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Oscillatory criteria of noncanonical even-order differential equations with a superlinear neutral term

Abstract

The oscillatory behavior of solutions of an even-order differential equation with a superlinear neutral term is considered using Riccati and generalized Riccati transformations, the integral averaging technique, and the theory of comparison. New sufficient conditions are established in the noncanonical case. An example is given to support our results.

1 Introduction

In this article, we are concerned with the oscillation property of solutions of the half-linear even-order differential equation with a superlinear neutral term of the form

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }+\sum_{i=1}^{m}q_{i} ( t ) x^{\delta } \bigl( \eta _{i} ( t ) \bigr) =0,\quad t\geq t_{0}\geq 0, $$
(1.1)

where \(w(t)=x ( t ) +p ( t ) x^{\beta } ( \xi ( t ) ) ,\beta \), γ, and δ are quotients of odd positive integers with \(\beta \geq 1\), \(\delta /\beta \leq \gamma \), and \(n\geq 4\) is an even integer under the noncanonical condition

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

Throughout the paper, we assume that

\(( H_{1} ) \) \(r ( t ) \in C^{1} ( [t_{0},\infty ), ( 0, \infty ) ) \), \(r^{\prime } ( t ) \geq 0\), \(p ( t ) \in C ( [t_{0},\infty ),R ) \), \(p ( t ) \geq 1\), \(p ( t ) \not \equiv 1\) for large t;

\(( H_{2} ) \) \(\xi ( t ) ,q_{i} ( t ) \in C ( [t_{0},\infty ),R ) \), \(\xi ( t ) \leq t,\xi \) is strictly increasing, \(\lim_{t\rightarrow \infty }\xi ( t ) =\infty \), \(q_{i} ( t ) \geq 0\), and \(q_{i} ( t ) \) are not equal to zero for large t, \(i=1,2,\ldots,m\);

\(( H_{3} ) \) \(\eta _{i} ( t ) \in C ( [t_{0}, \infty ),R ) \), there exists a function \(\eta ( t ) \in C ( [t_{0},\infty ),R ) \) such that \(\eta ( t ) \leq \eta _{i} ( t ) \) for \(i=1,2,\ldots,m\), \(\eta ( t ) <\xi ( t ) \) and \(\lim_{t\rightarrow \infty }\eta ( t ) =\infty \);

\(( H_{4} ) \) \(h ( t ) =\xi ^{-1} ( \eta ( t ) ) \), where \(\xi ^{-1}\) is the inverse function of ξ.

By a solution of (1.1), we mean a nontrivial function \(x ( t ) \in C ( [t_{x},\infty ) ) \), \(t_{x}\geq t_{0}\), which has the property \(r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma }\in C^{1} ( [t_{x},\infty ) ) \) and satisfies (1.1). We consider only those solutions \(x ( t ) \) of (1.1), which satisfy \(\sup \{ \vert x ( t ) \vert :t\geq t^{ \ast } \} >0\) for \(t^{\ast }\geq t_{x}\). A solution \(x ( t ) \) of (1.1) is termed oscillatory if it has arbitrarily large zeros on \([t_{x},\infty )\); otherwise, it is said to be nonoscillatory. Equation (1.1) is termed oscillatory if all its solutions oscillate.

It is notable that in recent years, the oscillation property of solutions of differential equations and their applications have been and still are receiving intensive attention (see [2, 57, 15, 1821]). In the natural sciences, technology, and population dynamics, differential equations find many application fields [1, 4, 8, 9, 13]. For particular applications of differential equations with neutral term they are often used for the study of distributed networks containing lossless transmission lines [10]. Moreover, for particular applications in superlinear wave equation [11]. Here, we mention some recent works concerned with special cases of (1.1), which motivated this work. Many authors have studied the oscillatory behavior of solutions of the differential equations

$$ \bigl( \bigl( y^{{ ( n-1 ) }} ( t ) \bigr) ^{ \alpha } \bigr) ^{\prime }+q ( t ) y^{B} \bigl( g ( t ) \bigr) =0, $$
(1.3)

where it is notable that some of their results can be extended to the following equations

$$ \bigl( r ( t ) \bigl( y^{{ ( n-1 ) }} ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) y^{B} \bigl( g ( t ) \bigr) =0. $$
(1.4)

In [24], Zhang et al. studied (1.4) in a noncanoncail case as (1.2). They established new oscillation criteria claiming that it could not be applied in case \(g ( t ) =t\). More recently, Zhang et al. [22] suggested some new oscillation criteria for the even-order delay differential equation (1.4) with the same noncanonical case for which they stressed that the study of oscillatory properties in this case brings in additional difficulties. Meanwhile, Li and Rogovchenko [14] discussed the oscillatory behavior of a class of even-order neutral differential equations of the form

$$ w^{{ ( n ) }} ( t ) +h ( t ) y \bigl( \xi ( t ) \bigr) =0, $$
(1.5)

with \(w ( t ) =y ( t ) +p ( t ) y ( g ( t ) ) \). Their new theorems complement and improve a number of results reported in the literature.

