1 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 [1–5, 7, 8, 10–20, 22–24, 26–39]. 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 [3–5, 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).

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 \(t\geq t_{1}\)

\(( S_{1} ) \):

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

\(( S_{2} ) \):

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

\(( S_{3} ) \):

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

3 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. □

4 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).

5 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.

6 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 \(z(t)=x ( t ) -\sum_{j=1}^{n}a_{j} ( t ) x^{ \alpha _{j}} ( \sigma _{j} ( t ) )\).