1 Introduction

The present paper deals with the investigation of the oscillatory behavior of fourth-order advanced differential equation

$$ \bigl( a ( t ) \bigl( y^{\prime \prime \prime } ( t ) \bigr) ^{\beta } \bigr) ^{\prime }+\sum_{i=1}^{j}q_{i} ( t ) g \bigl( y \bigl( \eta _{i} ( t ) \bigr) \bigr) =0,\quad t\geq t_{0}, $$
(1)

where \(j\geq 1\) and β is a quotient of odd positive integers. Throughout this work, we suppose that \(a\in C^{1} ( [t_{0}, \infty ),\mathbb{R} ) \), \(a ( t ) >0, a^{\prime } ( t ) \geq 0, q_{i},\eta _{i}\in C ( [t_{0},\infty ),\mathbb{R} ) \), \(q_{i} ( t ) \geq 0, \eta _{i} ( t ) \geq t\), \(\lim_{t\rightarrow \infty }\eta _{i} ( t ) = \infty, i=1,2,\ldots,j, g\in C ( \mathbb{R},\mathbb{R} ) \) such that \(g ( x ) /x^{\beta }\geq k>0\), for \(x\neq 0\) and under the condition

$$ \int _{t_{0}}^{\infty }\frac{1}{a^{1/\beta } ( s ) } \,\mathrm{d}s= \infty. $$
(2)

Definition 1.1

The function \(y\in C^{3}[t_{y},\infty ), t_{y}\geq t_{0} \) is called a solution of (1), if \(a ( t ) \times ( y^{\prime \prime \prime } ( t ) ) ^{\beta }\in C^{1}[t_{y},\infty )\), and \(y ( t ) \) satisfies (1) on \([t_{y},\infty )\). Moreover, Eq. (1) is oscillatory if all its solutions oscillate.

Definition 1.2

Let

$$ D= \bigl\{ ( t,s ) \in \mathbb{R} ^{2}:t\geq s\geq t_{0} \bigr\} \quad\text{and}\quad D_{0}= \bigl\{ ( t,s ) \in \mathbb{R} ^{2}:t>s\geq t_{0} \bigr\} . $$

A kernel function \(H_{i}\in C ( D,\mathbb{R} ) \) is said to belong to the function class ℑ, written \(H\in \Im \), if, for \(i=1,2\),

  1. (i)

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

  2. (ii)

    \(H_{i} ( t,s ) \) has a continuous and nonpositive partial derivative \(\partial H_{i}/\partial s\) on \(D_{0}\) and there exist functions \(\tau,\vartheta \in C^{1} ( [ t_{0},\infty ), ( 0,\infty ) ) \) and \(h_{i}\in C ( D_{0},\mathbb{R} ) \) such that

    $$ \frac{\partial }{\partial s}H_{1} ( t,s ) + \frac{\tau ^{\prime } ( s ) }{\tau ( s ) }H_{1} ( t,s ) =h_{1} ( t,s ) H_{1}^{\beta / ( \beta +1 ) } ( t,s ) $$
    (3)

    and

    ∂ ∂ s H 2 (t,s)+ ϑ ′ ( s ) ϑ ( s ) H 2 (t,s)= h 2 (t,s) H 2 ( t , s ) .
    (4)

In this paper the following methods were used:

  1. (a)

    The Riccati transformations technique.

  2. (b)

    The method of comparison with second-order differential equations.

  3. (c)

    The integral averaging technique.

From them we obtained new criteria for oscillation of Eq. (1).

Advanced differential equations can find application in dynamical systems, mathematics of networks, optimization, as well as, in the mathematical modeling of engineering problems, such as concerning electrical power systems, materials, energy; see [1–4].

During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations in the canonical case, see [5–7], and the noncanonical case, see [8–10]. One active area of research in this decade is the study of the qualitative behavior for oscillation of differential equations, see [11–32].

Our aim in this paper is to complement and improve results in [33–35]. To this end, the following results are presented.

