# Oscillation of super-linear fourth-order differential equations with several sub-linear neutral terms

## Abstract

In this paper, we discuss the oscillatory behavior of solutions of a class of Super-linear fourth-order differential equations with several sub-linear neutral terms using the Riccati and generalized Riccati transformations. Some Kamenev–Philos-type oscillation criteria are established. New oscillation criteria are deduced in both canonical and non-canonical cases. An illustrative example is given.

## Introduction

The aim of this paper is to discuss the oscillatory behavior of solutions of a class of super-linear fourth-order neutral differential equations of the type,

$$\bigl( r ( t ) \bigl( z^{{\prime \prime \prime }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }+\sum_{i=1}^{m}f_{i} \bigl( t,x \bigl( \tau _{i} ( t ) \bigr) \bigr) =0, \quad t\geq t_{0},$$
(1.1)

where $$z(t)=x ( t ) +\sum_{j=1}^{n}a_{j} ( t ) x^{ \alpha _{j}} ( \sigma _{j} ( t ) )$$, m, n are positive integers, and $$\alpha _{j}$$, γ are ratios of odd positive integers and $$0<\alpha _{j}\leq 1$$, $$\gamma\geq 1$$, under the conditions

$$R ( t_{0} ) = \int _{t_{0}}^{\infty} \frac{1}{r^{\frac{1}{\gamma}} ( t ) }\,dt=\infty ,$$
(1.2)

and

$$R ( t_{0} ) = \int _{t_{0}}^{\infty} \frac{1}{r^{\frac{1}{\gamma}} ( t ) }\,dt< \infty .$$
(1.3)

Throughout the paper, we assume the following assumptions

$$( A_{1} )$$:

$$r ( t ) \in C^{1} ( [t_{0},\infty ), ( 0, \infty ) )$$, $$r^{\prime} ( t ) \geq 0$$;

$$( A_{2} )$$:

$$a_{j} ( t ),\sigma _{j} ( t ) ,\tau _{i} ( t ) \in C[t_{0},\infty ))$$, $$\sigma _{j} ( t ) \leq t$$, $$\lim_{t\rightarrow \infty}\sigma _{j} ( t )=\infty$$;

$$( A_{3} )$$:

there exists a function $$\tau \in C^{1} ( [t_{0},\infty ), R )$$ such that $$\tau ( t ) \leq \tau _{i} ( t )$$ for $$i=1,2,\ldots,m$$, $$\tau ( t ) \leq t$$, $$\tau ^{\prime} ( t ) >0$$ and $$\lim_{t\rightarrow \infty}\tau ( t ) =\infty$$;

$$( A_{4} )$$:

$$0\leq a_{j} ( t ) \leq a_{0j} ( t )$$, $$\sum_{j=1}^{n}a_{0j} ( t ) <1$$, $$f_{i} ( t,x ) \in C ( [t_{0},\infty )\times R,R )$$ satisfy $$xf_{i} ( t,x )>0$$ for all $$x\neq 0$$, and there exist positive continuous functions $$q_{i} ( t )$$ defined on $$[t_{0},\infty )$$ such that $$\vert f_{i} ( t,x ) \vert \geq q_{i} ( t ) \vert x \vert ^{\gamma }$$.

By a solution of (1.1), we mean a nontrivial real function $$x ( t )$$ such that $$r ( t ) ( [ x ( t ) +\sum_{j=1}^{n}a_{j} ( t ) x^{\alpha _{j}} ( \sigma _{j} ( t ) ) ] ^{\prime \prime \prime} ) ^{\gamma}$$ is continuously differentiable satisfying (1.1) for any $$t_{1}\geq t_{0}$$.

A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

Oscillation phenomena take part in different models from real-world applications; see, e.g., paper [8] for more details. In the last three decades, there has been considerable interest in studying the oscillation of solutions of several kinds of differential equations [15, 7, 8, 1020, 2224, 2639]. The half-linear equations have numerous applications in the study of p-Laplace equations, non-Newtonian fluid theory, porous medium, etc.; see, e.g., papers [6, 21, 25] for more details. In particular, papers [11, 24] were concerned with the oscillation of various classes of half-linear differential equations, whereas the papers [35, 7, 10, 20, 26, 38] were concerned with the oscillatory behavior of the fourth-order differential equation (1.1) and its special cases. In what follows, we briefly comment on a number of closely related results which motivated our work. The authors in [3, 4, 26] discussed in their recent papers, the special case of (1.1) of the form,

$$\bigl( r ( t ) \bigl( \bigl[ x ( t ) +p ( t ) x \bigl( \tau ( t ) \bigr) \bigr] ^{ \prime \prime \prime} \bigr) ^{\alpha} \bigr) ^{\prime}+q ( t ) x^{\beta } \bigl( \delta ( t ) \bigr)=0.$$
(1.4)

Under the condition (1.2), Dassios and Bazighifan in [10] discussed the oscillation of the same equation under condition (1.3). In [20], Li et al. studied the oscillatory behavior of a class of fourth-order differential equations with the p-Laplacian-like operator of the type,

$$\bigl( r ( t ) \bigl\vert z^{{\prime \prime \prime }} ( t ) \bigr\vert ^{p-2}z^{{\prime \prime \prime }} ( t ) \bigr) ^{\prime }+\sum _{i=1}^{l}q_{i} ( t ) \bigl\vert x \bigl( \tau _{i} ( t ) \bigr) \bigr\vert ^{p-2}x \bigl( \tau _{i} ( t ) \bigr) =0,$$
(1.5)

where $$z(t)=x ( t ) +a ( t ) x ( \sigma ( t ) )$$. Under the condition $$\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{p-2}} ( t ) }\,dt<\infty$$, they used the Riccati transformation and integral averaging technique and presented a Kamenev-type oscillation criterion.

More recently, Bazighifan et al. [5] studied the asymptotic behavior of solutions of the fourth-order neutral differential equation with the continuously distributed delay of the form

$$\bigl( r ( t ) \bigl( \bigl[ x ( t ) +p ( t ) x \bigl( \phi ( t ) \bigr) \bigr] ^{ \prime \prime \prime } \bigr) ^{\alpha } \bigr) ^{\prime }+ \int _{a}^{b}q ( t,\theta ) x^{\beta } \bigl( \delta ( t,\theta ) \bigr)\,d\theta =0,$$
(1.6)

where α, β are quotients of odd positive integers, and $$\beta \geq \alpha$$ under the condition (1.2).

