New Conditions for Testing the Oscillation of Fourth-Order Differential Equations with Several Delays

Abstract

In this paper, we establish oscillation theorems for all solutions to fourth-order neutral differential equations using the Riccati transformation approach and some inequalities. Some new criteria are established that can be used in cases where known theorems fail to apply. The approach followed depends on finding conditions that guarantee the exclusion of positive solutions, and as a result of the symmetry between the positive and negative solutions of the studied equation, we therefore exclude negative solutions. An illustrative example is given.

1. Introduction

In this paper, we are concerned with the oscillation of solutions of the fourth-order neutral differential equation

$${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right)x^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)=0,$$

(1)

where $\mathfrak{r}\ge {\mathfrak{r}}_0$ and $V\left(\mathfrak{r}\right)=x\left(\mathfrak{r}\right)+p\left(\mathfrak{r}\right)x\left(\xi \left(\mathfrak{r}\right)\right)$. In this work, we assume $\alpha$ and $\beta$ are quotients of odd positive integers and $\beta \ge \alpha ,$ $\mathfrak{a},\xi \in C^1\left[{\mathfrak{r}}_0,\infty \right),$ $p,q_j,{\zeta }_j\in C\left[{\mathfrak{r}}_0,\infty \right),$ $\mathfrak{a}\left(\mathfrak{r}\right)>0,$ ${\mathfrak{a}}^{\prime }\left(\mathfrak{r}\right)\ge 0,$ $q_j\left(\mathfrak{r}\right)>0,$ $0\le p\left(\mathfrak{r}\right)<p_0<\infty ,$ ${\xi }^{\prime }\left(\mathfrak{r}\right)\ge {\xi }_0>0,$ ${\zeta }_j\circ \xi =\xi \circ {\zeta }_j,$ ${\zeta }_j\left(\mathfrak{r}\right)\le \xi \left(\mathfrak{r}\right)\le \mathfrak{r},$ ${lim}_{\mathfrak{r}\to \infty }\xi \left(\mathfrak{r}\right)={lim}_{\mathfrak{r}\to \infty }{\zeta }_j\left(\mathfrak{r}\right)=\infty$, and

$${\int }_{{\mathfrak{r}}_0}^{\infty }\frac{1}{{\mathfrak{a}}^{1/\alpha }\left(s\right)}\mathrm{d}s=\infty .$$

(2)

By a solution of (1) we mean a function x$\in C^3[{\mathfrak{r}}_x,\infty ),{\mathfrak{r}}_x\ge {\mathfrak{r}}_0,$ which has the property $\mathfrak{a}{\left(V^{‴}\right)}^{\alpha }\in C^1[{\mathfrak{r}}_x,\infty ),$ and satisfies (1) on $[{\mathfrak{r}}_x,\infty )$. We consider only those solutions x of (1) which satisfy $sup\{\left|x\left(\mathfrak{r}\right)\right|:\mathfrak{r}\ge \mathfrak{r}\}>0,$ for all $\mathfrak{r}\ge {\mathfrak{r}}_x$. A solution x of (1) is said to be nonoscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory.

The differential and functional differential equations arise in many applied problems in natural sciences and engineering; see Hale [1].

The oscillation theory has become a significant numerical mathematical tool for many disciplines and high technologies. The subject of finding oscillation criteria for certain functional DEs has been a highly active study area in recent decades; for example, see [2,3,4,5,6,7,8,9,10,11,12,13,14] and the references cited therein. In what follows, we briefly comment on some closely related results that motivated our study.

Baculikova et al. [15] studied the oscillatory behavior of solutions to the even-order neutral differential equation

$${\left({\left(V^{\left(n-1\right)}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+q\left(\mathfrak{r}\right)x^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)=0,$$

where $\alpha \ge 1.$ They established some oscillation results.

Agarwal et al. [16] concerned the even-order neutral differential equation

$${\left(V\left(\mathfrak{r}\right)\right)}^{\left(n\right)}+q\left(\mathfrak{r}\right)x\left(\zeta \left(\mathfrak{r}\right)\right)=0.$$

(3)

They established some sufficient conditions for oscillation by using the Riccati transformation technique.

Bazighifan et al. [17] investigated the oscillation of fourth-order nonlinear differential equation with neutral delay

$${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+q\left(\mathfrak{r}\right)x^{\beta }\left(\zeta \left(\mathfrak{r}\right)\right)=0.$$

They obtained some oscillation criteria for the equation by the theory of comparison.

Li and Rogovchenko [18] studied oscillation for (3). They used comparison with the first-order delay equation to obtain the following result:

Theorem 1.

Assume that there exist functions $\eta \in C\left[{\mathfrak{r}}_0,\infty \right)$ and $\delta \in C^1\left[{\mathfrak{r}}_0,\infty \right)$ satisfying

$$\eta \left(\mathfrak{r}\right)\le \zeta \left(\mathfrak{r}\right),\eta \left(\mathfrak{r}\right)<\xi \left(\mathfrak{r}\right),\delta \left(\mathfrak{r}\right)\le \zeta \left(\mathfrak{r}\right),\delta \left(\mathfrak{r}\right)<\xi \left(\mathfrak{r}\right),{\delta }^{\prime }\left(\mathfrak{r}\right)\ge 0$$

and

$$\underset{\mathfrak{r}\to \infty }{lim}\eta \left(\mathfrak{r}\right)=\underset{\mathfrak{r}\to \infty }{lim}\delta \left(\mathfrak{r}\right)=\infty .$$