In [7], Elabbasy et al. studied the even-order neutral differential equation with several delays

$$ \bigl( r ( t ) \bigl( v^{ ( n-1 ) } \bigr) ^{ \alpha } \bigr) ^{\prime }+\sum_{i=1}^{k}q_{i} ( t ) f \bigl( u \bigl( \eta _{i} ( t ) \bigr) \bigr) =0, $$
(1.6)

where \(v ( t ) =u ( t ) +p ( t ) u ( g ( t ) ) \), with \(r^{\prime } ( t ) \geq 0\), \(p ( t ) \in [ 0,p_{0} ] \), \(\eta _{i}^{\prime } ( t ) >0\) in the canonical case \(\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt=\infty \). They used the Riccati substitution technique and comparison with delay equations of the first order to establish new oscillation criteria, which simplify and complement some related results in the literature.

In [9], Grace et al. studied the oscillation of the higher-order dynamic equation with superlinear neutral term

$$ \bigl( a({\mu})y^{\Delta ^{n-1}}({ \mu}) \bigr) ^{\Delta }+q({\mu})x^{\beta } \bigl(\tau ({\mu})\bigr)=0, $$
(1.7)

where \(y ( t ) =x ( t ) +p ( t ) x^{ \alpha } ( g ( t ) ) \), \(a^{\Delta }({\mu})\geq 0\), \(\beta \leq \alpha \), and \(\alpha \geq 1\). Their proposed results provide a unified platform that adequately covers discrete and continuous equations and further sufficiently comments on the oscillation of a more general class of equations than the ones reported in the literature.

More recently, Dharuman et al. [3], were concerned with the oscillatory behavior of solutions of the even-order nonlinear differential equation with a superlinear neutral term

$$ \bigl(b(t)z^{(n-1)}(t)\bigr)^{\prime }+q(t)x^{\beta }\bigl( \sigma (t)\bigr)=0, $$
(1.8)

where \(z(t)=x(t)+p(t)x^{\alpha }(\tau (t))\), \(\alpha >1\), \(b^{\prime }(t) \geq 0\), in the canonical case \(\int _{t_{0}}^{\infty }\frac{1}{b ( t ) }\,dt=\infty \). They established new comparison theorems that compare the higher-order equation (1.8) with a couple of first-order delay differential equations. Moreover, as many results are available in the literature on the oscillation of first-order delay differential equations, it would be possible to formulate many criteria for the oscillation of (1.8) based on their results.

In this article, we study the oscillatory behavior of solutions of Eq. (1.1) in the noncanonical case (1.2) by applying the Riccati and generalized Riccati transformations, the integral averaging technique, and comparison theory.

2 Auxiliary lemmas

In this section, we outline some lemmas needed for our results.

Lemma 1

[1] Let \(w ( t ) \in C^{n}\ ( [t_{0},\infty ), ( 0, \infty ) ) ,w^{ ( n ) } ( t ) \) be of fixed sign, and \(\ w^{ ( n-1 ) } ( t ) w^{ ( n ) } ( t ) \leq 0\), for all \(t\geq t_{0}\). If \(\lim_{t\rightarrow \infty }w ( t ) \neq 0\), then for every \(\epsilon \in ( 0,1 ) \) there may exist \(t_{\epsilon }\) \(\geq t_{0}\) such that \(w ( t ) \geq \frac{\epsilon }{ ( n-1 ) !}t^{n-1} \vert w^{ ( n-1 ) } ( t ) \vert \) for \(t\geq t_{\epsilon }\).

Lemma 2

[12] Let the function \(w ( t ) \) satisfy \(w^{ ( i ) } ( t ) >0\), \(i=1,2,\ldots,n-1\) and \(w^{ ( n ) } ( t ) \leq 0\), then

$$ \frac{w ( t ) }{t^{n-1}/ ( n-1 ) !}\geq \frac{w^{\prime } ( t ) }{t^{n-2}/ ( n-2 ) !}. $$

Lemma 3

[22] Let \(w\in C^{I} ( [t_{0},\infty ),R^{+} ) \). If \(w^{ ( I ) } ( t ) \) is eventually of one sign for all large t, then there exists \(t_{1}\geq t_{0}\) and an integer \(i,0\leq i\leq I\) with \(I+i\) even for \(w^{ ( I ) } ( t ) \geq 0\), or \(I+i\) odd for \(w^{ ( I ) } ( t ) \leq 0\) such that

$$\begin{aligned}& i>0\textit{ yields}\quad w^{ ( j ) } ( t ) >0 \quad \textit{for }t\geq t_{1},j=0,1,\ldots,i-1 \quad \textit{and}\quad \\& i\leq I-1\textit{ yields}\quad ( -1 ) ^{i+j}w^{ ( j ) } ( \tau ) >0 \quad \textit{for }t\geq t_{1},j=i,i+1,\ldots,I-1. \end{aligned}$$

Lemma 4

[16] Let f\(( v ) =Av-B ( v-D ) ^{ \frac{\gamma +1}{\gamma }}\), where \(B>0\), A, and D are constants. Then, the maximum value of f on R at \(v^{\ast }=D+ ( \gamma A/ ( \gamma +1 ) B ) ^{ \gamma }\) is

$$ \max_{v\in R}f ( v ) =f \bigl( v^{\ast } \bigr) =AD+ \frac{\gamma ^{\gamma }}{ ( \gamma +1 ) ^{\gamma +1}} \frac{A^{\gamma +1}}{B^{\gamma }}. $$