## Preliminaries

The following preliminary results will be needed for our proofs.

### Lemma 1

([9])

Let $$h>0$$. Then

$$h^{\alpha}\leq \alpha h+ ( 1-\alpha ) ,\quad 0< \alpha \leq 1.$$

### Lemma 2

([28])

Let $$z ( t )$$ be a positive and n-times differentiable function on an interval $$[T,\infty )$$ with non-positive nth derivative $$z^{ ( n ) } ( t )$$ on $$[T,\infty )$$, which is not identically zero on any interval of the form $$[T^{\prime},\infty )$$, $$T^{\prime}\geq T$$ and such that $$z^{ ( n-1 ) } ( t ) z^{ ( n ) } ( t ) \leq 0$$. Then, there exist constants $$0<\theta <1$$ and $$N>0$$ such that $$z^{\prime} ( \theta t ) \geq Nt^{n-2}z^{ ( n-1 ) } ( t )$$ for all sufficient large t.

### Lemma 3

([26])

Let $$z^{ ( n ) } ( t )$$ be of fixed sign and $$z^{ ( n-1 ) } ( t ) z^{ ( n ) } ( t ) \leq 0$$ for all $$t\geq t_{1}$$. If $$\lim_{t \rightarrow \infty}z ( t ) \neq 0$$, then for every $$\lambda \in ( 0,1 )$$, there exists $$t_{\lambda}$$t such that $$z ( t ) \geq \frac{\lambda}{ ( n-1 ) !}t^{n-1} \vert z^{ ( n-1 ) } ( t ) \vert$$ for $$t\geq t_{\lambda}$$.

### Lemma 4

([2])

Let α is a ratio of two odd numbers. Suppose that U, V are constants with $$V>0$$. Then, $$UY-VY^{\frac{ ( \gamma +1 ) }{\gamma}}\leq \frac{\gamma ^{\gamma}}{ ( \gamma +1 ) ^{\gamma +1}} \frac{U^{\gamma +1}}{V^{\gamma}}$$.

### Lemma 5

Assume that $$x ( t )$$ is an eventually positive solution of (1.1), $$z^{\prime} ( t ) >0$$, and there exists a positive decreasing function $$\delta ( t ) \in C ( [t_{0},\infty ) )$$ tending to zero such that $$\theta ( \tau _{i} ( t ) ) >0$$ for $$t\geq t_{0}$$ where $$\theta ( t ) =1-\sum_{j=1}^{n}\alpha _{j}a_{j} ( t ) -\frac{1}{\delta ( t ) }\sum_{j=1}^{n} ( 1- \alpha _{j} ) a_{j} ( t )$$. Then,

$$\bigl( r ( t ) \bigl( z^{{\prime \prime \prime}} ( t ) \bigr) ^{\gamma} \bigr) ^{\prime}\leq -\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) z^{\gamma} \bigl( \tau ( t ) \bigr) .$$
(2.1)

### Proof

Let x be an eventually positive solution of Eq. (1.1). Then, there exists a $$t_{1}\geq t_{0}$$ such that $$x ( t ) >0$$, $$x ( \sigma _{j} ( t ) ) >0$$ and $$x ( \tau _{i} ( t ) )>0$$ for $$t\geq t_{1}$$. Now from the definition of z, we have

$$x ( t ) =z ( t ) -\sum_{j=1}^{n}a_{j} ( t ) x^{\alpha _{j}} \bigl( \sigma _{j} ( t ) \bigr) \geq z ( t ) - \sum_{j=1}^{n}a_{j} ( t ) z^{ \alpha _{j}} \bigl( \sigma _{j} ( t ) \bigr) \geq z ( t ) -\sum _{j=1}^{n}a_{j} ( t ) z^{\alpha _{j}} ( t ) .$$

Then, by Lemma 1, we have

$$x ( t ) \geq \Biggl( 1-\sum_{j=1}^{n}\alpha _{j}a_{j} ( t ) \Biggr) z ( t ) -\sum _{j=1}^{n} ( 1- \alpha _{j} ) a_{j} ( t ) .$$

Now since $$z ( t )$$ is positive and increasing, and $$\delta ( t )$$ is a positive decreasing function tending to zero, then there exists a $$t_{2}\geq t_{1}$$ such that $$z ( t ) \geq \delta ( t )$$, and

$$x ( t ) \geq \Biggl[ 1-\sum_{j=1}^{n}\alpha _{j}a_{j} ( t ) -\frac{1}{\delta ( t ) }\sum _{j=1}^{n} ( 1-\alpha _{j} ) a_{j} ( t ) \Biggr] z ( t ),\quad \text{for } t\geq t_{2}.$$

That is $$x ( t ) \geq \theta ( t ) z ( t )$$. Therefore, from (1.1), it follows that

$$\bigl( r ( t ) \bigl( z^{{\prime \prime \prime}} ( t ) \bigr) ^{\gamma} \bigr) ^{\prime}\leq -\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) z^{\gamma} \bigl( \tau _{i} ( t ) \bigr) \leq - \sum _{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) z^{\gamma} \bigl( \tau ( t ) \bigr) .$$

Thus, the proof is completed. □

The following two auxiliary results are very similar to those reported in [3] and [10].

### Lemma 6

Let $$x ( t )$$ be a positive solution of (1.1). If (1.2) is satisfied, then there exists $$t\geq t_{1}$$ such that

\begin{aligned}& z ( t ) >0,\qquad z^{\prime} ( t ) >0,\qquad z^{\prime \prime \prime } ( t ) >0, \qquad z^{ ( 4 ) } ( t ) < 0, \qquad \bigl( r ( t ) \bigl( z^{{\prime \prime \prime}} ( t ) \bigr) ^{\gamma} \bigr) ^{\prime}\leq 0. \end{aligned}

### Lemma 7

Let $$x ( t )$$ be a positive solution of (1.1). If (1.3) is satisfied, then there exist three possible cases for sufficiently large $$t\geq t_{1}$$

$$( S_{1} )$$:

$$z ( t ) >0$$, $$z^{\prime} ( t ) >0$$, $$z^{\prime \prime \prime} ( t ) >0$$, $$z^{ ( 4 ) } ( t ) \leq 0$$;

$$( S_{2} )$$:

$$z ( t ) >0$$, $$z^{\prime} ( t ) >0$$, $$z^{\prime \prime} ( t )>0$$, $$z^{\prime \prime \prime} ( t )<0$$;

$$( S_{3} )$$:

$$z ( t ) >0$$, $$z^{\prime} ( t ) <0$$, $$z^{\prime \prime} ( t ) >0$$, $$z^{\prime \prime \prime} ( t )<0$$.

