}-1\right).$$

Thus, the condition of Theorem 1 hold, and therefore, each solution of (34) is oscillatory.

Example 3.

Let the equation

$${\left(e^y{\left({\xi }_1\left(y\right)+e^{-y}{\xi }_1\left({\epsilon }_{1}y\right)\right)}^{\prime }\right)}^{\prime }+e^{ay}{\xi }_1^{r_2}\left({\epsilon }_{2}y\right)=0,y>0,$$

(35)

where ${\epsilon }_1,{\epsilon }_2\in \left(0,1\right)$ and $r_2>1.$ Let, $r_1=1,\varsigma \left(y\right)=e^y,{\alpha }_1\left(y\right)=e^{-y},$$\zeta \left(y\right)=e^{ay},{\alpha }_2\left(y\right)={\epsilon }_{1}y$ and ${\beta }_{ı}\left(y\right)={\epsilon }_{2}y$. This gives that $\varpi \left(y\right)=e^{-y}$ and

$$X\left(y\right)=e^{ay}{\left(1-e^{-{\epsilon }_1{\epsilon }_{2}y}\right)}^{r_2}.$$

Moreover, we find

$$\begin{array}{ccc}& & {\int }_{y_0}^{\infty }{\left(\frac{1}{\varsigma \left(x\right)}{\int }_{y_0}^x{\varpi }^{r_2}\left({\beta }_{ı}\left(s\right)\right)X\left(s\right)\mathrm{d}s\right)}^{1/r_1}\mathrm{d}x\hfill \\ & & ={\int }_{y_0}^{\infty }\left(e^{-x}{\int }_{y_0}^x{e}^{ay}{\left[e^{-{\epsilon }_{2}y}\left(1-e^{-{\epsilon }_1{\epsilon }_{2}y}\right)\right]}^{r_2}\mathrm{d}s\right)\mathrm{d}x=\infty ,\hfill \end{array}$$

if $a>r_2{\epsilon }_2+1$. Thus, by Theorem 1, every solution of (35) oscillates if $a>r_2{\epsilon }_2+1$.

Remark 1.

We see that ${\alpha }_1^{\prime }\left(y\right) <0$ in Example 3; thus, the results in Reference [13,17,18] cannot be applied in Equation (35).

4. Conclusions

In this work, we have presented many of the oscillatory properties of the second-order neutral differential equation (1). Our results improve, unify, and extend some known results for differential equations with neutral term. In future work, we will discuss oscillation conditions of these equations by using integral averaging technique and Riccati technique.