Lemma 5

Assume that \(x ( t ) \) is an eventually positive solution of (1.1). Then, there exists \(t_{1}\geq t_{0}\) such that for \(t\geq t_{1}\) the corresponding function w satisfies one of the following four cases

$$\begin{aligned} c_{1})\ & w>0,w^{\prime }>0,w^{ ( n-2 ) }>0,w^{ ( n-1 ) }>0,w^{ ( n ) } \leq 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{\prime }\leq 0,\\ c_{2})\ & w>0,w^{\prime }>0,w^{ ( n-2 ) }< 0,w^{ ( n-1 ) }>0,w^{ ( n ) } \leq 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{\prime }\leq 0,\\ c_{3})\ & w>0,w^{\prime }>0,w^{ ( n-2 ) }>0,w^{ ( n-1 ) }< 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{ \gamma } \bigr) ^{\prime }\leq 0, \\ c_{4})\ & w>0,w^{ ( i ) }< 0,w^{ ( i+1 ) }>0 \textit{ for every odd integer }i\in \{ 1,2,\ldots,n-3 \} \textit{ and } \\ & w^{ ( n-1 ) }< 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{\prime }\leq 0. \end{aligned}$$

Proof

Assume that \(x ( t ) \) is an eventually positive solution of (1.1), then for any \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for all \(t\geq t_{1}\). Hence, using the definition of \(w ( t ) \), we have \(w ( t ) >0\). It follows from (1.1) that \(( r ( w^{ ( n-1 ) } ) ^{\gamma } ) ^{ \prime }\leq 0\). Next following the proof of Theorem 2.1 of [22] with Lemma 3, we have three cases

$$\begin{aligned} 1)\ & w>0,w^{\prime }>0,w^{ ( n-1 ) }>0,w^{ ( n ) } \leq 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{\prime }\leq 0,\\ 2)\ & w>0,w^{\prime }>0,w^{ ( n-2 ) }>0,w^{ ( n-1 ) }< 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{ \prime }\leq 0,\\ 3)\ & w>0,w^{ ( i ) }< 0,w^{ ( i+1 ) }>0\text{ for every odd integer } i\in \{ 1,2,\ldots,n-3 \} \text{ and } \\ & w^{ ( n-1 ) }< 0, \quad \bigl( r \bigl( w^{ ( n-1 ) } \bigr) ^{\gamma } \bigr) ^{\prime }\leq 0. \end{aligned}$$

Now going through as in [23], case 1 \(w^{ ( n-2 ) }\) has two possibilities either \(w^{ ( n-2 ) }>0\) or \(w^{ ( n-2 ) }<0 \). This completes the proof. □

3 Main results

Lemma 6

Assume that \(x ( t ) \) is an eventually positive solution of (1.1) with w satisfying case 1 of Lemma 5, then for all constants \(k_{1}>0\),

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }\leq -\sum_{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) w^{\frac{\delta }{\beta }} \bigl( h ( t ) \bigr) , $$
(3.1)

where \(P_{1} ( t ) = \frac{1}{p ( \xi ^{-1} ( t ) ) } [ 1- [ \frac{\xi ^{-1} ( t ) }{\xi ^{-1} ( \xi ^{-1} ( t ) ) } ] ^{\frac{1-n}{\beta }} \frac{k_{1}^{\frac{1}{\beta }-1}}{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) } ] \geq 0\).

Proof

Since \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). From the definition of \(w ( t ) \)

$$ x^{\beta } \bigl( \xi ( t ) \bigr) = \frac{1}{p ( t ) } \bigl[ w ( t ) -x ( t ) \bigr] \leq \frac{w ( t ) }{p ( t ) }, $$

which implies that

$$ x \bigl( \xi ^{-1} ( t ) \bigr) \leq \frac{w^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) }{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) }. $$

Using the above inequality in the definition of \(w ( t )\), we get

$$\begin{aligned} x^{\beta } ( t ) &= \frac{1}{p ( \xi ^{-1} ( t ) ) } \bigl[ w \bigl( \xi ^{-1} ( t ) \bigr) -x \bigl( \xi ^{-1} ( t ) \bigr) \bigr] \\ & \geq \frac{1}{p ( \xi ^{-1} ( t ) ) } \biggl[ w \bigl( \xi ^{-1} ( t ) \bigr) - \frac{w^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) }{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) } \biggr] . \end{aligned}$$
(3.2)

Since \(\xi ( t ) \leq t\) \(,\xi ( t ) \) is strictly increasing and \(t\leq \xi ^{-1} ( t )\), then

$$ \xi ^{-1} ( t ) \leq \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) . $$
(3.3)

Since \(w ( t ) \) satisfies case 1 of Lemma 5, so from Lemma 2, we obtain

$$ \frac{w ( t ) }{t^{n-1}/ ( n-1 ) !}\geq \frac{w^{\prime } ( t ) }{t^{n-2}/ ( n-2 ) !}, $$

i.e.,

$$ w ( t ) \geq \frac{tw^{\prime } ( t ) }{n-1}, $$

which implies

$$ \bigl( t^{-n+1}w ( t ) \bigr) ^{\prime }=t^{-n+1}w^{ \prime } ( t ) + ( 1-n ) t^{-n}w ( t ) =t^{-n} \bigl[ tw^{\prime } ( t ) + ( 1-n ) w ( t ) \bigr] \leq 0. $$

The function \(t^{-n+1}w ( t ) \) is nonincreasing, which with (3.3) leads to

$$ \bigl[ \xi ^{-1} ( t ) \bigr] ^{1-n}w \bigl( \xi ^{-1} ( t ) \bigr) \geq \bigl[ \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr] ^{1-n}w \bigl( \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr). $$