## Main results

We first consider the case $$R ( t_{0} ) =\infty$$.

### Theorem 8

If there exist $$\eta ( t ) \in C^{1} ( [t_{0},\infty ), ( 0, \infty ) )$$, $$b ( t ) \in C^{1} ( [t_{0},\infty ),[0,\infty ) )$$, $$\zeta \in ( 0,1 )$$ and $$\epsilon >0$$ such that

$$\underset{t\rightarrow \infty }{\lim \sup } \int _{t_{0}}^{t} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime } ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds=\infty ,$$
(3.1)

then (1.1) is oscillatory, where $$Q ( t ) =\eta ( t ) \sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma } ( \tau _{i} ( t ) ) -\eta ( t ) [ r ( t ) b ( t ) ] ^{\prime}+\zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) r ( t ) \eta ( t ) b^{1+\frac{1}{\gamma}} ( t )$$.

### Proof

Suppose for the contrary that x is an eventually positive solution of (1.1). Then there exists a $$t_{1}\geq t_{0}$$ such that $$x ( t )>0$$, $$x ( \sigma _{j} ( t ) )>0$$ and $$x ( \tau _{i} ( t ) )>0$$ for $$t\geq t_{1}$$. Using Lemma 5, we obtain (2.1). Define

$$\psi ( t ) =\eta ( t ) \biggl[ \frac{r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}}{z^{{\gamma}} ( \zeta \tau ( t ) ) }+r ( t ) b ( t ) \biggr] ,\quad \mathbf{t}\geq \mathbf{t}_{1}.$$
(3.2)

It is clear that $$\psi ( t ) >0$$ for $$t\geq t_{1}$$, and

\begin{aligned} \psi ^{\prime} ( t ) = {}&\frac{\eta ^{\prime} ( t ) }{\eta ( t ) }\psi ( t ) +\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}+\eta ( t ) \frac{ ( r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma} ) ^{\prime}}{z^{\gamma} ( \zeta \tau ( t ) ) } \\ &{} -\eta ( t ) \frac{\gamma \zeta r ( t ) \tau ^{\prime } ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}z^{\prime} ( \zeta \tau ( t ) ) }{z^{\gamma +1} ( \zeta \tau ( t ) ) }. \end{aligned}

Thus, by (2.1), it follows that

\begin{aligned} \psi ^{\prime} ( t ) \leq{}& \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }\psi ( t ) +\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}-\eta ( t ) \frac{\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} ( \tau _{i} ( t ) ) z^{\gamma} ( \tau ( t ) ) }{z^{\gamma} ( \zeta \tau ( t ) ) } \\ &{} -\eta ( t ) \frac{\gamma \zeta r ( t ) \tau ^{\prime } ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}z^{\prime} ( \zeta \tau ( t ) ) }{z^{\gamma +1} ( \zeta \tau ( t ) ) }. \end{aligned}

By Lemma 2, we have

$$z^{\prime} \bigl( \zeta \tau ( t ) \bigr) \geq \epsilon \tau ^{2} ( t ) z^{{\prime \prime \prime}} \bigl( \tau ( t ) \bigr) \geq \epsilon \tau ^{2} ( t ) z^{{\prime \prime \prime}} ( t ) .$$

However, since $$z ( t )$$ is increasing, then $$z ( \tau ( t ) ) \geq z ( \zeta \tau ( t ) )$$. Therefore,

\begin{aligned} \psi ^{\prime} ( t ) \leq {}&\frac{\eta ^{\prime} ( t ) }{\eta ( t ) }\psi ( t ) +\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}-\eta ( t ) \sum _{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) \\ &{} -\eta ( t ) \frac{\gamma \zeta \epsilon r ( t ) \tau ^{\prime} ( t ) \tau ^{2} ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma +1}}{z^{\alpha +1} ( \zeta \tau ( t ) ) }. \end{aligned}

Moreover, since from (3.2), we have

$$\frac{z^{{\prime \prime \prime}} ( t ) }{z ( \zeta \tau ( t ) ) }=\frac{1}{r^{\frac{1}{\gamma}} ( t ) } \biggl[ \frac{\psi ( t ) }{\eta ( t ) }- \bigl[ r ( t ) b ( t ) \bigr] \biggr] ^{ \frac{1}{\gamma}},$$

then

\begin{aligned} \psi ^{\prime} ( t ) \leq& \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }\psi ( t ) +\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}-\eta ( t ) \sum _{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) \\ &{}-\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) \frac{\eta ( t ) }{r^{\frac {1}{\gamma}} ( t ) } \biggl( \frac{\psi ( t ) }{\eta ( t ) }- \bigl[ r ( t ) b ( t ) \bigr] \biggr) ^{\frac{\gamma +1}{\gamma}}. \end{aligned}
(3.3)

As in [35], we use the inequality

$$M^{1+\frac{1}{\gamma}}- ( M-N ) ^{1+\frac{1}{\gamma}}\leq N^{ \frac{1}{\gamma}} \biggl[ \biggl( 1+\frac{1}{\gamma} \biggr) M- \frac {1}{\gamma}N \biggr] , \quad MN\geq 0, \gamma \geq 1,$$

with

$$M=\frac{\psi ( t ) }{\eta ( t ) }\quad \text{and}\quad N=r ( t ) b ( t ) ,$$

to get

\begin{aligned} \biggl( \frac{\psi ( t ) }{\eta ( t ) }- \bigl[ r ( t ) b ( t ) \bigr] \biggr) ^{ \frac{\gamma +1}{\gamma}} \geq& \biggl[ \frac{\psi ( t ) }{\eta ( t ) } \biggr] ^{1+\frac{1}{\gamma}}+ \frac{1}{\gamma} \bigl[ r ( t ) b ( t ) \bigr] ^{1+\frac{1}{\gamma}} \\ &{}- \biggl( 1+ \frac{1}{\gamma } \biggr) \frac{ [ r ( t ) b ( t ) ] ^{\frac{1}{\gamma}}}{\eta ( t ) }\psi ( t ). \end{aligned}
(3.4)