In particular, by using the comparison technique, the equation

$$ \bigl( \bigl( y^{ ( \kappa -1 ) } ( t ) \bigr) ^{\beta } \bigr) ^{\prime }+q ( t ) y^{\beta } \bigl( \eta ( t ) \bigr) =0 $$
(5)

has been studied by Agarwal and Grace [33]. They proved that it is oscillatory, if

$$ \mathop{\lim \inf }_{t\rightarrow \infty } \int _{t}^{\eta ( t ) } \bigl( \eta ( s ) -s \bigr) ^{\kappa -2} \biggl( \int _{s}^{\infty }q ( t ) \,\mathrm{d}t \biggr) ^{1/\beta } \,\mathrm{d}s> \frac{ ( \kappa -2 ) !}{e}. $$
(6)

Agarwal et al. in [34] extended the Riccati transformation to obtain new oscillatory criteria for (5) under the condition

$$ \mathop{\lim \sup }_{t\rightarrow \infty }t^{\beta ( \kappa -1 ) } \int _{t}^{\infty }q ( s ) \,\mathrm{d}s> \bigl( ( \kappa -1 ) ! \bigr) ^{\beta }. $$
(7)

Authors in [35] studied the oscillatory behavior of (5), for \(\beta =1\). Also, they proved it to be oscillatory, if there exists a function \(\tau \in C^{1} ( [ t_{0},\infty ), ( 0, \infty ) ) \), by using the Riccati transformation. If

$$ \int _{t_{0}}^{\infty } \biggl( \tau ( s ) q ( s ) - \frac{ ( \kappa -2 ) ! ( \tau ^{\prime } ( s ) ) ^{2}}{2^{3-2\kappa }s^{\kappa -2}\tau ( s ) } \biggr) \,\mathrm{d}s= \infty. $$
(8)

To prove this, we apply the previous results to the equation

$$ y^{ ( \kappa ) } ( t ) +by ( rt ) =0, \quad t\geq 1, $$
(9)

where \(\kappa =4, b=q_{0}/t^{4} \) and \(r=3\), and we find:

  1. 1.

    By applying condition (6) in [33], we get

    $$ q_{0}>13.6. $$
  2. 2.

    By applying condition (7) in [34], we get

    $$ q_{0}>18. $$
  3. 3.

    By applying condition (8) in [35], we get

    $$ q_{0}>576. $$

    From the above we find that the results in [34] improve the results in [35]. Moreover, the results in [33] improve results [34, 35].

Our aim in the present paper is to employ the Riccati technique, the integral averaging technique and the theory of comparison to establish some new conditions for the oscillation of all solutions of Eq. (1) under the condition (2). Our results essentially improve and complement the results in [33–35]. Some examples are provided to illustrate the main results.

2 Some auxiliary lemmas

The proofs of our main results are essentially based on the following lemmas.

Lemma 2.1

([36])

Suppose that\(y\in C^{\kappa } ( [ t_{0},\infty ), ( 0, \infty ) ) \), \(y^{ ( \kappa ) }\)is of a fixed sign on\([ t_{0},\infty ) \), \(y^{ ( \kappa ) }\)not identically zero and there exists a\(t_{1}\geq t_{0}\)such that

$$ y^{ ( \kappa -1 ) } ( t ) y^{ ( \kappa ) } ( t ) \leq 0, $$

for all\(t\geq t_{1}\). If we have\(\lim_{t\rightarrow \infty }y ( t ) \neq 0\), then there exists\(t_{\theta }\geq t_{1}\)such that

$$ y ( t ) \geq \frac{\theta }{ ( \kappa -1 ) !}t^{ \kappa -1} \bigl\vert y^{ ( \kappa -1 ) } ( t ) \bigr\vert , $$

for every\(\theta \in ( 0,1 ) \)and\(t\geq t_{\theta }\).

Lemma 2.2

([13])

Letβbe a ratio of two odd numbers, \(V>0\)andUare constants. Then

$$ Ux-Vx^{ ( \beta +1 ) /\beta }\leq \frac{\beta ^{\beta }}{(\beta +1)^{\beta +1}} \frac{U^{\beta +1}}{V^{\beta }},\quad V>0. $$

Lemma 2.3

([15])

Suppose thatyis an eventually positive solution of (1). Then, there exist two possible cases:

\(( \mathbf{S}_{1} )\):

\(y ( t ) >0, y^{\prime } ( t ) >0\), \(y^{\prime \prime } ( t ) >0, y^{\prime \prime \prime } ( t ) >0, y^{ ( 4 ) } ( t ) <0\),

\(( \mathbf{S}_{2} ) \):

\(y ( t ) >0, y^{\prime } ( t ) >0\), \(y^{\prime \prime } ( t ) <0, y^{\prime \prime \prime } ( t ) >0, y^{ ( 4 ) } ( t ) <0\),

for\(t\geq t_{1}\), where\(t_{1}\geq t_{0} \)is sufficiently large.