If

$$\frac{1}{\left(n-1\right)!}\underset{\mathfrak{r}\to \infty }{liminf}{\int }_{{\xi }^{-1}\left(\eta \left(\mathfrak{r}\right)\right)}^{\mathfrak{r}}q\left(s\right)p_{*}\left(\zeta \left(s\right)\right){\left({\xi }^{-1}\left(\eta \left(s\right)\right)\right)}^{n-1}\mathrm{d}s>\frac{1}{\mathrm{e}}$$

and

$$\frac{1}{\left(n-3\right)!}\underset{\mathfrak{r}\to \infty }{liminf}{\int }_{{\xi }^{-1}\left(\delta \left(\mathfrak{r}\right)\right)}^{\mathfrak{r}}\left({\int }_s^{\infty }{\left(\varkappa -s\right)}^{n-3}q\left(\varkappa \right)p*\left(\zeta \left(\varkappa \right)\right)\mathrm{d}\varkappa \right){\xi }^{-1}\left(\delta \left(s\right)\right)\mathrm{d}s>\frac{1}{\mathrm{e}},$$

then (3) is oscillatory, where

$$p_{*}\left(\mathfrak{r}\right):=\frac{1}{p\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)}\left(1-\frac{{\left({\xi }^{-1}\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)\right)}^{n-1}}{{\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)}^{n-1}p\left({\xi }^{-1}\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)\right)}\right)$$

and

$$p*\left(\mathfrak{r}\right)=\frac{1}{p\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)}\left(1-\frac{{\xi }^{-1}\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)}{{\xi }^{-1}\left(\mathfrak{r}\right)p\left({\xi }^{-1}\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)\right)}\right).$$

The purpose of this article is to give sufficient conditions for the oscillatory behavior of (1). Based on introducing a new Riccati substitution, we obtain an improved criteria without requiring the existence of the unknown function.

We will need the following lemmas to discuss our main results:

Lemma 1 ([19]).

If the function x satisfies $x^{\left(i\right)}\left(\mathfrak{r}\right)>0,$ $i=0,1,...,n,$ and $x^{\left(n+1\right)}\left(\mathfrak{r}\right)<0,$ then

$$x\left(\mathfrak{r}\right)\ge \frac{\lambda }{n}\mathfrak{r}{x}^{\prime }\left(\mathfrak{r}\right),$$

for every $\lambda \in \left(0,1\right)$ eventually.

Lemma 2.

Assume that $\varkappa ,\varrho \ge 0$ and β is a positive real number. Then

$${\left(\varkappa +\varrho \right)}^{\beta }\le 2^{\beta -1}\left({\varkappa }^{\beta }+{\varrho }^{\beta }\right),\mathrm{for}\beta \ge 1$$

and

$${\left(\varkappa +\varrho \right)}^{\beta }\le \left({\varkappa }^{\beta }+{\varrho }^{\beta }\right),\mathrm{for}\beta \le 1.$$

Lemma 3 ([20]).

Let α bea ratio of two odd positive integers. Then

$$K\varrho -L{\varrho }^{{}^{\left(\alpha +1\right)/\alpha }}\le \frac{{\alpha }^{\alpha }}{{(\alpha +1)}^{\alpha +1}}\frac{K^{\alpha +1}}{L^{\alpha }},L>0.$$

Lemma 4.

Assume that (2) holds and $x$ is an eventually positive solution of (1). Then, $(\mathfrak{a}\left(\mathfrak{r}\right)$ ${\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }{)}^{\prime }$ $<0$ and there are the following two possible cases eventually:

$$\begin{array}{ccc}\hfill \left({\mathbf{C}}_1\right)V\left(\mathfrak{r}\right)& >& 0,V^{\prime }\left(\mathfrak{r}\right)>0,V^{″}\left(\mathfrak{r}\right)>0,V^{‴}\left(\mathfrak{r}\right)>0,V^{\left(4\right)}\left(\mathfrak{r}\right)<0,\hfill \\ \hfill \left({\mathbf{C}}_2\right)V\left(\mathfrak{r}\right)& >& 0,V^{\prime }\left(\mathfrak{r}\right)>0,V^{″}\left(\mathfrak{r}\right)<0,V^{‴}\left(\mathfrak{r}\right)>0.\hfill \end{array}$$

2. Main Results

In the sequel, we will adopt the following notation:

$${\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right)={\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\frac{1}{{\mathfrak{a}}^{1/\alpha }\left(s\right)}\mathrm{d}s,$$

$${\eta }_{k+1}\left(\mathfrak{r},{\mathfrak{r}}_1\right)={\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\eta }_k\left(\varkappa ,{\mathfrak{r}}_1\right)\mathrm{d}\varkappa ,k=1,2,$$

$$\zeta \left(\mathfrak{r}\right):=min\left\{{\zeta }_j\left(\mathfrak{r}\right):j=1,2,...,m\right\}$$

and

$$Q\left(\mathfrak{r}\right)=min\left\{q_j\left(\mathfrak{r}\right),q_j\left(\xi \left(\mathfrak{r}\right)\right):j=1,2,...,m\right\}.$$

Lemma 5.

Assume that x is a positive solution of (1). Then

$${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\beta }}{{\xi }_0}{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{Q\left(\mathfrak{r}\right)}{2^{\beta -1}}\sum _{j=1}^m{V}^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)\le 0.$$

(4)

Proof. 