Using inequalities (3.3) and (3.4), for $$t\geq T$$, we have

\begin{aligned} \psi ^{\prime} ( t ) \leq{} & \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }\psi ( t ) +\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}-\eta ( t ) \sum _{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) \\ & {}+\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) \frac{\eta ( t ) }{r^{\frac{1}{\gamma}} ( t ) } \biggl[ \biggl( 1+\frac{1}{\gamma} \biggr) \frac{ [ r ( t ) b ( t ) ] ^{\frac{1}{\gamma}}}{\eta ( t ) }\psi ( t ) \\ &{}-\frac{1}{\gamma} \bigl[ r ( t ) b ( t ) \bigr] ^{1+\frac{1}{\gamma}}- \frac{\psi ^{1+\frac{1}{\gamma}} ( t ) }{\eta ^{1+\frac{1}{\gamma}} ( t ) } \biggr]. \end{aligned}

Then,

\begin{aligned} \psi ^{\prime} ( t ) \leq {}&\eta ( t ) \Biggl( \bigl[ r ( t ) b ( t ) \bigr] ^{ \prime}-\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) \Biggr) \\ &{} + \biggl[ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime } ( t ) \tau ^{2} ( t ) b^{ \frac{1}{\gamma}} ( t ) \biggr] \psi ( t ) \\ &{} - \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \eta ^{\frac{1}{\gamma}} ( t ) }\psi ^{1+\frac{1}{\gamma}} ( t ) - \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) r ( t ) \eta ( t ) b^{1+ \frac{1}{\gamma}} ( t ), \end{aligned}

i.e.

\begin{aligned} \psi ^{\prime} ( t ) \leq& -Q ( t ) + \biggl[ \frac {\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) \biggr] \psi ( t ) \\ &{}- \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \eta ^{\frac{1}{\gamma}} ( t ) }\psi ^{1+\frac{1}{\gamma}} ( t ). \end{aligned}
(3.5)

Now let

\begin{aligned}& U=\frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) , \\& V= \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \eta ^{\frac{1}{\gamma}} ( t ) } \quad \text{and}\quad Y=\psi ( t ). \end{aligned}

Then, by Lemma 4, we obtain

\begin{aligned} & \biggl[ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) \biggr] \psi ( t ) - \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \eta ^{\frac{1}{\gamma}} ( t ) } \psi ^{1+\frac{1}{\gamma}} ( t ) \\ &\quad \leq \frac{\gamma ^{\gamma}r ( t ) \eta ( t ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) ] ^{{\gamma +1}}}{\gamma ^{\gamma} [ \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) ] ^{\gamma}}. \end{aligned}

Thus, we have

$$\psi ^{\prime} ( t ) \leq -Q ( t ) + \frac{r ( t ) \eta ( t ) }{ ( \gamma +1 ) ^{\gamma +1}}\frac{ [ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) ] ^{\gamma}}.$$
(3.6)

Integrating (3.6) from T to t, we get

$$\int _{T}^{t} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime } ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds\leq \psi ( T ),$$

which contradicts (3.1), and this completes the proof. □

The following result deals with the Kamenev-type oscillation for Eq. (1.1) under the condition (1.2).

### Theorem 9

If

\begin{aligned}& \underset{t\rightarrow \infty }{\lim \sup }\frac{1}{t^{n}}\int _{t_{0}}^{t} ( t-s ) ^{n} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds \\& \quad =\infty , \end{aligned}
(3.7)

then (1.1) is oscillatory.

### Proof

Let x be a nonoscillatory solution of (1.1) on $$[t_{0},\infty )$$. Without loss of generality, we may assume that x is an eventually positive solution. Define $$\psi ( t )$$ as in (3.2). Then, following the same steps as in the proof of Theorem 8, we arrive at (3.6). Multiplying (3.6) by $$( t-s ) ^{n}$$ and integrating the resulting inequality from $$t_{0}$$ to t, we have

\begin{aligned}& -{ \int _{t_{0}}^{t}} ( t-s ) ^{n} \psi ^{\prime} ( s )\,ds \\& \quad \geq { \int _{t_{0}}^{t}} ( t-s ) ^{n} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds. \end{aligned}
(3.8)

However, since

$${ \int _{t_{0}}^{t}} ( t-s ) ^{n} \psi ^{\prime} ( s )\,ds=n{ \int _{t_{0}}^{t}} ( t-s ) ^{n-1} \psi ( s ) \,ds- ( t-t_{0} ) ^{n}\psi ( t_{0} ) ,$$

then from (3.8), we get

\begin{aligned} & ( t-t_{0} ) ^{n}\psi ( t_{0} ) -n{ \int _{t_{0}}^{t}} ( t-s ) ^{n-1} \psi ( s ) \,ds \\ &\quad \geq { \int _{t_{0}}^{t}} ( t-s ) ^{n} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds. \end{aligned}

Hence,

\begin{aligned}& \frac{1}{t^{n}}{ \int _{t_{0}}^{t}} ( t-s ) ^{n} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds \\& \quad \leq \biggl( \frac{t-t_{0}}{t} \biggr) ^{n}\psi ( t_{0} ), \end{aligned}

and so

\begin{aligned}& \underset{t\rightarrow \infty }{\lim \sup }\frac{1}{t^{n}}{ \int _{t_{0}}^{t}} ( t-s ) ^{n} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime} ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma}} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) ] ^{\gamma}} \biggr]\,ds \\& \quad \rightarrow \psi ( t_{0} ), \end{aligned}

which contradicts (3.7), and this completes the proof. □

Now we are going to discuss the so called Philos-type oscillation criteria for Eq. (1.1) under condition (1.2), but we first outline the following definition.