3 Main results

In this section, we shall establish some oscillation criteria for Eq. (1).

Remark 3.1

([37])

It is well known that the differential equation

$$ \bigl[ a ( t ) \bigl( y^{\prime } ( t ) \bigr) ^{\beta } \bigr] ^{\prime }+q ( t ) y^{\beta } \bigl( g ( t ) \bigr) =0, $$
(10)

where \(\beta >0 \) is the ratio of odd positive integers, a, \(q \in C ( [t_{0},\infty ),\mathbb{R} ^{+} ) \), is nonoscillatory if and only if there exist a number \(t\geq t_{0}\), and a function \(\varsigma \in C^{1} ( [t,\infty ),\mathbb{R} ) \), satisfying the inequality

$$ \varsigma ^{\prime } ( t ) +\gamma a^{-1/\beta } ( t ) \bigl( \varsigma ( t ) \bigr) ^{ ( 1+ \beta ) /\beta }+q ( t ) \leq 0. $$

Theorem 3.1

Assume that (2) holds. If the differential equations

$$ \biggl( \frac{2a^{\frac{1}{\beta }} ( t ) }{ ( \theta t^{2} ) ^{\beta }} \bigl( y^{\prime } ( t ) \bigr) ^{\beta } \biggr) ^{ \prime }+k\sum_{i=1}^{j}q_{i} ( t ) y^{\beta } ( t ) =0 $$
(11)

and

$$ y^{\prime \prime } ( t ) +y ( t ) \int _{t}^{ \infty } \Biggl( \frac{1}{a ( \varsigma ) } \int _{ \varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }\,\mathrm{d}\varsigma =0 $$
(12)

are oscillatory, then every solution of (1) is oscillatory.

Proof

Assume, for the sake of contradiction, that y is a positive solution of (1). Then, we can suppose that \(y ( t ) \) and \(y ( \eta _{i} ( t ) ) \) are positive for all \(t\geq t_{1}\) sufficiently large. From Lemma 2.3, we have two possible cases, \(( \mathbf{S}_{1} ) \) and \(( \mathbf{S}_{2} ) \).

Let case \(( \mathbf{S}_{1} ) \) hold. Using Lemma 2.1, we find

$$ y^{\prime } ( t ) \geq \frac{\theta }{2}t^{2}y^{\prime \prime \prime } ( t ), $$
(13)

for every \(\theta \in (0,1)\) and for all large t.

Defining

$$ \varphi ( t ):=\tau ( t ) \biggl( \frac{a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta }}{y^{\beta } ( t ) } \biggr), $$
(14)

we see that \(\varphi ( t ) >0 \) for \(t\geq t_{1}\), where \(\tau \in C^{1} ( [ t_{0},\infty ), ( 0, \infty ) ) \) and

$$\begin{aligned} \varphi ^{\prime } ( t ) ={}&\tau ^{\prime } ( t ) \frac{a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta }}{y^{\beta } ( t ) }+ \tau ( t ) \frac{ ( a ( y^{\prime \prime \prime } ) ^{\beta } ) ^{\prime } ( t ) }{y^{\beta } ( t ) } \\ &{}-\beta \tau ( t ) \frac{y^{\beta -1} ( t ) y^{\prime } ( t ) a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta }}{y^{2\beta } ( t ) }. \end{aligned}$$