Assume that x is a positive solution of (1). From (1), we obtain

$$\begin{array}{ccc}\hfill 0& =& {\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\beta }}{{\xi }_0}{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right)x^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)\hfill \\ & & +p_0^{\beta }\sum _{j=1}^m{q}_j\left(\xi \left(\mathfrak{r}\right)\right)x^{\beta }\left({\zeta }_j\left(\xi \left(\mathfrak{r}\right)\right)\right)\hfill \\ & \ge & {\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\beta }}{{\xi }_0}{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }\hfill \\ & & +Q\left(\mathfrak{r}\right)\sum _{j=1}^m\left(x^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)+p_0^{\beta }{x}^{\beta }\left({\zeta }_j\left(\xi \left(\mathfrak{r}\right)\right)\right)\right),\hfill \end{array}$$

which follows from Lemma 2 and ${\zeta }_j\circ \xi =\xi \circ {\zeta }_j$ that

$${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\frac{p_0^{\beta }}{{\xi }_0}{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }+\frac{Q\left(\mathfrak{r}\right)}{2^{\beta -1}}\sum _{j=1}^m{V}^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)\le 0.$$

The proof is complete. □

Lemma 6.

Assume that x is a positive solution of (1). If $\left(C_1\right)$ holds, then

$$V\left(\mathfrak{r}\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right){\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right).$$

(5)

Proof. 

Assume that x is a positive solution of (1). Let $\left(C_1\right)$ hold. Since ${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }\le 0$. Then we get

$$\begin{array}{ccc}\hfill V^{″}\left(\mathfrak{r}\right)& \ge & V^{″}\left(\mathfrak{r}\right)-V^{″}\left({\mathfrak{r}}_1\right)={\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\frac{{\left(\mathfrak{a}\left(s\right){\left(V^{‴}\left(s\right)\right)}^{\alpha }\right)}^{1/\alpha }}{{\mathfrak{a}}^{1/\alpha }\left(s\right)}\mathrm{d}s\hfill \\ & \ge & {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right){\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right),\hfill \end{array}$$

integrating the above inequality from ${\mathfrak{r}}_1$ to $\mathfrak{r},$ we have

$$V^{\prime }\left(\mathfrak{r}\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right){\eta }_2\left(\mathfrak{r},{\mathfrak{r}}_1\right),$$

(6)

integrating (6) from ${\mathfrak{r}}_1$ to $\mathfrak{r},$ we get

$$V\left(\mathfrak{r}\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right){\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right).$$

The proof is complete. □

Lemma 7.

Assume that x is a positive solution of (1). If $\left(C_2\right)$ holds, then

$$V^{″}\left(\mathfrak{r}\right)\le -{\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{V}^{\beta /\alpha }\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}{\int }_{{\xi }^{-1}\left(\varkappa \right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}\varkappa .$$

(7)

Proof. 

Assume that x is a positive solution of (1). Integrating (4) from $\mathfrak{r}$ to ∞ and using ${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }\le 0,$ we obtain

$$-\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }-\frac{p_0^{\beta }}{{\xi }_0}\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\le -{\int }_{\mathfrak{r}}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{V}^{\beta }\left({\zeta }_j\left(s\right)\right)\mathrm{d}s.$$

(8)

From Lemma 1 and (8), we have

$$-\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }-\frac{p_0^{\beta }}{{\xi }_0}\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\le -{\int }_{\mathfrak{r}}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}{V}^{\beta }\left(s\right)\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s,$$

that is,

$$\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }+\frac{p_0^{\beta }}{{\xi }_0}\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\ge V^{\beta }\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s,$$

since $\xi \left(\mathfrak{r}\right)\le \mathfrak{r}$ and ${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }\le 0$, then, we have

$$\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }+\frac{p_0^{\beta }}{{\xi }_0}\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\ge V^{\beta }\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s,$$

that is,

$$\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\ge \left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)V^{\beta }\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s$$

(9)

or

$$\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\ge \left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)V^{\beta }\left({\xi }^{-1}\left(\mathfrak{r}\right)\right){\int }_{{\xi }^{-1}\left(\mathfrak{r}\right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s,$$

since ${\xi }^{-1}\left(\mathfrak{r}\right)>\mathfrak{r}$ then $V\left({\xi }^{-1}\left(\mathfrak{r}\right)\right)>V\left(\mathfrak{r}\right).$ From the above inequality, we have

$$\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\ge \left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)V^{\beta }\left(\mathfrak{r}\right){\int }_{{\xi }^{-1}\left(\mathfrak{r}\right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s.$$

Integrating the above inequality from $\mathfrak{r}$ to $\infty ,$ we obtain

$$V^{″}\left(\mathfrak{r}\right)\le -{\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{V}^{\beta /\alpha }\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}{\int }_{{\xi }^{-1}\left(\varkappa \right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}\varkappa .$$

The proof is complete. □

Theorem 2.

Let $\beta \ge 1$, ${\zeta }_j\left(\mathfrak{r}\right)\in C^1\left(\left[{\mathfrak{r}}_0,\infty \right)\right)$, ${\zeta }_j^{\prime }>0$, and ${\zeta }_j\left(\mathfrak{r}\right)\le \xi \left(\mathfrak{r}\right)$. Assume that there exists a functions $\rho \left(\mathfrak{r}\right),\theta \left(\mathfrak{r}\right)\in C^1\left(\left[{\mathfrak{r}}_0,\infty \right),\left(0,\infty \right)\right)$, for all sufficiently large ${\mathfrak{r}}_1\ge {\mathfrak{r}}_0$, there is a ${\mathfrak{r}}_2>{\mathfrak{r}}_1$ such that

$$\underset{\mathfrak{r}\to \infty }{limsup}{\int }_{{\mathfrak{r}}_2}^{\mathfrak{r}}\left(m\rho \left(s\right)\frac{Q\left(s\right)}{2^{\beta -1}}{M}^{\beta -\alpha }-\left(1+\frac{p_0^{\beta }}{{\xi }_0}\right)\frac{{\left(\alpha +1\right)}^{-\left(\alpha +1\right)}{\left({\rho }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\rho \left(s\right){\eta }_2\left(\zeta \left(s\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(s\right)\right)}^{\alpha }}\right)\mathrm{d}s=\infty$$