### Definition 10

Let $$D= \{ ( t,s ) \in R^{2}:t\geq s\geq t_{0} \}$$ and $$D_{0}= \{ ( t,s ) \in R^{2}:t>s \geq t_{0} \}$$. The functions $$K_{i} ( t,s ) \in C ( D,R )$$, $$i=1,2$$ are said to belong to the class X (written $$K_{i}\in X$$) if they satisfy

1. (I)

$$K_{i} ( t,t ) =0$$ for $$t\geq t_{0}$$, $$K_{i} ( t,s ) >0$$, $$( t,s ) \in D_{0}$$

2. (II)

$$\frac{\partial K_{i} ( t,s ) }{\partial s}\leq 0$$, and there exist $$\rho ( t ) \in C^{1} ( [t_{0},\infty ), ( 0,\infty ) )$$ and $$L_{i} ( t,s ) \in C ( D,R )$$ such that

$$-\frac{\partial K_{1} ( t,s ) }{\partial s}=K_{1} ( t,s ) \biggl[ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) \biggr] +L_{1} ( t,s ),$$

and

$$\frac{\partial K_{2} ( t,s ) }{\partial s}+ \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }K_{2} ( t,s ) = \frac{L_{2} ( t,s ) }{\rho ( t ) } \bigl[ K_{2} ( t,s ) \bigr] ^{\frac{\gamma}{\gamma +1}}.$$

### Theorem 11

Assume that there exists a function $$K_{1}\in X$$ such that

$$\underset{t\rightarrow \infty }{\lim \sup } \frac{1}{K_{1} ( t,t_{0} ) } \int _{t_{0}}^{t} \biggl[ K_{1} ( t,s ) Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \vert L_{1} ( t,s ) \vert ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) K_{1} ( t,s ) ] ^{\gamma}} \biggr]\,ds=\infty .$$
(3.9)

Then, Eq. (1.1) is oscillatory.

### Proof

Let x be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is an eventually positive solution of (1.1). Now define $$\psi ( t )$$ as in (3.2). Following the same steps as in the proof of Theorem 8, we arrive at (3.5). Multiplying (3.5) by $$K_{1} ( t,s )$$ and integrating the resulting inequality from T to t, we have

$${ \int _{T}^{t}} K_{1} ( t,s ) Q ( s )\,ds \leq { \int _{T}^{t}} K_{1} ( t,s ) \bigl[- \psi ^{\prime} ( s ) +A ( s ) \psi ( s ) -B ( s ) \psi ^{1+\frac{1}{\gamma}} ( s ) \bigr]\,ds,$$

where

$$A ( t ) = \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) ,\qquad B ( t ) = \frac{\zeta \gamma \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \eta ^{\frac{1}{\gamma}} ( t ) }.$$

Then, we have

\begin{aligned} { \int _{T}^{t}} K_{1} ( t,s ) Q ( s )\,ds \leq{}& K_{1} ( t,T ) \psi ( T ) +{ \int _{T}^{t}} \biggl[ \frac{\partial K_{1} ( t,s ) }{\partial s}+K_{1} ( t,s ) A ( s ) \biggr] \psi ( s )\,ds \\ &{} -{ \int _{T}^{t}} K_{1} ( t,s ) B ( s ) \psi ^{1+\frac{1}{\gamma}} ( s )\,ds \\ ={}&K_{1} ( t,T ) \psi ( T ) -{ \int _{T}^{t}} L_{1} ( t,s ) \psi ( s ) \,ds-{ \int _{T}^{t}} K_{1} ( t,s ) B ( s ) \psi ^{1+\frac{1}{\gamma}} ( s )\,ds \\ \leq{}& K_{1} ( t,T ) \psi ( T ) +{ \int _{T}^{t}} \bigl[ \bigl\vert L_{1} ( t,s ) \bigr\vert \psi ( s ) -K_{1} ( t,s ) B ( s ) \psi ^{1+\frac{1}{\gamma}} ( s ) \bigr]\,ds. \end{aligned}

Putting $$U= \vert L_{1} ( t,s ) \vert$$, $$V=K_{1} ( t,s ) B ( s )$$ and then using Lemma 4, we obtain

$$\bigl\vert L_{1} ( t,s ) \bigr\vert \psi ( s ) -K_{1} ( t,s ) B ( s ) \psi ^{1+ \frac{1}{\gamma}} ( s ) \leq \frac{\gamma ^{\gamma}}{ ( \gamma +1 ) ^{\gamma +1}} \frac{ \vert L_{1} ( t,s ) \vert ^{\gamma +1}}{ [ K_{1} ( t,s ) B ( s ) ] ^{\gamma}}.$$

Then,

$${ \int _{T}^{t}} K_{1} ( t,s ) Q ( s )\,ds \leq K_{1} ( t,T ) \psi ( T ) +{ \int _{T}^{t}} \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \vert L_{1} ( t,s ) \vert ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) K_{1} ( t,s ) ] ^{\gamma}}\,ds.$$

Hence,

$$\frac{1}{K_{1} ( t,T ) }{ \int _{T}^{t}} \biggl[ K_{1} ( t,s ) Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \vert L_{1} ( t,s ) \vert ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime} ( s ) \tau ^{2} ( s ) K_{1} ( t,s ) ] ^{\gamma}} \biggr]\,ds\leq \psi ( T ) ,$$

for all sufficiently large t, which contradicts (3.9). □

### Theorem 12

Assume that

$$\underset{t\rightarrow \infty }{\lim \inf } \frac{1}{\phi _{1}^{\ast} ( t ) } \int _{t}^{\infty}\phi _{2} ( s ) \bigl[ \phi _{1}^{ \ast} ( s ) \bigr] ^{\frac{\gamma +1}{\gamma}}\,ds> \frac{\gamma}{ ( \gamma +1 ) ^{\frac{\gamma +1}{\gamma}}}$$
(3.10)

where

\begin{aligned}& \phi _{1} ( t ) =\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) ,\qquad \phi _{2} ( t ) = \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) }{r^{\frac{1}{\gamma}} ( t ) },\quad \textit{and} \\& \phi _{1}^{\ast} ( t ) = \int _{t}^{\infty}\phi _{1} ( s )\,ds. \end{aligned}

Then, (1.1) is oscillatory.