Combining (13) and (14), we obtain

$$\begin{aligned} \varphi ^{\prime } ( t ) \leq {}&\frac{\tau _{+}^{\prime } ( t ) }{\tau ( t ) }\varphi ( t ) +\tau ( t ) \frac{ ( a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta } ) ^{\prime }}{y^{\beta } ( t ) } \\ &{}-\beta \tau ( t ) \frac{\theta }{2}t^{2} \frac{a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta +1}}{y^{\beta +1} ( t ) } \\ \leq{} &\frac{\tau ^{\prime } ( t ) }{\tau ( t ) } \varphi ( t ) +\tau ( t ) \frac{ ( a ( t ) ( y^{\prime \prime \prime } ( t ) ) ^{\beta } ) ^{\prime }}{y^{\beta } ( t ) } \\ &{} - \frac{\beta \theta t^{2}}{2 ( \tau ( t ) a ( t ) ) ^{\frac{1}{\beta }}}\varphi ^{ \frac{\beta +1}{\beta }} ( t ). \end{aligned}$$
(15)

From (1) and (15), we get

$$ \varphi ^{\prime } ( t ) \leq \frac{\tau ^{\prime } ( t ) }{\tau ( t ) }\varphi ( t ) -k\tau ( t ) \frac{\sum_{i=1}^{j}q_{i} ( t ) y^{\beta } ( \eta _{i} ( t ) ) }{y^{\beta } ( t ) }- \frac{\beta \theta t^{2}}{2 ( \tau ( t ) a ( t ) ) ^{\frac{1}{\beta }}} \varphi ^{\frac{\beta +1}{\beta }} ( t ). $$

Note that \(y^{\prime } ( t ) >0\) and \(\eta _{i} ( t ) \geq t\). Thus

$$ \varphi ^{\prime } ( t ) \leq \frac{\tau ^{\prime } ( t ) }{\tau ( t ) }\varphi ( t ) -k\tau ( t ) \sum_{i=1}^{j}q_{i} ( t ) - \frac{\beta \theta t^{2}}{2 ( \tau ( t ) a ( t ) ) ^{\frac{1}{\beta }}} \varphi ( t ) ^{\frac{\beta +1}{\beta }}. $$
(16)

If we set \(\tau ( t ) =k=1\) in (16), we obtain

$$ \varphi ^{\prime } ( t ) + \frac{\beta \theta t^{2}}{2a^{\frac{1}{\beta }} ( t ) }\varphi ^{\frac{\beta +1}{\beta }} ( t ) + \sum_{i=1}^{j}q_{i} ( t ) \leq 0. $$

Thus, we can see that Eq. (11) is nonoscillatory, which is a contradiction.

Let case \(( \mathbf{S}_{2} ) \) hold. Defining

$$ \psi ( t ):=\vartheta ( t ) \frac{y^{\prime } ( t ) }{y ( t ) }, $$

we see that \(\psi ( t ) >0 \) for \(t\geq t_{1}\), where \(\vartheta \in C^{1} ( [ t_{0},\infty ), ( 0, \infty ) ) \). By differentiating \(\psi ( t ) \), we find

$$ \psi ^{\prime } ( t ) = \frac{\vartheta ^{\prime } ( t ) }{\vartheta ( t ) }\psi ( t ) +\vartheta ( t ) \frac{y^{\prime \prime } ( t ) }{y ( t ) }- \frac{1}{\vartheta ( t ) }\psi ^{2} ( t ). $$
(17)

Now, by integrating (1) from t to m and using \(y^{\prime } ( t ) >0\), we have

$$ a ( m ) \bigl( y^{\prime \prime \prime } ( m ) \bigr) ^{\beta }-a ( t ) \bigl( y^{\prime \prime \prime } ( t ) \bigr) ^{\beta }=- \int _{t}^{m}\sum_{i=1}^{j}q_{i} ( s ) g \bigl( y \bigl( \eta _{i} ( s ) \bigr) \bigr) \,\mathrm{d}s. $$

By virtue of \(y^{\prime } ( t ) >0\) and \(\eta _{i} ( t ) \geq t\), we get

$$ a ( m ) \bigl( y^{\prime \prime \prime } ( m ) \bigr) ^{\beta }-a ( t ) \bigl( y^{\prime \prime \prime } ( t ) \bigr) ^{\beta }\leq -ky^{\beta } ( t ) \int _{t}^{u}\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s. $$