(10)

and

$$\underset{\mathfrak{r}\to \infty }{limsup}{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\left(\theta \left(\varrho \right){\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{M}^{\left(\beta /\alpha \right)-1}{\int }_{\varrho }^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}\Phi \left(\varkappa \right)\right)}^{1/\alpha }\mathrm{d}\varkappa -\frac{{\left({\theta }_{+}^{\prime }\left(\varrho \right)\right)}^2}{4\theta \left(\varrho \right)}\right)\mathrm{d}\varrho =\infty ,$$

(11)

where

$$\Phi \left(\varkappa \right)={\int }_{{\xi }^{-1}\left(\varkappa \right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s,$$

${\rho }_{+}^{\prime }\left(\mathfrak{r}\right)=max\left\{0,{\rho }^{\prime }\left(\mathfrak{r}\right)\right\}$ and ${\theta }_{+}^{\prime }\left(\mathfrak{r}\right)=max\left\{0,{\theta }^{\prime }\left(\mathfrak{r}\right)\right\}.$ Then (1) is oscillatory.

Proof. 

Assume that x is a positive solution of (1). It follows from Lemma 4 that there exists two possible cases $\left(C_1\right)$ and $\left(C_2\right).$ Let $\left(C_1\right)$ hold. We define a function $\omega \left(\mathfrak{r}\right)$ by

$$\omega \left(\mathfrak{r}\right)=\rho \left(\mathfrak{r}\right)\frac{\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)},$$

(12)

then $\omega \left(\mathfrak{r}\right)>0.$ Differentiating (12), we have

$${\omega }^{\prime }\left(\mathfrak{r}\right)=\frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\omega \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\rho \left(\mathfrak{r}\right)\frac{\alpha \mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }{V}^{\prime }\left(\zeta \left(\mathfrak{r}\right)\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{V^{\alpha +1}\left(\zeta \left(\mathfrak{r}\right)\right)},$$

(13)

from (6) and $\zeta \left(\mathfrak{r}\right)\le {\zeta }_j\left(\mathfrak{r}\right)\le \mathfrak{r},$ we get

$$V^{\prime }\left(\zeta \left(\mathfrak{r}\right)\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)V^{‴}\left(\zeta \left(\mathfrak{r}\right)\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right)$$

(14)

and so, (13) can be written as

$${\omega }^{\prime }\left(\mathfrak{r}\right)\le \frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\omega \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha +1}{\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{-1}\left(\mathfrak{r}\right){\mathfrak{a}}^{-\left(\alpha +1\right)/\alpha }\left(\mathfrak{r}\right)V^{\alpha +1}\left(\zeta \left(\mathfrak{r}\right)\right)}.$$

(15)

It follows from (12) and (15) that

$${\omega }^{\prime }\left(\mathfrak{r}\right)\le \frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\omega \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(\mathfrak{r}\right).$$

(16)

Similarly, define another function $\psi$ by

$$\psi \left(\mathfrak{r}\right)=\rho \left(\mathfrak{r}\right)\frac{\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)},$$

(17)

then $\psi \left(\mathfrak{r}\right)>0.$ Differentiating (17), we have

$${\psi }^{\prime }\left(\mathfrak{r}\right)=\frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\psi \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }{V}^{\prime }\left(\zeta \left(\mathfrak{r}\right)\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{-1}\left(\mathfrak{r}\right){\mathfrak{a}}^{-1}\left(\xi \left(\mathfrak{r}\right)\right)V^{\alpha +1}\left(\zeta \left(\mathfrak{r}\right)\right)},$$

(18)

from (6) and $\zeta \left(\mathfrak{r}\right)\le {\zeta }_j\left(\mathfrak{r}\right)\le \xi \left(\mathfrak{r}\right),$ we get

$$V^{\prime }\left(\zeta \left(\mathfrak{r}\right)\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)V^{‴}\left(\zeta \left(\mathfrak{r}\right)\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\xi \left(\mathfrak{r}\right)\right)V^{‴}\left(\xi \left(\mathfrak{r}\right)\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right)$$

(19)

and so, (18) can be written as

$${\psi }^{\prime }\left(\mathfrak{r}\right)\le \frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\psi \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha +1}{\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{-1}\left(\mathfrak{r}\right){\mathfrak{a}}^{-\left(1+\alpha \right)/\alpha }\left(\xi \left(\mathfrak{r}\right)\right)V^{\alpha +1}\left(\zeta \left(\mathfrak{r}\right)\right)}.$$

(20)

It follows from (17) and (20) that

$${\psi }^{\prime }\left(\mathfrak{r}\right)\le \frac{{\rho }^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\psi \left(\mathfrak{r}\right)+\rho \left(\mathfrak{r}\right)\frac{{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\psi }^{\left(1+\alpha \right)/\alpha }\left(\mathfrak{r}\right).$$

(21)

