Skip to main content

Oscillatory and asymptotic behavior of advanced differential equations

Abstract

In this paper, a class of fourth-order differential equations with advanced type is studied. Applying the generalized Riccati transformation, integral averaging technique and the theory of comparison, a set of new criteria for oscillation or certain asymptotic behavior of solutions of this equations is given. Our results essentially improve and complement some earlier publications. Some examples are presented to demonstrate the main results.

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 , 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 [14].

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 [57], and the noncanonical case, see [810]. One active area of research in this decade is the study of the qualitative behavior for oscillation of differential equations, see [1132].

Our aim in this paper is to complement and improve results in [3335]. 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 [3335]. Some examples are provided to illustrate the main results.

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.

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. □

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 [3335].

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 [3335].

References

  1. 1.

    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  Google Scholar 

  2. 2.

    Dzurina, J., Grace, S.R., Jadlovska, I., Li, T.: Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293(5), 910–922 (2020)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Li, T., Rogovchenko, Y.V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 105, Article ID 106293 (2020)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70, Article ID 86 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: A new approach in the study of oscillatory behavior of even-order neutral delay differential equations. Appl. Math. Comput. 225, 787–794 (2013)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Li, T., Rogovchenko, Y.V.: Oscillation criteria for even-order neutral differential equations. Appl. Math. Lett. 61, 35–41 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143–160 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Even-order half-linear advanced differential equations: improved criteria in oscillatory and asymptotic properties. Appl. Math. Comput. 266, 481–490 (2015)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Li, T., Rogovchenko, Y.V.: On asymptotic behavior of solutions to higher-order sublinear Emden–Fowler delay differential equations. Appl. Math. Lett. 67, 53–59 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zhang, C., Li, T., Agarwal, R.P., Bohner, M.: Oscillation results for fourth-order nonlinear dynamic equations. Appl. Math. Lett. 25(12), 2058–2065 (2012)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bazighifan, O., Dassios, I.: Riccati technique and asymptotic behavior of fourth-order advanced differential equations. Mathematics 8, 1–11 (2020)

    Google Scholar 

  12. 12.

    Bazighifan, O., Dassios, I.: On the asymptotic behavior of advanced differential equations with a non-canonical operator. Appl. Sci. 10, 3130 (2020)

    Article  Google Scholar 

  13. 13.

    Bazighifan, O., Ahmed, H., Yao, S.: New oscillation criteria for advanced differential equations of fourth order. Mathematics 8, 1–11 (2020)

    Google Scholar 

  14. 14.

    Bazighifan, O., Ramos, H.: On the asymptotic and oscillatory behavior of the solutions of a class of higher-order differential equations with middle term. Appl. Math. Lett. 107, 106431 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Bazighifan, O.: Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations. Adv. Differ. Equ. 2020, 201 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Bazighifan, O.: On the oscillation of certain fourth-order differential equations with p-Laplacian like operator. Appl. Math. Comput. 386, 125475 (2020)

    MathSciNet  Google Scholar 

  17. 17.

    Bazighifan, O.: Oscillatory applications of some fourth-order differential equations. Math. Methods Appl. Sci.. https://doi.org/10.1002/mma.6694

  18. 18.

    Chatzarakis, G.E., Moaaz, O., Li, T., Qaraad, B.: Some oscillation theorems for nonlinear second-order differential equations with an advanced argument. Adv. Differ. Equ. 2020, 160 (2020)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Chatzarakis, G.E., Elabbasy, E.M., Bazighifan, O.: An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay. Adv. Differ. Equ. 2019, 336 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Grace, S., Dzurina, J., Jadlovska, I., Li, T.: On the oscillation of fourth order delay differential equations. Adv. Differ. Equ. 2019, 118 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford (1991)

    MATH  Google Scholar 

  22. 22.

    Li, T., Baculikova, B., Dzurina, J., Zhang, C.: Oscillation of fourth order neutral differential equations with p-Laplacian like operators. Bound. Value Probl. 2014, 56 (2014)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Moaaz, O., Elabbasy, E.M., Bazighifan, O.: On the asymptotic behavior of fourth-order functional differential equations. Adv. Differ. Equ. 2017, 261 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Moaaz, O., Dassios, I., Bazighifan, O., Muhib, A.: Oscillation theorems for nonlinear differential equations of fourth-order. Mathematics 520, 8 (2020)

    Google Scholar 

  25. 25.

    Moaaz, O., Elabbasy, E.M., Muhib, A.: Oscillation criteria for even-order neutral differential equations with distributed deviating arguments. Adv. Differ. Equ. 2019, 297 (2019)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Nehari, Z.: Oscillation criteria for second order linear differential equations. Trans. Am. Math. Soc. 85, 428–445 (1957)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Philos, C.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay. Arch. Math. (Basel) 36, 168–178 (1981)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Rehak, P.: How the constants in Hille–Nehari theorems depend on time scales. Adv. Differ. Equ. 2006, 06453 (2006)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: New results for oscillatory behavior of even-order half-linear delay differential equations. Appl. Math. Lett. 26, 179–183 (2013)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Agarwal, R., Grace, S., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Acad. Publ., Dordrecht (2000)

    Book  Google Scholar 

  31. 31.

    Agarwal, R., Shieh, S.L., Yeh, C.C.: Oscillation criteria for second order retarded differential equations. Math. Comput. Model. 26, 1–11 (1997)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Baculikova, B., Dzurina, J., Graef, J.R.: On the oscillation of higher-order delay differential equations. Math. Slovaca 187, 387–400 (2012)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Agarwal, R., Grace, S.R.: Oscillation theorems for certain functional differential equations of higher order. Math. Comput. Model. 39, 1185–1194 (2004)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Agarwal, R., Grace, S.R., O’Regan, D.: Oscillation criteria for certain n th order differential equations with deviating arguments. J. Math. Anal. Appl. 262, 601–622 (2001)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Grace, S.R., Lalli, B.S.: Oscillation theorems for nth-order differential equations with deviating arguments. Proc. Am. Math. Soc. 90, 65–70 (1984)

    MATH  Google Scholar 

  36. 36.

    Zhang, C., Li, T., Sun, B., Thandapani, E.: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 24, 1618–1621 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Agarwal, R., Shieh, S.L., Yeh, C.C.: Oscillation criteria for second-order retarded differential equations. Math. Comput. Model. 26, 1–11 (1997)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their work and constructive comments that improved the manuscript.

Availability of data and materials

Not applicable.

Funding

The authors received no direct funding for this work.

Author information

Affiliations

Authors

Contributions

The authors declare that they have read and approved the final manuscript.

Corresponding author

Correspondence to Omar Bazighifan.

Ethics declarations

Competing interests

The authors declares that there are no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bazighifan, O., Chatzarakis, G.E. Oscillatory and asymptotic behavior of advanced differential equations. Adv Differ Equ 2020, 414 (2020). https://doi.org/10.1186/s13662-020-02875-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-020-02875-8

MSC

  • 34C10
  • 34K11

Keywords

  • Oscillation
  • Nonoscillatory solutions
  • Fourth-order
  • Advanced differential equations