Thus, using (3.2), we obtain

$$ x^{\beta } ( t ) \geq \frac{w ( \xi ^{-1} ( t ) ) }{p ( \xi ^{-1} ( t ) ) } \biggl[ 1- \biggl[ \frac{\xi ^{-1} ( t ) }{\xi ^{-1} ( \xi ^{-1} ( t ) ) } \biggr] ^{\frac{1-n}{\beta }} \frac{w^{\frac{1}{\beta }-1} ( \xi ^{-1} ( t ) ) }{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) } \biggr] . $$

Hence since \(w ( t ) \) is positive and increasing, there exists a positive constant \(k_{1}\) such that \(w ( t ) \geq k_{1}\). Thus, we have

$$ x^{\beta } ( t ) \geq P_{1} ( t ) w \bigl( \xi ^{-1} ( t ) \bigr) . $$

Substituting into (1.1), we have

$$ \bigl( r \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }\leq -\sum_{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) w^{ \frac{\delta }{\beta }} \bigl( \xi ^{-1} \bigl( \eta _{i} ( t ) \bigr) \bigr) , $$

and since \(\xi ^{-1} ( t ) \), w \(( t ) \) are increasing functions, then we have

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }\leq -\sum_{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) w^{\frac{\delta }{\beta }} \bigl( h ( t ) \bigr). $$

 □

Lemma 7

Assume that \(x ( t ) \) is an eventually positive solution of (1.1) with w satisfying case 2 of Lemma 5, then for all constants \(k_{2}>0\),

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }\leq -\sum_{i=1}^{m}q_{i} ( t ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) w^{\frac{\delta }{\beta }} \bigl( h ( t ) \bigr) , $$
(3.4)

where \(P_{2} ( t ) = \frac{1}{p ( \xi ^{-1} ( t ) ) } [ 1- [ \frac{\xi ^{-1} ( t ) }{\xi ^{-1} ( \xi ^{-1} ( t ) ) } ] ^{\frac{3-n}{\beta }} \frac{k_{2}^{\frac{1}{\beta }-1}}{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) } ] \geq 0\).

Proof

Since \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). Suppose that \(w ( t ) \) satisfies case 2 of Lemma 5, then by Lemma 2, we obtain

$$ \frac{w ( t ) }{t^{n-3}/ ( n-3 ) !}\geq \frac{w^{\prime } ( t ) }{t^{n-4}/ ( n-4 ) !}, $$

i.e., \(w ( t ) \geq \frac{tw^{\prime } ( t ) }{n-3}\), which implies that

$$ \bigl( t^{-n+3}w ( t ) \bigr) ^{\prime }=t^{-n+3}w^{ \prime } ( t ) + ( 3-n ) t^{-n+2}w ( t ) =t^{-n+2} \bigl[ tw^{\prime } ( t ) + ( 3-n ) w ( t ) \bigr] \leq 0, $$

and hence the function \(t^{3-n}w ( t ) \) is nonincreasing and by (3.3)

$$ \bigl[ \xi ^{-1} ( t ) \bigr] ^{3-n}w \bigl( \xi ^{-1} ( t ) \bigr) \geq \bigl[ \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr] ^{3-n}w \bigl( \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr) . $$

Going through as in the proof of Lemma 6, we obtain (3.4). □

Lemma 8

Suppose that \(x ( t ) \) is an eventually positive solution of (1.1) with w satisfying case 3 of Lemma 5, then for all constants \(k_{3}>0\),

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }\leq -\sum_{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) w^{\frac{\delta }{\beta }} \bigl( h ( t ) \bigr) , $$
(3.5)

where \(P_{3} ( t ) = \frac{1}{p ( \xi ^{-1} ( t ) ) } [ 1- [ \frac{\xi ^{-1} ( t ) }{\xi ^{-1} ( \xi ^{-1} ( t ) ) } ] ^{\frac{2-n}{\beta }} \frac{k_{3}^{\frac{1}{\beta }-1}}{p^{\frac{1}{\beta }} ( \xi ^{-1} ( \xi ^{-1} ( t ) ) ) } ] \geq 0\).

Proof

Since \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). Suppose that \(w ( t ) \) satisfies case 3 of Lemma 5, then by Lemma 2, we obtain

$$ \frac{w ( t ) }{t^{n-2}/ ( n-2 ) !}\geq \frac{w^{\prime } ( t ) }{t^{n-3}/ ( n-3 ) !}, $$

i.e., \(w ( t ) \geq \frac{t}{n-2}w^{\prime } ( t ) \), which implies that

$$ \bigl( t^{-n+2}w ( t ) \bigr) ^{\prime }=t^{-n+2}w^{ \prime } ( t ) + ( 2-n ) t^{-n+1}w ( t ) =t^{-n+1} \bigl[ tw^{\prime } ( t ) + ( 2-n ) w ( t ) \bigr] \leq 0, $$

and hence the function \(t^{2-n}w ( t ) \) is nonincreasing and by (3.3)

$$ \bigl[ \xi ^{-1} ( t ) \bigr] ^{2-n}w \bigl( \xi ^{-1} ( t ) \bigr) \geq \bigl[ \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr] ^{2-n}w \bigl( \xi ^{-1} \bigl( \xi ^{-1} ( t ) \bigr) \bigr) . $$