Using (16) and (21), we get

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\mathfrak{r}\right)+\frac{p_0^{\beta }}{{\xi }_0}{\psi }^{\prime }\left(\mathfrak{r}\right)& \le & \rho \left(\mathfrak{r}\right)\left(\frac{{\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}+\frac{p_0^{\beta }}{{\xi }_0}\frac{{\left(\mathfrak{a}\left(\xi \left(\mathfrak{r}\right)\right){\left(V^{‴}\left(\xi \left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)}^{\prime }}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}\right)\hfill \\ & & +\frac{{\rho }_{+}^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\omega \left(\mathfrak{r}\right)-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(\mathfrak{r}\right)\hfill \\ & & +\frac{p_0^{\beta }}{{\xi }_0}\left(\frac{{\rho }_{+}^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\psi \left(\mathfrak{r}\right)-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\psi }^{\left(1+\alpha \right)/\alpha }\left(\mathfrak{r}\right)\right).\hfill \end{array}$$

(22)

By (4) and (23), we obtain

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\mathfrak{r}\right)+\frac{p_0^{\beta }}{{\xi }_0}{\psi }^{\prime }\left(\mathfrak{r}\right)& \le & -\rho \left(\mathfrak{r}\right)\frac{Q\left(\mathfrak{r}\right)}{2^{\beta -1}}\frac{{\sum }_{j=1}^m{V}^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)}{V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)}-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\omega }^{\left(\alpha +1\right)/\alpha }\left(\mathfrak{r}\right)\hfill \\ & & +\frac{p_0^{\beta }}{{\xi }_0}\left(\frac{{\rho }_{+}^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\psi \left(\mathfrak{r}\right)-\frac{\alpha {\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)}{{\rho }^{1/\alpha }\left(\mathfrak{r}\right)}{\psi }^{\left(1+\alpha \right)/\alpha }\left(\mathfrak{r}\right)\right)\hfill \\ & & +\frac{{\rho }_{+}^{\prime }\left(\mathfrak{r}\right)}{\rho \left(\mathfrak{r}\right)}\omega \left(\mathfrak{r}\right),\hfill \end{array}$$

(23)

from Lemma 3 and (24), we have

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\mathfrak{r}\right)+\frac{p_0^{\beta }}{{\xi }_0}{\psi }^{\prime }\left(\mathfrak{r}\right)& \le & -m\rho \left(\mathfrak{r}\right)\frac{Q\left(\mathfrak{r}\right)}{2^{\beta -1}}{V}^{\beta -\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)+\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }_{+}^{\prime }\left(\mathfrak{r}\right)\right)}^{\alpha +1}}{{\left(\rho \left(\mathfrak{r}\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)\right)}^{\alpha }}\hfill \\ & & +\frac{p_0^{\beta }}{{\xi }_0}\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }_{+}^{\prime }\left(\mathfrak{r}\right)\right)}^{\alpha +1}}{{\left(\rho \left(\mathfrak{r}\right){\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(\mathfrak{r}\right)\right)}^{\alpha }}.\hfill \end{array}$$

(24)

Since $V^{\prime }\left(\mathfrak{r}\right)>0$, there exists a ${\mathfrak{r}}_2\ge {\mathfrak{r}}_1$ and a constant $M>0$ such that

$$V\left(\mathfrak{r}\right)>M,\mathrm{for}\mathrm{all}\mathfrak{r}\ge {\mathfrak{r}}_2,$$

(25)

by using (25) and integrating (25) from ${\mathfrak{r}}_2$ $\left({\mathfrak{r}}_2\ge {\mathfrak{r}}_1\right)$ to $\mathfrak{r}$, we get

$$\begin{array}{ccc}\hfill {\int }_{{\mathfrak{r}}_2}^{\mathfrak{r}}\left(m\rho \left(s\right)\frac{Q\left(s\right)}{2^{\beta -1}}{M}^{\beta -\alpha }-\left(1+\frac{p_0^{\beta }}{{\xi }_0}\right)\frac{{\left(\alpha +1\right)}^{-\left(\alpha +1\right)}{\left({\rho }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\rho \left(s\right){\eta }_2\left(\zeta \left(s\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(s\right)\right)}^{\alpha }}\right)\mathrm{d}s& \le & \omega \left({\mathfrak{r}}_2\right)\hfill \\ & & +\frac{p_0^{\beta }}{{\xi }_0}\psi \left({\mathfrak{r}}_2\right),\hfill \end{array}$$

which contradicts (10).

Let $\left(C_2\right)$ hold. We define a function $\phi \left(\mathfrak{r}\right)$ by

$$\phi \left(\mathfrak{r}\right)=\theta \left(\mathfrak{r}\right)\frac{V^{\prime }\left(\mathfrak{r}\right)}{V\left(\mathfrak{r}\right)},$$

(26)

then $\phi \left(\mathfrak{r}\right)>0.$ Differentiating (26), we have

$${\phi }^{\prime }\left(\mathfrak{r}\right)=\frac{{\theta }^{\prime }\left(\mathfrak{r}\right)}{\theta \left(\mathfrak{r}\right)}\phi \left(\mathfrak{r}\right)+\theta \left(\mathfrak{r}\right)\frac{V^{″}\left(\mathfrak{r}\right)}{V\left(\mathfrak{r}\right)}-\theta \left(\mathfrak{r}\right)\frac{{\left(V^{\prime }\left(\mathfrak{r}\right)\right)}^2}{V^2\left(\mathfrak{r}\right)},$$

(27)

from (26) and (27), we have

$${\phi }^{\prime }\left(\mathfrak{r}\right)=\frac{{\theta }^{\prime }\left(\mathfrak{r}\right)}{\theta \left(\mathfrak{r}\right)}\phi \left(\mathfrak{r}\right)+\theta \left(\mathfrak{r}\right)\frac{V^{″}\left(\mathfrak{r}\right)}{V\left(\mathfrak{r}\right)}-\frac{1}{\theta \left(\mathfrak{r}\right)}{\phi }^2\left(\mathfrak{r}\right),$$