### Proof

Assume that $$x ( t )$$ is an eventually positive solution of (1.1). Then, there exists a $$t_{1}\geq t_{0}$$ such that $$x ( t ) >0$$, $$x ( \sigma _{j} ( t ) ) >0$$ and $$x ( \tau _{i} ( t ) ) >0$$ for $$t\geq t_{1}$$. Using Lemma 5, we arrive at (2.1). Define

$$\omega ( t ) = \frac{r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}}{z^{{\gamma}} ( \zeta \tau ( t ) ) }.$$

Then, it is clear by (2.1) that

$$\omega ^{\prime} ( t ) \leq - \frac{\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} ( \tau _{i} ( t ) ) z^{{\gamma}} ( \tau ( t ) ) }{z^{{\gamma}} ( \zeta \tau ( t ) ) }- \frac{\gamma \zeta \tau ^{\prime} ( t ) r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}z^{{\prime}} ( \zeta \tau ( t ) ) }{z^{{\gamma +1}} ( \zeta \tau ( t ) ) }$$

Since, by Lemma 2, we have

$$z^{{\prime}} \bigl( \zeta \tau ( t ) \bigr) \geq \epsilon \tau ^{2} ( t ) z^{{\prime \prime \prime}} \bigl( \tau ( t ) \bigr) \geq \epsilon \tau ^{2} ( t ) z^{{\prime \prime \prime}} ( t ) ,$$

then

$$\omega ^{\prime} ( t ) \leq -\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) - \frac{\gamma \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma +1}}{z^{{\gamma +1}} ( \zeta \tau ( t ) ) }$$

i.e.

$$\omega ^{\prime} ( t ) +\phi _{1} ( t ) +\phi _{2} ( t ) \omega ^{\frac{{\gamma +1}}{\gamma}} ( t ) \leq 0.$$

Integrating the above inequality from t to l, we get

$$\omega ( l ) -\omega ( t ) +{ \int _{t}^{l}} \phi _{1} ( s ) \,ds+{ \int _{t}^{l}} \phi _{2} ( s ) \omega ^{\frac{{\gamma +1}}{\gamma}} ( s )\,ds\leq 0.$$

Letting $$l\rightarrow \infty$$ and using the fact that $$\omega ( t )$$ is positive and decreasing, we get

$$\frac{\omega ( t ) }{\phi _{1}^{\ast} ( t ) } \geq 1+\frac{1}{\phi _{1}^{\ast} ( t ) }{ \int _{t}^{\infty}} \phi _{2} ( s ) \bigl[ \phi _{1}^{\ast} ( s ) \bigr] ^{ \frac{{\gamma +1}}{\gamma}} \biggl[ \frac{\omega ( s ) }{\phi _{1}^{\ast} ( s ) } \biggr] ^{\frac{{\gamma +1}}{\gamma}}\,ds.$$
(3.11)

Let $$\delta =\inf_{t\geq T} \frac{\omega ( t ) }{\phi _{1}^{\ast } ( t ) }$$. Then obviously $$\delta \geq 1$$, and by (3.10) and (3.11), it follows that

$$\delta \geq 1+\gamma \biggl( \frac{\delta}{\gamma +1} \biggr) ^{ \frac{\gamma +1}{\gamma}},$$

which contradicts the admissible values of $$\delta \geq 1$$ and $$\gamma \geq 1$$. Therefore, the proof is completed. □

## The case $$R ( t_{0} ) <\infty$$

Now we are going to discuss the oscillatory behavior of Eq. (1.1) under the condition (1.3). First we need the following lemma.

### Lemma 13

Assume that x is an eventually positive solution of Eq. (1.1) and $$( S_{2} )$$ holds. If

$$\vartheta ( t ) =\rho ( t ) \frac{r ( t ) [ z^{{\prime \prime \prime}} ( t ) ] ^{\gamma}}{ [ z^{\prime \prime} ( t ) ] ^{\gamma}},$$
(4.1)

then

$$\vartheta ^{\prime} ( t ) \leq \frac{\rho ^{\prime} ( t ) }{\rho ( t ) }\vartheta ( t ) -\rho ( t ) \biggl[ \frac{\lambda}{2}\tau ^{2} ( t ) \biggr] ^{ \gamma}\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) - \frac{\gamma \vartheta ^{\gamma +1} ( t ) }{r^{\frac{1}{\gamma}} ( t ) \rho ^{\frac{1}{\gamma}} ( t ) },\quad \lambda \in ( 0,1 ).$$
(4.2)

### Proof

Since x is an eventually positive solution of Eq. (1.1) and $$( S_{2} )$$ holds, then using Lemma 5, we obtain (2.1). Now from Eq. (4.1), we see that $$\vartheta ( t ) <0$$ for $$t\geq t_{1}$$, and

$$\vartheta ^{\prime} ( t ) = \frac{\rho ^{\prime} ( t ) }{\rho ( t ) }\vartheta ( t ) +\rho ( t ) \frac{ [ r ( t ) [ z^{{\prime \prime \prime}} ( t ) ] ^{\gamma} ] ^{\prime}}{ [ z^{\prime \prime } ( t ) ] ^{\gamma}}- \frac{\gamma \rho ( t ) r ( t ) [ z^{{\prime \prime \prime}} ( t ) ] ^{\gamma +1}}{ [ z^{\prime \prime} ( t ) ] ^{\gamma +1}}.$$

This with (2.1) and (4.1) leads to

$$\vartheta ^{\prime} ( t ) \leq \frac{\rho ^{\prime} ( t ) }{\rho ( t ) }\vartheta ( t ) -\rho ( t ) \frac{\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} ( \tau _{i} ( t ) ) z^{{\gamma}} ( \tau ( t ) ) }{ [ z^{\prime \prime} ( t ) ] ^{\gamma}}- \frac{\gamma [ \vartheta ( t ) ] ^{\gamma +1}}{r^{\frac{1}{\gamma}} ( t ) \rho ^{\frac{1}{\gamma}} ( t ) },$$

i.e.

$$\vartheta ^{\prime} ( t ) \leq \frac{\rho ^{\prime} ( t ) }{\rho ( t ) }\vartheta ( t ) -\rho ( t ) \frac{\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} ( \tau _{i} ( t ) ) z^{{\gamma}} ( \tau ( t ) ) [ z^{\prime \prime} ( \tau ( t ) ) ] ^{\gamma}}{ [ z^{\prime \prime} ( \tau ( t ) ) ] ^{\gamma} [ z^{\prime \prime} ( t ) ] ^{\gamma}}- \frac{\gamma [ \vartheta ( t ) ] ^{\gamma +1}}{r^{\frac{1}{\gamma}} ( t ) \rho ^{\frac{1}{\gamma}} ( t ) }.$$