Going through as in the proof of Lemma 6, we obtain (3.5). □

Theorem 9

Assume that there exist \(\upsilon ( \tau ) \in C^{1} ( [\tau _{0},\infty ), ( 0,\infty ) ) \), \(c_{1}>0,k\) \(\in ( 0,1 ) \) such that

$$ \int _{t_{0}}^{\infty } \Biggl[ \upsilon ( s ) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-1 ) \delta /\beta }\sum_{i=1}^{m}q_{i} ( s ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{L [ \upsilon ^{\prime } ( s ) ] ^{2}r ( s ) }{s^{ ( n-1 ) \delta /\beta -1}\upsilon ( s ) } \Biggr] \,ds=\infty , $$
(3.6)

\(L= [ \frac{ ( n-1 ) !}{k} ] ^{\delta /\beta } \frac{\beta }{4\delta ( n-1 ) c_{1}^{\delta /\beta -\gamma }}\). Assume further that the equation

$$ z^{\prime } ( t ) +\sum_{i=1}^{m}q_{i} ( t ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \biggl[ \frac{kh^{n-1} ( t ) }{ ( n-1 ) ! [ r ( h ( t ) ) ] ^{1/\gamma }} \biggr] ^{\frac{\delta }{\beta }}z^{ \delta /\beta \gamma } \bigl( h ( t ) \bigr) =0 $$
(3.7)

is oscillatory if

$$\begin{aligned}& \int _{t_{0}}^{\infty } \Biggl[ E^{\gamma } ( s ) c_{2}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{Nh^{n-2} ( s ) }{ ( n-2 ) !} \biggr] ^{\delta /\beta }\sum_{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{\gamma ^{\gamma +1}}{ ( \gamma +1 ) ^{\gamma +1}E ( s ) r^{1/\gamma } ( s ) } \Biggr] \,ds \\& \quad =\infty , \end{aligned}$$
(3.8)

for some constants \(c_{2}>0,N\) \(\in ( 0,1 )\), and

$$ \int _{t}^{\infty }E ( s ) \,ds=\infty . $$
(3.9)

Then, every solution of (1.1) is oscillatory.

Proof

Without loss of generality, we may assume that \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). Then, as in Lemma 5, there exist four possible cases. Suppose that \(w ( t ) \) satisfies case 1, then, as in Lemma 6, the function \(t^{1-n}w ( t ) \) is nonincreasing and since \(h ( t ) < t\), we have

$$ \bigl[ h ( t ) \bigr] ^{1-n}w \bigl( h ( t ) \bigr) \geq t^{1-n}w ( t ) . $$
(3.10)

Define

$$ \omega ( t ) =\upsilon ( t ) \frac{r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma }}{w^{\delta /\beta } ( t ) }, $$

then \(\omega ( t ) >0 \), and

$$ \omega ^{\prime } ( t ) = \frac{\upsilon ^{\prime } ( t ) }{\upsilon ( t ) }\omega ( t ) +\upsilon ( t ) \frac{ [ r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma } ] ^{\prime }}{w^{\delta /\beta } ( t ) }-\upsilon ( t ) \frac{\delta /\beta r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma }w^{\prime } ( t ) w^{\delta /\beta -1} ( t ) }{w^{2\delta /\beta } ( t ) }. $$

Using (3.1), (3.10), we have

$$\begin{aligned} \omega ^{\prime } ( t ) \leq & \frac{\upsilon ^{\prime } ( t ) }{\upsilon ( t ) }\omega ( t ) - \upsilon ( t ) \biggl[ \frac{h ( t ) }{t} \biggr] ^{ ( n-1 ) \delta /\beta }\sum _{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \\ &{}-\upsilon ( t ) \frac{\delta /\beta r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma }w^{\prime } ( t ) w^{\delta /\beta -1} ( t ) }{w^{2\delta /\beta } ( t ) }. \end{aligned}$$

Now from Lemma 1, we have

$$ w ( t ) \geq \frac{k}{ ( n-1 ) !}t^{n-1}w^{{ ( n-1 ) }} ( t ) ,\quad k\in ( 0,1 ), $$
(3.11)

and \(w^{\prime } ( t ) \geq \frac{k}{ ( n-2 ) !}t^{n-2}w^{{ ( n-1 ) }} ( t ) \), then we have

$$\begin{aligned} \omega ^{\prime } ( t ) \leq & \frac{\upsilon ^{\prime } ( t ) }{\upsilon ( t ) }\omega ( t ) - \upsilon ( t ) \biggl[ \frac{h ( t ) }{t} \biggr] ^{ ( n-1 ) \delta /\beta }\sum _{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \\ &{} - \biggl[ \frac{k}{ ( n-1 ) !} \biggr] ^{\delta /\beta } \frac{\delta /\beta ( n-1 ) r ( t ) \upsilon ( t ) t^{ ( n-1 ) \delta /\beta -1} ( w^{{ ( n-1 ) }} ( t ) ) ^{2\gamma } ( w^{{ ( n-1 ) }} ( t ) ) ^{\delta /\beta -\gamma }}{w^{2\delta /\beta } ( t ) }. \end{aligned}$$