Letting \(m\rightarrow \infty \), we see that

$$ a ( t ) \bigl( y^{\prime \prime \prime } ( t ) \bigr) ^{\beta }\geq ky^{\beta } ( t ) \int _{t}^{ \infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s $$

and so

$$ y^{\prime \prime \prime } ( t ) \geq y ( t ) \Biggl( \frac{k}{a ( t ) } \int _{t}^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }. $$

Integrating again from t to ∞, we get

$$ y^{\prime \prime } ( t ) +y ( t ) \int _{t}^{ \infty } \Biggl( \frac{k}{a ( \varsigma ) } \int _{ \varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }\,\mathrm{d}\varsigma \leq 0. $$
(18)

Combining (17) and (18), we obtain

$$ \psi ^{\prime } ( t ) \leq \frac{\vartheta ^{\prime } ( t ) }{\vartheta ( t ) }\psi ( t ) - \vartheta ( t ) \int _{t}^{\infty } \Biggl( \frac{k}{a ( \varsigma ) }\int _{\varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }\,\mathrm{d}\varsigma - \frac{1}{\vartheta ( t ) }\psi ^{2} ( t ). $$
(19)

If \(\vartheta ( t ) =k=1\) in (19), then we get

$$ \psi ^{\prime } ( t ) +\psi ^{2} ( t ) + \int _{t}^{ \infty } \Biggl( \frac{1}{a ( \varsigma ) } \int _{ \varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }\,\mathrm{d}\varsigma \leq 0. $$

Hence, we see that Eq. (12) is nonoscillatory, which is a contradiction. The proof of the theorem is complete. □

Based on the above results and Theorem 3.1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with \(\beta =1\).

Theorem 3.2

Let\(\beta =k=1\). Assume that

$$ \int _{t_{0}}^{\infty }\frac{\theta t^{2}}{2a ( t ) } \,\mathrm{d}t=\infty $$

and

$$ \mathop{\lim \inf }_{t\rightarrow \infty } \biggl( \int _{t_{0}}^{t} \frac{\theta s^{2}}{2a ( s ) }\,\mathrm{d}s \biggr) \int _{t}^{ \infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s>\frac{1}{4}, $$
(20)

for some constant\(\theta \in ( 0,1 ) \),

$$ \mathop{\lim \inf }_{t\rightarrow \infty }t \int _{t_{0}}^{t} \int _{v}^{ \infty } \Biggl( \frac{1}{a ( \varsigma ) } \int _{ \varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) \,\mathrm{d}\varsigma \,\mathrm{d}v>\frac{1}{4}, $$
(21)

then all solutions of (1) are oscillatory.

In this theorem, we employ the integral averaging technique to establish an oscillation criterion for (1).

Theorem 3.3

Let (2) hold. If there exist positive functions\(\tau,\vartheta \in C^{1} ( [ t_{0},\infty ),\mathbb{R} ) \)such that

$$ \mathop{\lim \sup }_{t\rightarrow \infty } \frac{1}{H_{1} ( t,t_{1} ) } \int _{t_{1}}^{t} \Biggl( H_{1} ( t,s ) k \tau ( s ) \sum_{i=1}^{j}q_{i} ( s ) -\pi ( s ) \Biggr) \,\mathrm{d}s=\infty $$
(22)

and

$$ \mathop{\lim \sup }_{t\rightarrow \infty } \frac{1}{H_{2} ( t,t_{1} ) } \int _{t_{1}}^{t} \biggl( H_{2} ( t,s ) \vartheta ( s ) \varpi ( s ) - \frac{\vartheta ( s ) h_{2}^{2} ( t,s ) }{4} \biggr) \,\mathrm{d}s=\infty, $$
(23)

where

$$ \pi ( s ) = \frac{h_{1}^{\beta +1} ( t,s ) H_{1}^{\beta } ( t,s ) }{ ( \beta +1 ) ^{\beta +1}} \frac{2^{\beta }\tau ( s ) a ( s ) }{ ( \theta s^{2} ) ^{\beta }}, $$

for all\(\theta \in ( 0,1 )\), and

$$ \varpi ( s ) = \int _{t}^{\infty } \Biggl( \frac{k}{a ( \varsigma ) } \int _{\varsigma }^{\infty } \sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta } \,\mathrm{d}\varsigma, $$

then (1) is oscillatory.