Now since $$z^{\prime \prime} ( t )$$ is decreasing, then it follows that $$- \frac{z^{\prime \prime} ( \tau ( t ) ) }{z^{\prime \prime} ( t ) }\leq -1$$. Consequently, by Lemma 3, we have $$z ( \tau ( t ) ) \geq \frac{\lambda}{2}\tau ^{2} ( t ) z^{\prime \prime} ( \tau ( t ) )$$. Then

$$\vartheta ^{\prime} ( t ) \leq \frac{\rho ^{\prime} ( t ) }{\rho ( t ) }\vartheta ( t ) -\rho ( t ) \biggl[ \frac{\lambda}{2}\tau ^{2} ( t ) \biggr] ^{ \gamma}\sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma} \bigl( \tau _{i} ( t ) \bigr) - \frac{\gamma [ \vartheta ( t ) ] ^{\gamma +1}}{r^{\frac{1}{\gamma}} ( t ) \rho ^{\frac{1}{\gamma}} ( t ) }.$$

The proof is completed. □

### Theorem 14

Suppose that (3.9) holds, and

\begin{aligned}& \underset{t\rightarrow \infty }{\lim \sup } \int _{t_{0}}^{t} \Biggl[ K_{2} ( t,s ) \rho ( s ) \biggl[ \frac{\lambda }{2} \tau ^{2} ( s ) \biggr] ^{\gamma }\sum_{i=1}^{m}q_{i} ( s ) \theta ^{\gamma } \bigl( \tau _{i} ( s ) \bigr) \\& \quad {} - \frac{r ( s ) }{ ( \gamma +1 ) ^{\gamma +1}\rho ^{\gamma } ( s ) } \bigl[ L_{2} ( t,s ) \bigr] ^{\gamma +1} \Biggr]\,ds>0. \end{aligned}
(4.3)

If

$$\int _{t_{0}}^{\infty }R ( s )\,ds=\infty ,$$
(4.4)

or

$$\int _{t_{0}}^{\infty } \int _{u}^{\infty }R ( s )\,ds\,du= \infty ,$$
(4.5)

then Eq. (1.1) is oscillatory.

### Proof

Suppose for the contrary that there exists a nonoscillatory solution $$x ( t ) >0$$ of (1.1). Then, we have one of the three possible cases of Lemma 7. We first assume that $$( S_{1} )$$ holds. Then by Theorem 11, if (3.9) holds, Eq. (1.1) is oscillatory. Secondly, if $$( S_{2} )$$ holds, then by Lemma 13, we get (4.2). Multiplying (4.2) by $$K_{2} ( t,s )$$ and integrating from $$t_{1}$$ to t, we obtain

\begin{aligned} & { \int _{t_{1}}^{t}} K_{2} ( t,s ) \rho ( s ) \biggl[ \frac{\lambda}{2}\tau ^{2} ( s ) \biggr] ^{\gamma}\sum_{i=1}^{m}q_{i} ( s ) \theta ^{\gamma} \bigl( \tau _{i} ( s ) \bigr)\,ds \\ &\quad \leq K_{2} ( t,t_{1} ) \omega ( t_{1} ) +{ \int _{t_{1}}^{t}} \biggl[ \frac{\partial K_{2} ( t,s ) }{\partial s}+ \frac {\rho ^{\prime} ( s ) }{\rho ( s ) }K_{2} ( t,s ) \biggr] \omega ( s )\,ds- \gamma { \int _{t_{1}}^{t}} K_{2} ( t,s ) \frac{\omega ^{\frac{\gamma +1}{\gamma}} ( s ) }{r^{\frac{1}{\gamma}} ( s ) \rho ^{\frac{1}{\gamma}} ( s ) }\,ds \\ & \quad =K_{2} ( t,t_{1} ) \omega ( t_{1} ) +{ \int _{t_{1}}^{t}} \frac{L_{2} ( t,s ) }{\rho ( s ) } \bigl[ K_{2} ( t,s ) \bigr] ^{\frac{\gamma}{\gamma +1}}\omega ( s )\,ds- \gamma { \int _{t_{1}}^{t}} K_{2} ( t,s ) \frac{\omega ^{\frac{\gamma +1}{\gamma}} ( s ) }{r^{\frac{1}{\gamma}} ( s ) \rho ^{\frac{1}{\gamma}} ( s ) }\,ds. \end{aligned}

Setting

$$V= \frac{\gamma K_{2} ( t,s ) }{r^{\frac{1}{\gamma}} ( s ) \rho ^{\frac{1}{\gamma}} ( s ) }, \qquad U= \frac {L_{2} ( t,s ) }{\rho ( s ) } \bigl[ K_{2} ( t,s ) \bigr] ^{\frac{\gamma}{\gamma +1}}\quad \text{and}\quad Y= \omega ( s ) .$$

Then, by Lemma 4, we have

\begin{aligned} & \frac{L_{2} ( t,s ) }{\rho ( s ) } \bigl[ K_{2} ( t,s ) \bigr] ^{\frac{\gamma}{\gamma +1}} \omega ( s ) - \frac{\gamma K_{2} ( t,s ) \omega ^{\frac{\gamma +1}{\gamma}} ( s ) }{r^{\frac{1}{\gamma}} ( s ) \rho ^{\frac{1}{\gamma}} ( s ) } \\ &\quad \leq \frac{1}{ ( \gamma +1 ) ^{\gamma +1}} \bigl[ L_{2} ( t,s ) \bigr] ^{ ( \gamma +1 ) } \frac{r ( s ) }{\rho ^{\gamma} ( s ) }. \end{aligned}

Hence,

\begin{aligned} & { \int _{t_{1}}^{t}} \Biggl[ K_{2} ( t,s ) \rho ( s ) \biggl[ \frac{\lambda }{2}\tau ^{2} ( s ) \biggr] ^{\gamma}\sum_{i=1}^{m}q_{i} ( s ) \theta ^{\gamma} \bigl( \tau _{i} ( s ) \bigr) - \frac{r ( s ) }{ ( \gamma +1 ) ^{\gamma +1}\rho ^{\gamma } ( s ) } \bigl[ L_{2} ( t,s ) \bigr] ^{\gamma +1} \Biggr] \,ds \\ &\quad \leq K_{2} ( t,t_{1} ) \omega ( t_{1} ) < 0. \end{aligned}

This contradicts (4.3). Finally, assume the case $$( S_{3} )$$. Hence, since $$r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}$$ is nonincreasing, then for $$s\geq t\geq t_{1,}$$ we have

$$r^{{\frac{1}{\gamma}}} ( s ) \bigl( z^{{\prime \prime \prime}} ( s ) \bigr) \leq r^{{\frac{1}{\gamma}}} ( t ) \bigl( z^{{\prime \prime \prime}} ( t ) \bigr).$$

Going through as in the proof of Theorem 2.3 case 1 in [20], we get a contradiction with (4.4) and (4.5), and so the proof is completed. □

### Remark 15

Theorem 14 remains true if we used (3.1), or (3.7), or (3.10) instead of (3.9).