Since \(w^{{ ( n-1 ) }} ( t ) \) is positive and nonincreasing function, there exists a positive constant \(c_{1}\) such that \(w^{{ ( n-1 ) }} ( t ) \leq c_{1}\). Then,

$$\begin{aligned} \omega ^{\prime } ( t ) \leq & \frac{\upsilon ^{\prime } ( t ) }{\upsilon ( t ) }\omega ( t ) - \upsilon ( t ) \biggl[ \frac{h ( t ) }{t} \biggr] ^{ ( n-1 ) \delta /\beta }\sum _{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \\ &{}- \biggl[ \frac{k}{ ( n-1 ) !} \biggr] ^{\delta /\beta } \frac{ ( n-1 ) \delta /\beta c_{1}^{\delta /\beta -\gamma }t^{ ( n-1 ) \delta /\beta -1}\omega ^{2} ( t ) }{\upsilon ( t ) r ( t ) }. \end{aligned}$$

By completing the squares

$$ \omega ^{\prime } ( t ) \leq -\upsilon ( t ) \biggl[ \frac{h ( t ) }{t} \biggr] ^{ ( n-1 ) \delta /\beta } \sum_{i=1}^{m}q_{i} ( t ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) +L \frac{ [ \upsilon ^{\prime } ( t ) ] ^{2}r ( t ) }{t^{ ( n-1 ) \delta /\beta -1}\upsilon ( t ) }. $$

Integrating from \(t_{2}\) to t, we get

$$ 0< \omega ( t ) \leq \omega ( t_{2} ) - \int _{t_{2}}^{t} \Biggl( \upsilon ( s ) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-1 ) \delta /\beta }\sum_{i=1}^{m}q_{i} ( s ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) -L \frac{ [ \upsilon ^{\prime } ( s ) ] ^{2}r ( s ) }{s^{ ( n-1 ) \delta /\beta -1}\upsilon ( s ) } \Biggr) \,ds. $$

This is a contradiction with (3.6). Assume that we have case 2, using Lemma 1, we find (3.11). Thus, using (3.4), we have

$$ \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime }+ \biggl[ \frac{k}{ ( n-1 ) !}h^{n-1} ( t ) w^{{ ( n-1 ) }} \bigl( h ( t ) \bigr) \biggr] ^{\frac{\delta }{\beta }}\sum _{i=1}^{m}q_{i} ( t ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \leq 0, $$

then

$$\begin{aligned}& \bigl( r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma } \bigr) ^{\prime } \\& \quad{} + \biggl[ \frac{k}{ ( n-1 ) ! [ r ( h ( t ) ) ] ^{\frac{1}{\gamma }}}h^{n-1} ( t ) \biggr] ^{\frac{\delta }{\beta }}\sum_{i=1}^{m}q_{i} ( t ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \bigl[ \bigl[ r \bigl( h ( t ) \bigr) \bigr] ^{\frac{1}{\gamma }}w^{{ ( n-1 ) }} \bigl( h ( t ) \bigr) \bigr] ^{\frac{\delta }{\beta }}\leq 0, \end{aligned}$$

we see that \(z ( t ) =r ( t ) ( w^{{ ( n-1 ) }} ( t ) ) ^{\gamma }\) is a positive solution of the differential inequality

$$ z^{\prime } ( t ) +\sum_{i=1}^{m}q_{i} ( t ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) \biggl[ \frac{kh^{n-1} ( t ) }{ ( n-1 ) ! [ r ( h ( t ) ) ] ^{1/\gamma }} \biggr] ^{\frac{\delta }{\beta }}z^{ \delta /\beta \gamma } \bigl( h ( t ) \bigr) \leq 0, $$

using [[17], Corollary 1], we see that (3.7) has a positive solution, and this is a contradiction. Assume that we have case 3, then, as in Lemma 8, we have that \(t^{2-n}w ( t ) \) is nonincreasing, and since \(h ( t ) < t\), we have \([ h ( t ) ] ^{2-n}w ( h ( t ) ) >t^{2-n}w ( t ) \). Now, we define

$$ \phi ( t ) = \frac{r ( t ) [ w^{ ( n-1 ) } ( t ) ] ^{\gamma }}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}, $$

then \(\phi ( t ) <0\), and by using (3.5),

$$ \phi ^{\prime } ( t ) \leq - \frac{\sum_{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} ( \eta _{i} ( t ) ) w^{\frac{\delta }{\beta }} ( h ( t ) ) }{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}- \gamma \frac{\phi ^{\frac{\gamma +1}{\gamma }} ( t ) }{r^{\frac{1}{\gamma }} ( t ) }. $$

On the other hand, using Lemma 1, we get

$$ w ( t ) \geq \frac{N}{ ( n-2 ) !}t^{n-2}w^{ ( n-2 ) } ( t ) $$
(3.12)

for every \(N\in ( 0,1 ) \), and all sufficiently large t, then

$$ \phi ^{\prime } ( t ) \leq - \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( t ) \biggr] ^{\frac{\delta }{\beta }} \frac{\sum_{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} ( \eta _{i} ( t ) ) [ w^{ ( n-2 ) } ( h ( t ) ) ] ^{\frac{\delta }{\beta }}}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}-\gamma \frac{\phi ^{\frac{\gamma +1}{\gamma }} ( t ) }{r^{\frac{1}{\gamma }} ( t ) }, $$

since \(w^{ ( n-2 ) }\) is positive and decreasing then \(w^{ ( n-2 ) } ( h ( t ) ) \geq w^{ ( n-2 ) } ( t ) \). There exists a positive constant \(c_{2}\) such that \(w^{ ( n-2 ) } ( t ) \leq c_{2}\), then we have

$$ \phi ^{\prime } ( t ) \leq -c_{2}^{\frac{\delta }{\beta }- \gamma } \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( t ) \biggr] ^{\frac{\delta }{\beta }}\sum _{i=1}^{m}q_{i} ( t ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) - \gamma \frac{\phi ^{\frac{\gamma +1}{\gamma }} ( t ) }{r^{\frac{1}{\gamma }} ( t ) }. $$
(3.13)