(28)

from (7) and (28), we have

$$\begin{array}{ccc}\hfill {\phi }^{\prime }\left(\mathfrak{r}\right)& \le & -\theta \left(\mathfrak{r}\right){\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{V}^{\left(\beta /\alpha \right)-1}\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}\Phi \left(\varkappa \right)\right)}^{1/\alpha }\mathrm{d}\varkappa \hfill \\ & & +\frac{{\theta }^{\prime }\left(\mathfrak{r}\right)}{\theta \left(\mathfrak{r}\right)}\phi \left(\mathfrak{r}\right)-\frac{1}{\theta \left(\mathfrak{r}\right)}{\phi }^2\left(\mathfrak{r}\right).\hfill \end{array}$$

Thus, we obtain

$${\phi }^{\prime }\left(\mathfrak{r}\right)\le -\theta \left(\mathfrak{r}\right){\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{V}^{\left(\beta /\alpha \right)-1}\left(\mathfrak{r}\right){\int }_{\mathfrak{r}}^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}\Phi \left(\varkappa \right)\right)}^{1/\alpha }\mathrm{d}\varkappa +\frac{{\left({\theta }_{+}^{\prime }\left(\mathfrak{r}\right)\right)}^2}{4\theta \left(\mathfrak{r}\right)},$$

(29)

by using (25) and integrating (29) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we get

$$\phi \left({\mathfrak{r}}_1\right)\ge {\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\left(\theta \left(\varrho \right){\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{M}^{\left(\beta /\alpha \right)-1}{\int }_{\varrho }^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}\Phi \left(\varkappa \right)\right)}^{1/\alpha }\mathrm{d}\varkappa -\frac{{\left({\theta }_{+}^{\prime }\left(\varrho \right)\right)}^2}{4\theta \left(\varrho \right)}\right)\mathrm{d}\varrho ,$$

which contradicts (11). This completes the proof. □

Theorem 3.

Assume that $0\le p\left(\mathfrak{r}\right)<1$ and $\alpha =\beta$. If (11) holds and

$$\underset{\mathfrak{r}\to \infty }{limsup}{\int }_{\zeta \left(\mathfrak{r}\right)}^{\mathfrak{r}}{G}^{\alpha }\left(\zeta \left(s\right)\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s>1,\zeta \mathrm{is}\mathrm{nondecreasing}$$

(30)

or

$$\underset{\mathfrak{r}\to \infty }{liminf}{\int }_{\zeta \left(\mathfrak{r}\right)}^{\mathfrak{r}}{G}^{\alpha }\left(\zeta \left(s\right)\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s>\frac{1}{\mathrm{e}},$$

(31)

where

$$G\left(\mathfrak{r}\right)={\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^v{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\eta }_3^{\alpha }\left(\zeta \left(s\right),{\mathfrak{r}}_1\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u\mathrm{d}v,$$

then (1) is oscillatory.

Proof. 

Assume that x is a positive solution of (1). It follows from Lemma 4 that there exists two possible cases $\left(C_1\right)$ and $\left(C_2\right).$ Let $\left(C_1\right)$ hold. Using the definition of $V\left(t\right)$, we get

$$x\left(\mathfrak{r}\right)=V\left(\mathfrak{r}\right)-p\left(\mathfrak{r}\right)x\left(\xi \left(\mathfrak{r}\right)\right)\ge V\left(\mathfrak{r}\right)-p\left(\mathfrak{r}\right)V\left(\xi \left(\mathfrak{r}\right)\right)\ge \left(1-p\left(\mathfrak{r}\right)\right)V\left(\mathfrak{r}\right),$$

(32)

from (1) and (32), we have

$$\begin{array}{ccc}\hfill {\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }& \le & -\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }{V}^{\alpha }\left({\zeta }_j\left(\mathfrak{r}\right)\right)\hfill \\ & \le & -V^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }.\hfill \end{array}$$

(33)

On the other hand, it follows from the monotonicity of ${\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)$ that

$$V^{″}\left(\mathfrak{r}\right)=V^{″}\left({\mathfrak{r}}_1\right)+{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\frac{1}{{\mathfrak{a}}^{1/\alpha }\left(s\right)}{\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\mathrm{d}s\ge {\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right).$$

(34)

Integrating (34) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we have

$$V^{\prime }\left(\mathfrak{r}\right)\ge {\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\eta }_1\left(s,{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\mathrm{d}s\ge {\eta }_2\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right).$$

(35)

Integrating (35) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we have

$$V\left(\mathfrak{r}\right)\ge {\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\mathrm{d}s\mathrm{d}u={\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right).$$

(36)

A simple computation shows that

$${\left(V^{″}\left(\mathfrak{r}\right)-{\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\prime }=-{\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\prime }.$$

(37)

Applying the chain rule, it is easy to see that

$${\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }=\alpha {\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha -1}{\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\prime }.$$

By virtue of (33), the latter equality yields

$$\begin{array}{ccc}\hfill {\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\prime }& \le & -\frac{1}{\alpha }{\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{1-\alpha }{V}^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)\hfill \\ & & \times \sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }.\hfill \end{array}$$

(38)

Combining (37) and (39), we obtain

$$\begin{array}{ccc}\hfill {\left(V^{″}\left(\mathfrak{r}\right)-{\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{\prime }& \ge & \frac{1}{\alpha }{\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)\right)}^{1-\alpha }{V}^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)\hfill \\ & & \times \sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }.\hfill \end{array}$$

(39)