## Example

### Example 16

Consider the fourth-order differential equation

$$\biggl( t \biggl[ x ( t ) +\frac{1}{t^{3}}x^{\frac{1}{3}} ( t-2 ) + \frac{1}{t^{4}}x^{\frac{1}{5}} ( t-3 ) \biggr] ^{\prime \prime \prime } \biggr) ^{\prime }+\frac{3}{t}x ( t ) +\frac{1}{t^{3}}x ( 2t ) =0,\quad t \geq 2.$$
(5.1)

Here $$\gamma =1$$, $$r ( t ) =t$$, $$a_{1}=\frac{1}{t^{3}}$$, $$a_{2}= \frac{1}{t^{4}}$$, $$\alpha _{1}=\frac{1}{3}$$, $$\alpha _{2}=\frac{1}{5}$$, $$q_{1}= \frac{3}{t}$$, $$q_{2}=\frac{1}{t^{3}}$$, $$\tau _{1} ( t ) =t$$, $$\tau _{2} ( t ) =2t$$. Let $$\tau ( t ) =\frac{t}{2}\rightarrow$$ $$\tau ( t ) \leq$$ $$\tau _{i} ( t )$$, $$\lim_{t\rightarrow \infty }$$ $$\tau ( t ) =\infty$$, $$\tau ^{\prime } ( t ) =\frac{1}{2}>0$$. Therefore, the conditions $$( A_{1} ) - ( A_{5} )$$ and (1.2) are satisfied. Choosing $$\delta ( t ) =\frac{1}{t}$$. Then $$\delta ( t ) \rightarrow 0$$ for $$t\rightarrow \infty$$. Moreover, $$\theta ( \tau _{1} ( t ) ) =\theta ( t ) = [ 1-\frac{2}{3t^{2}}-\frac{17}{15t^{3}}-\frac{1}{5t^{4}} ] >0$$ for $$t\geq 2$$, and $$\theta ( \tau _{2} ( t ) ) =\theta ( 2t ) = [ 1-\frac{1}{6t^{2}}-\frac{17}{120t^{3}}- \frac{1}{80t^{4}} ] >0$$ for $$t\geq 2$$. Choosing $$\eta ( t ) =1$$, $$b ( t ) =\frac{1}{t^{2}}$$, we have

\begin{aligned}& Q ( t ) = \eta ( t ) \sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma } \bigl( \tau _{i} ( t ) \bigr) -\eta ( t ) \bigl[ r ( t ) b ( t ) \bigr] ^{\prime }+\zeta \epsilon \tau ^{\prime } ( t ) \tau ^{2} ( t ) r ( t ) \eta ( t ) b^{1+\frac{1}{\gamma }} ( t ) \\& \hphantom{Q ( t )} = \frac{1}{t} \biggl[ \biggl( 3+\frac{\zeta \epsilon }{8} \biggr) + \frac{1}{t}-\frac{1}{t^{2}}-\frac{17}{5t^{3}}- \frac{23}{30t^{4}}- \frac{17}{120t^{5}}-\frac{1}{80t^{6}} \biggr] , \\& \underset{t\rightarrow \infty }{\lim \sup } \int _{t_{0}}^{t} \biggl[ Q ( s ) - \frac{r ( s ) \eta ( s ) }{ ( \gamma +1 ) ^{\gamma +1}} \frac{ [ \frac{\eta ^{\prime } ( s ) }{\eta ( s ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime } ( s ) \tau ^{2} ( s ) b^{\frac{1}{\gamma }} ( s ) ] ^{\gamma +1}}{ [ \zeta \epsilon \tau ^{\prime } ( s ) \tau ^{2} ( s ) ] ^{\gamma }} \biggr]\,ds \\& \quad =\underset{t\rightarrow \infty }{\lim \sup } \int _{2}^{t} \frac{1}{s} \biggl[ 3+ \frac{1}{s}-\frac{1}{s^{2}}-\frac{17}{5s^{3}}- \frac{23}{30s^{4}}- \frac{17}{120s^{5}}-\frac{1}{80s^{6}} \biggr]\,ds=\infty . \end{aligned}

Therefore, by Theorem 8, every solution of (5.1) is oscillatory.

## Conclusions

In this paper, we consider a general class of super-linear fourth-order differential equations with several sub-linear neutral terms of the type (1.1). Using the Riccati and generalized Riccati transformations, we establish new oscillation criteria in both cases of canonical case $$\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt=\infty$$ and non-canonical case $$\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt<\infty$$. With the help of the methods given in this paper, we derive some the Kamenev–Philos-type oscillation criteria for (1.1). An illustrative example is given. For interested researchers, there is a good deal of finding new results for (1.1) when $$z(t)=x ( t ) -\sum_{j=1}^{n}a_{j} ( t ) x^{ \alpha _{j}} ( \sigma _{j} ( t ) )$$.

Not applicable.

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## Acknowledgements

The authors of the paper are grateful to the editorial board and reviewers for the careful reading and helpful suggestions, which led to an improvement of our original manuscript.

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This research was not supported by any project.

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El-Gaber, A.A., El-Sheikh, M.M.A. & El-Saedy, E.I. Oscillation of super-linear fourth-order differential equations with several sub-linear neutral terms. Bound Value Probl 2022, 41 (2022). https://doi.org/10.1186/s13661-022-01620-2

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• DOI: https://doi.org/10.1186/s13661-022-01620-2

• 34C10
• 34K11

### Keywords

• Oscillation
• Fourth order
• Neutral differential equations