Proof

Assume, for the sake of contradiction, that y is a positive solution of (1). Then, we can suppose that \(y ( t ) \) and \(y ( \eta _{i} ( t ) ) \) are positive for all \(t\geq t_{1}\) sufficiently large. From Lemma 2.3, we have two possible cases, \(( \mathbf{S}_{1} ) \) and \(( \mathbf{S}_{2} ) \).

Assume that \(( \mathbf{S}_{1} ) \) holds. From Theorem 3.1, we find that (16) holds. Multiplying (16) by \(H_{1} ( t,s ) \) and integrating the resulting inequality from \(t_{1}\) to t, we find that

$$\begin{aligned} \int _{t_{1}}^{t}H_{1} ( t,s ) k\tau ( s ) \sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \leq {}&\varphi ( t_{1} ) H_{1} ( t,t_{1} ) + \int _{t_{1}}^{t} \biggl( \frac{\partial }{\partial s}H_{1} ( t,s ) + \frac{\tau ^{\prime } ( s ) }{\tau ( s ) }H_{1} ( t,s ) \biggr) \varphi ( s ) \,\mathrm{d}s \\ &{}- \int _{t_{1}}^{t} \frac{\beta \theta s^{2}}{2 ( \tau ( s ) a ( s ) ) ^{\frac{1}{\beta }}}H_{1} ( t,s ) \varphi ^{\frac{\beta +1}{\beta }} ( s ) \,\mathrm{d}s. \end{aligned}$$

From (3), we get

$$\begin{aligned} \int _{t_{1}}^{t}H_{1} ( t,s ) k\tau ( s ) \sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \leq{} &\varphi ( t_{1} ) H_{1} ( t,t_{1} ) + \int _{t_{1}}^{t}h_{1} ( t,s ) H_{1}^{\beta / ( \beta +1 ) } ( t,s ) \varphi ( s ) \,\mathrm{d}s \\ &{}- \int _{t_{1}}^{t} \frac{\beta \theta s^{2}}{2 ( \tau ( s ) a ( s ) ) ^{\frac{1}{\beta }}}H_{1} ( t,s ) \varphi ^{\frac{\beta +1}{\beta }} ( s ) \,\mathrm{d}s. \end{aligned}$$
(24)

Using Lemma 2.2 with \(V=\beta \theta s^{2}/ ( 2 ( \tau ( s ) a ( s ) ) ^{\frac{1}{\beta }} ) H_{1} ( t,s ), U=h_{1} ( t,s ) H_{1}^{\beta / ( \beta +1 ) } ( t,s ) \) and \(y=\varphi ( s ) \), we get

$$\begin{aligned} &h_{1} ( t,s ) H_{1}^{\beta / ( \beta +1 ) } ( t,s ) \varphi ( s ) - \frac{\beta \theta s^{2}}{2 ( \tau ( s ) a ( s ) ) ^{\frac{1}{\beta }}}H_{1} ( t,s ) \varphi ^{\frac{\beta +1}{\beta }} ( s ) \\ &\quad\leq \frac{h_{1}^{\beta +1} ( t,s ) H_{1}^{\beta } ( t,s ) }{ ( \beta +1 ) ^{\beta +1}} \frac{2^{\beta }\tau ( s ) a ( s ) }{ ( \theta s^{2} ) ^{\beta }}, \end{aligned}$$

which with (24) gives

$$ \frac{1}{H_{1} ( t,t_{1} ) } \int _{t_{1}}^{t} \Biggl( H_{1} ( t,s ) k\tau ( s ) \sum_{i=1}^{j}q_{i} ( s ) - \pi ( s ) \Biggr) \,\mathrm{d}s\leq \varphi ( t_{1} ). $$

This contradicts (22).