Integrating (40) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we have

$$\begin{array}{ccc}\hfill V^{″}\left(\mathfrak{r}\right)& \ge & {\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\eta }_1\left(s,{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\right)}^{1-\alpha }{V}^{\beta }\left(\zeta \left(s\right)\right)\hfill \\ & & \times \sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\beta }\mathrm{d}s.\hfill \end{array}$$

(40)

Integrating (40) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we have

$$\begin{array}{ccc}\hfill V^{\prime }\left(\mathfrak{r}\right)& \ge & {\eta }_2\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\right)}^{1-\alpha }\hfill \\ & & \times V^{\alpha }\left(\zeta \left(s\right)\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u.\hfill \end{array}$$

(41)

Integrating (41) from ${\mathfrak{r}}_1$ to $\mathfrak{r}$, we have

$$\begin{array}{ccc}\hfill V\left(\mathfrak{r}\right)& \ge & {\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^v{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\right)}^{1-\alpha }\hfill \\ & & \times V^{\alpha }\left(\zeta \left(s\right)\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u\mathrm{d}v.\hfill \end{array}$$

Taking (36) and the monotonicity of ${\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)$ into account, we arrive at

$$\begin{array}{ccc}\hfill V\left(\mathfrak{r}\right)& \ge & {\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^v{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\right)}^{1-\alpha }\hfill \\ & & \times {\eta }_3^{\alpha }\left(\zeta \left(s\right),{\mathfrak{r}}_1\right)\mathfrak{a}\left(\zeta \left(s\right)\right){\left(V^{‴}\left(\zeta \left(s\right)\right)\right)}^{\alpha }\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u\mathrm{d}v\hfill \\ & \ge & {\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right){\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^v{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\left({\mathfrak{a}}^{1/\alpha }\left(s\right)V^{‴}\left(s\right)\right)}^{1-\alpha }\hfill \\ & & \times {\eta }_3^{\alpha }\left(\zeta \left(s\right),{\mathfrak{r}}_1\right)\mathfrak{a}\left(s\right){\left(V^{‴}\left(s\right)\right)}^{\alpha }\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u\mathrm{d}v\hfill \\ & \ge & \left({\eta }_3\left(\mathfrak{r},{\mathfrak{r}}_1\right)+\frac{1}{\alpha }{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}{\int }_{{\mathfrak{r}}_1}^v{\int }_{{\mathfrak{r}}_1}^u{\eta }_1\left(s,{\mathfrak{r}}_1\right){\eta }_3^{\alpha }\left(\zeta \left(s\right),{\mathfrak{r}}_1\right)\sum _{j=1}^m{q}_j\left(s\right){\left(1-p\left({\zeta }_j\left(s\right)\right)\right)}^{\alpha }\mathrm{d}s\mathrm{d}u\mathrm{d}v\right)\hfill \\ & & \times {\mathfrak{a}}^{1/\alpha }\left(\mathfrak{r}\right)V^{‴}\left(\mathfrak{r}\right).\hfill \end{array}$$

(42)

Thus, we conclude that

$$V\left(\zeta \left(\mathfrak{r}\right)\right)\ge {\mathfrak{a}}^{1/\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)V^{‴}\left(\zeta \left(\mathfrak{r}\right)\right)G\left(\zeta \left(\mathfrak{r}\right)\right).$$

(43)

Using (43) in (33), by virtue of $\left(C_1\right)$, one can see that $y\left(\mathfrak{r}\right):=\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }$ is a positive solution of the first-order delay differential inequality

$$y^{\prime }\left(\mathfrak{r}\right)+\left(G^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)y\left(\zeta \left(\mathfrak{r}\right)\right)\le 0.$$

(44)

In view of ([21], Theorem 1), the associated delay differential equation

$$y^{\prime }\left(\mathfrak{r}\right)+\left(G^{\alpha }\left(\zeta \left(\mathfrak{r}\right)\right)\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right){\left(1-p\left({\zeta }_j\left(\mathfrak{r}\right)\right)\right)}^{\alpha }\right)y\left(\zeta \left(\mathfrak{r}\right)\right)=0,$$

(45)

also has a positive solution. However, it is well known that condition (30) or condition (31) ensures oscillation of (45), which is a contradiction.

Let $\left(C_2\right)$ hold. One proceeds as in the proof of Theorem 2. Therefore, the proof is complete. □

Example 1.

Consider the fourth-order neutral differential equation

$${\left(x\left(\mathfrak{r}\right)+6x\left(\frac{1}{2}\mathfrak{r}\right)\right)}^{\left(4\right)}+\frac{q_0}{{\mathfrak{r}}^4}x\left(\frac{1}{3}\mathfrak{r}\right)+\frac{2q_0}{{\mathfrak{r}}^4}x\left(\frac{1}{4}\mathfrak{r}\right)=0,$$

(46)

where $q_0>0$. We note that $p\left(\mathfrak{r}\right)=6,$ $\xi \left(\mathfrak{r}\right)=\mathfrak{r}/2,$ ${\zeta }_1\left(\mathfrak{r}\right)=\mathfrak{r}/3,$ ${\zeta }_2\left(\mathfrak{r}\right)=\mathfrak{r}/4,$ $q_1\left(\mathfrak{r}\right)=q_0/{\mathfrak{r}}^4$, and $q_2\left(\mathfrak{r}\right)=2q_0/{\mathfrak{r}}^4$. It is easy to verify that