Multiplying (3.13) by \(E^{\gamma } ( t ) \) and integrating it from \(t_{1}\) to t, we have

$$\begin{aligned}& E^{\gamma } ( t ) \phi ( t ) -E^{\gamma } ( t_{1} ) \phi ( t_{1} ) -\gamma \int _{t_{1}}^{t}E^{ \gamma -1} ( s ) \phi ( s ) E^{\prime } ( s ) \,ds\\& \qquad{} + \int _{t_{1}}^{t}E^{\gamma } ( s ) c_{2}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}\sum_{i=1}^{m}q_{i} ( s ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \,ds + \gamma \int _{t_{1}}^{t}E^{\gamma } ( s ) \frac{\phi ^{\frac{\gamma +1}{\gamma }} ( s ) }{r^{\frac{1}{\gamma }} ( s ) }\,ds \\& \quad \leq 0. \end{aligned}$$

Now using the inequality

$$ Av^{\frac{\gamma +1}{\gamma }}-Bv\geq - \frac{\gamma ^{\gamma }}{ ( \gamma +1 ) ^{\gamma +1}}\frac{B^{\gamma +1}}{A^{\gamma }},\quad A,B>0 $$

with

$$ A= \frac{E^{\gamma } ( s ) }{r^{\frac{1}{\gamma }} ( s ) },\qquad B=r^{-\frac{1}{\gamma }} ( s ) E^{\gamma -1} ( s ) \quad \text{and}\quad v=-\phi ( s ) , $$

we have

$$\begin{aligned}& \int _{t_{1}}^{t}E^{\gamma } ( s ) c_{2}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}\sum_{i=1}^{m}q_{i} ( s ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{\gamma ^{\gamma }}{ ( \gamma +1 ) ^{\gamma +1}E ( s ) r^{\frac{1}{\gamma }} ( s ) }\,ds \\& \quad \leq -E^{\gamma } ( t ) \phi ( t ) +E^{\gamma } ( t_{1} ) \phi ( t_{1} ) . \end{aligned}$$
(3.14)

Noting that \(r [ w^{ ( n-1 ) } ] ^{\gamma }\) is nonincreasing, we have

$$ r^{\frac{1}{\gamma }} ( s ) w^{ ( n-1 ) } ( s ) \leq r^{\frac{1}{\gamma }} ( t ) w^{ ( n-1 ) } ( t ) ,\quad s\geq t. $$

Dividing by \(r^{\frac{1}{\gamma }} ( s ) \) and integrating from t to ϰ gives

$$ w^{ ( n-2 ) } ( \varkappa ) \leq w^{ ( n-2 ) } ( t ) +r^{\frac{1}{\gamma }} ( t ) w^{ ( n-1 ) } ( t ) \int _{t}^{\varkappa } \frac{1}{r^{\frac{1}{\gamma }} ( s ) }\,ds. $$

Letting \(\varkappa \rightarrow \infty \), we have

$$ w^{ ( n-2 ) } ( t ) \geq -E ( t ) r^{ \frac{1}{\gamma }} ( t ) w^{ ( n-1 ) } ( t ) , $$
(3.15)

then

$$ - \frac{E ( t ) r^{\frac{1}{\gamma }} ( t ) w^{ ( n-1 ) } ( t ) }{w^{ ( n-2 ) } ( t ) } \leq 1, $$

i.e.,

$$ -E ( t ) \phi ^{\frac{1}{\gamma }} ( t ) \leq 1. $$

By substituting into (3.14), we have

$$\begin{aligned}& \int _{t_{1}}^{t}E^{\gamma } ( s ) c_{2}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}\sum_{i=1}^{m}q_{i} ( s ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{\gamma ^{\gamma }}{ ( \gamma +1 ) ^{\gamma +1}E ( s ) r^{\frac{1}{\gamma }} ( s ) }\,ds \\& \quad \leq E^{\gamma } ( t_{1} ) \phi ( t_{1} ) +1, \end{aligned}$$

which contradicts (3.8). Assume that we have case 4. Since \(r [ w^{ ( n-1 ) } ] ^{\gamma }\) is nonincreasing, as in the proof of case 3, we get (3.15). Hence, there exists a constant \(c_{3}>0\), such that

$$ w^{ ( n-2 ) } ( t ) \geq c_{3}E ( t ) . $$

Integrating from \(t_{1}\) to t provides

$$ -w^{ ( n-3 ) } ( t_{1} ) \geq c_{3} \int _{t_{1}}^{t}E ( \varrho ) \,d\varrho , $$

which contradicts (3.9), and this completes the proof. □

Theorem 10

Assume that (3.6), (3.7), and (3.9) hold. If there exist \(d\in C^{1} ( [ t_{0},\infty ] ,R^{+} ) \), \(c_{4}\) is a positive constant such that

$$\begin{aligned}& \lim \sup_{t\rightarrow \infty } \frac{E^{\gamma } ( t ) }{d ( t ) } \int _{t_{0}}^{t} \Biggl[d ( s ) \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{ \frac{\delta }{\beta }}c_{4}^{\frac{\delta }{\beta }-\gamma } \sum_{i=1}^{m}q_{i} ( s ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \\& \quad{} - \frac{r ( s ) ( d^{\prime } ( s ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( s ) } \Biggr] \,ds> 1, \end{aligned}$$
(3.16)

then every solution of (1.1) oscillates.