Assume that \(( \mathbf{S}_{2} ) \) holds. From Theorem 3.1, (19) holds. Multiplying (19) by \(H_{2} ( t,s ) \) and integrating the resulting inequality from \(t_{1}\) to t, we obtain

$$\begin{aligned} \int _{t_{1}}^{t}H_{2} ( t,s ) \vartheta ( s ) \varpi ( s ) \,\mathrm{d}s \leq {}&\psi ( t_{1} ) H_{2} ( t,t_{1} ) \\ &{}+ \int _{t_{1}}^{t} \biggl( \frac{\partial }{\partial s}H_{2} ( t,s ) + \frac{\vartheta ^{\prime } ( s ) }{\vartheta ( s ) }H_{2} ( t,s ) \biggr) \psi ( s ) \,\mathrm{d}s \\ &{}- \int _{t_{1}}^{t}\frac{1}{\vartheta ( s ) }H_{2} ( t,s ) \psi ^{2} ( s ) \,\mathrm{d}s. \end{aligned}$$

Thus, from (4), we get

∫ t 1 t H 2 ( t , s ) ϑ ( s ) ϖ ( s ) d s ≤ ψ ( t 1 ) H 2 ( t , t 1 ) + ∫ t 1 t h 2 ( t , s ) H 2 ( t , s ) ψ ( s ) d s − ∫ t 1 t 1 ϑ ( s ) H 2 ( t , s ) ψ 2 ( s ) d s ≤ ψ ( t 1 ) H 2 ( t , t 1 ) + ∫ t 1 t ϑ ( s ) h 2 2 ( t , s ) 4 d s

and so

$$ \frac{1}{H_{2} ( t,t_{1} ) } \int _{t_{1}}^{t} \biggl( H_{2} ( t,s ) \vartheta ( s ) \varpi ( s ) - \frac{\vartheta ( s ) h_{2}^{2} ( t,s ) }{4} \biggr) \,\mathrm{d}s\leq \psi ( t_{1} ), $$

which contradicts (23). The proof of the theorem is complete. □

4 Examples

In this section, we give the following examples.

Example 4.1

Consider the equation

$$ y^{ ( 4 ) } ( t ) +\frac{q_{0}}{t^{4}}y ( 2t ) =0,\quad t\geq 1, $$
(25)

where \(q_{0}>0\) is a constant. Note that \(\beta =1, \kappa =4, a ( t ) =1, q ( t ) =q_{0}/t^{4}\) and \(\eta ( t ) =2t\). If we set \(k=1\), then conditions (20) and (21) become

$$\begin{aligned} \mathop{\lim \inf }_{t\rightarrow \infty } \biggl( \int _{t_{0}}^{t} \frac{\theta s^{2}}{2a ( s ) }\,\mathrm{d}s \biggr) \int _{t}^{ \infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s &=\mathop{\lim \inf }_{t\rightarrow \infty } \biggl( \frac{t^{3}}{3} \biggr) \int _{t}^{\infty } \frac{q_{0}}{s^{4}}\,\mathrm{d}s \\ &=\frac{q_{0}}{9}>\frac{1}{4} \end{aligned}$$

and

$$\begin{aligned} \mathop{\lim \inf }_{t\rightarrow \infty }t \int _{t_{0}}^{t} \int _{v}^{ \infty } \Biggl( \frac{1}{a ( \varsigma ) } \int _{ \varsigma }^{\infty }\sum_{i=1}^{j}q_{i} ( s ) \,\mathrm{d}s \Biggr) ^{1/\beta }\,\mathrm{d}\varsigma \,\mathrm{d}v &=\mathop{\lim \inf }_{t\rightarrow \infty }t \biggl( \frac{q_{0}}{6t} \biggr), \\ &=\frac{q_{0}}{6}>\frac{1}{4}, \end{aligned}$$

respectively. From Theorem 3.2, all solutions of (25) are oscillatory, if \(q_{0}>2.25\).

Remark 4.1

We compare our result with the known related criteria

Therefore, our result improves results [33–35].

Example 4.2

Consider the differential equation (9), where \(q_{0}>0\) is a constant. Note that \(\beta =1, \kappa =4, a ( t ) =1, q ( t ) =q_{0}/t^{4}\) and \(\eta ( t ) =3t\). If we set \(k=1\), then condition (20) becomes

$$ \frac{q_{0}}{9}>\frac{1}{4}. $$

Therefore, from Theorem 3.2, all solutions of (9) are oscillatory, if \(q_{0}>2.25\).

Remark 4.2

Our result improves results [33–35].