$$\begin{array}{ccc}\hfill Q\left(\mathfrak{r}\right)& =& \frac{q_0}{{\mathfrak{r}}^4},\hfill \\ \hfill \Phi \left(\mathfrak{r}\right)& =& {\int }_{{\xi }^{-1}\left(\mathfrak{r}\right)}^{\infty }\frac{Q\left(s\right)}{2^{\beta -1}}\sum _{j=1}^m{\left(\frac{{\zeta }_j\left(s\right)}{s}\right)}^{\beta /\lambda }\mathrm{d}s\hfill \\ & =& q_0\left(\left(\frac{1}{3^{1/\lambda }}\right)+\left(\frac{1}{4^{1/\lambda }}\right)\right)\frac{1}{3{\left(2\mathfrak{r}\right)}^3},\hfill \end{array}$$

$${\eta }_1\left(\mathfrak{r},{\mathfrak{r}}_1\right)={\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\frac{1}{{\mathfrak{a}}^{1/\alpha }\left(s\right)}\mathrm{d}s,=\left(\mathfrak{r}-{\mathfrak{r}}_1\right)$$

and

$${\eta }_2\left(\zeta \left(\mathfrak{r}\right),{\mathfrak{r}}_1\right)=\frac{{\left(\left(\mathfrak{r}/4\right)-{\mathfrak{r}}_1\right)}^2}{2}.$$

By choosing $\rho \left(\mathfrak{r}\right)={\mathfrak{r}}^3$ and $\theta \left(\mathfrak{r}\right)=\mathfrak{r},$ we obtain

$$\begin{array}{ccc}& & \underset{\mathfrak{r}\to \infty }{limsup}{\int }_{{\mathfrak{r}}_2}^{\mathfrak{r}}\left(m\rho \left(s\right)\frac{Q\left(s\right)}{2^{\beta -1}}{M}^{\beta -\alpha }-\left(1+\frac{p_0^{\beta }}{{\xi }_0}\right)\frac{{\left(\alpha +1\right)}^{-\left(\alpha +1\right)}{\left({\rho }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\rho \left(s\right){\eta }_2\left(\zeta \left(s\right),{\mathfrak{r}}_1\right){\zeta }^{\prime }\left(s\right)\right)}^{\alpha }}\right)\mathrm{d}s\hfill \\ & & \underset{\mathfrak{r}\to \infty }{=limsup}{\int }_{{\mathfrak{r}}_2}^{\mathfrak{r}}\left(2s^3\frac{q_0}{s^4}-\left(1+\frac{6}{1/2}\right)\frac{{\left(2\right)}^{-\left(2\right)}{\left(3s^2\right)}^2}{\left(s^3\frac{{\left(\left(\mathfrak{r}/4\right)-{\mathfrak{r}}_1\right)}^2}{2}\frac{1}{4}\right)}\right)\mathrm{d}s\hfill \\ & & \underset{\mathfrak{r}\to \infty }{=limsup}{\int }_{{\mathfrak{r}}_2}^{\mathfrak{r}}\left(2\frac{q_0}{s}-\left(1+\frac{6}{1/2}\right)\frac{4^3{3}^2}{2s}\right)\mathrm{d}s\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \underset{\mathfrak{r}\to \infty }{limsup}{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\left(\theta \left(\varrho \right){\left(\frac{{\xi }_0}{{\xi }_0+p_0^{\beta }}\right)}^{1/\alpha }{M}^{\left(\beta /\alpha \right)-1}{\int }_{\varrho }^{\infty }{\left(\frac{1}{\mathfrak{a}\left(\varkappa \right)}\Phi \left(\varkappa \right)\right)}^{1/\alpha }\mathrm{d}\varkappa -\frac{{\left({\theta }_{+}^{\prime }\left(\varrho \right)\right)}^2}{4\theta \left(\varrho \right)}\right)\mathrm{d}\varrho \hfill \\ & & \underset{\mathfrak{r}\to \infty }{=limsup}{\int }_{{\mathfrak{r}}_1}^{\mathfrak{r}}\left(\frac{1}{\varrho }\left(\frac{\left(1/2\right)}{\left(1/2\right)+6}\frac{q_0}{2^{3}6}\left(\left(\frac{1}{3^{1/\lambda }}\right)+\left(\frac{1}{4^{1/\lambda }}\right)\right)\right)-\frac{1}{4\varrho }\right)\mathrm{d}\varrho .\hfill \end{array}$$

Thus, conditions (10) and (11) are satisfied if $q_0>1872$ and $q_0>267.43,$ respectively. Therefore, we see that (46) is oscillatory if $q_0>1872.$

Remark 1.

Consider the special case

$${\left(x\left(\mathfrak{r}\right)+6x\left(\frac{1}{2}\mathfrak{r}\right)\right)}^{\left(4\right)}+\frac{q_0}{{\mathfrak{r}}^4}x\left(\frac{1}{6}\mathfrak{r}\right)=0.$$

(47)

From Theorem 2, we see that (47) is oscillatory if $q_0>12,636.$ However, choosing $\eta \left(\mathfrak{r}\right)=\mathfrak{r}/6$ and using Theorem 1 we observe that cannot be applied to (47) because $p=6<8.$ Therefore, our results improve results in [18].

3. Conclusions

In the canonical case, we established oscillation theorems for all solutions to fourth-order neutral differential equations using the Riccati transformation approach and some inequalities in the case where $0\le p\left(\mathfrak{r}\right)<p_0<\infty$. We obtain improved criteria without requiring the existence of the unknown function. It would also be of interest to study the equation

$${\left(\mathfrak{a}\left(\mathfrak{r}\right){\left(V^{‴}\left(\mathfrak{r}\right)\right)}^{\alpha }\right)}^{\prime }+\sum _{j=1}^m{q}_j\left(\mathfrak{r}\right)x^{\beta }\left({\zeta }_j\left(\mathfrak{r}\right)\right)=0$$

in the non-canonical case.