Proof

Without loss of generality, we may assume that \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). Then, as in Lemma 5, there exist four possible cases. The proofs in the three cases 1, 2, and 4 are the same as in Theorem 9. Now assume that case 3 holds. Since \(r [ w^{ ( n-1 ) } ] ^{\gamma }\) is nonincreasing, as in Theorem 9, we have (3.15). Define

$$ \varphi ( t ) =d ( t ) \biggl( \frac{r ( t ) [ w^{ ( n-1 ) } ( t ) ] ^{\gamma }}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}+ \frac{1}{E^{\gamma } ( t ) } \biggr) . $$
(3.17)

From (3.15), \(\varphi ( t ) >0\) for \(\ t\geq t_{1}\). Therefore, we have

$$ \varphi ^{\prime } ( t ) = \frac{d^{\prime } ( t ) }{d ( t ) }\varphi ( t ) +d ( t ) \frac{ [ r ( t ) [ w^{ ( n-1 ) } ( t ) ] ^{\gamma } ] ^{\prime }}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}-d ( t ) \frac{\gamma r ( t ) [ w^{ ( n-1 ) } ( t ) ] ^{\gamma +1}}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma +1}}- \frac{\gamma d ( t ) E^{\prime } ( t ) }{E^{\gamma +1} ( t ) }, $$

by (3.5) and (3.17)

$$\begin{aligned} \varphi ^{\prime } ( t ) & \leq \frac{d^{\prime } ( t ) }{d ( t ) }\varphi ( t ) -d ( t ) \frac{\sum_{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} ( \eta _{i} ( t ) ) w^{\frac{\delta }{\beta }} ( h ( t ) ) }{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}- \frac{\gamma }{d^{\frac{1}{\gamma }} ( t ) r^{\frac{1}{\gamma }} ( t ) } \biggl[ \varphi ( t ) - \frac{d ( t ) }{E^{\gamma } ( t ) } \biggr] ^{\frac{{\gamma +1}}{\gamma }} \\ &\quad{} - \frac{\gamma d ( t ) E^{\prime } ( t ) }{E^{\gamma +1} ( t ) }. \end{aligned}$$

As in Theorem 2.2 of [16], using Lemma 4 with \(A=\frac{d^{\prime } ( t ) }{d ( t ) }\), \(B= \frac{\gamma }{d^{\frac{1}{\gamma }} ( t ) r^{\frac{1}{\gamma }} ( t ) }\), \(D= \frac{d ( t ) }{E^{\gamma } ( t ) }\) and \(v=\varphi \), we have

$$ \varphi ^{\prime } ( t ) \leq -d ( t ) \frac{\sum_{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} ( \eta _{i} ( t ) ) w^{\frac{\delta }{\beta }} ( h ( t ) ) }{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}+ \biggl[ \frac{d ( t ) }{E^{\gamma } ( t ) } \biggr] ^{\prime }+ \frac{r ( t ) ( d^{\prime } ( t ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( t ) }. $$

Now using Lemma 1, we have (3.12). Since \(w^{ ( n-2 ) } ( t ) \) is positive and decreasing \(w^{ ( n-2 ) } ( h ( t ) ) \geq w^{ ( n-2 ) } ( t ) \), and there exists a positive constant \(c_{4}\) such that \(w^{ ( n-2 ) } ( t ) \leq c_{4}\), we have

$$ \varphi ^{\prime } ( t ) \leq -d ( t ) c_{4}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( t ) \biggr] ^{ \frac{\delta }{\beta }}\sum _{i=1}^{m}q_{i} ( t ) P_{3}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( t ) \bigr) + \biggl[ \frac{d ( t ) }{E^{\gamma } ( t ) } \biggr] ^{\prime }+ \frac{r ( t ) ( d^{\prime } ( t ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( t ) }. $$

Integrating the above inequality from \(t_{1}\) to t, we find

$$\begin{aligned}& \varphi ( t ) -\varphi ( t_{1} ) \\& \quad \leq - \int _{t_{1}}^{t} \Biggl( d ( s ) \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}c_{4}^{ \frac{\delta }{\beta }-\gamma } \sum_{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{r ( s ) ( d^{\prime } ( s ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( s ) } \Biggr) \,ds \\& \qquad{} +\frac{d ( t ) }{E^{\gamma } ( t ) }- \frac{d ( t_{1} ) }{E^{\gamma } ( t_{1} ) }. \end{aligned}$$

From the definition of \(\varphi ( t ) \), we see that

$$\begin{aligned}& \int _{t_{1}}^{t} \Biggl( d ( s ) \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}c_{4}^{ \frac{\delta }{\beta }-\gamma } \sum_{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{r ( s ) ( d^{\prime } ( s ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}} d^{\gamma } ( s ) } \Biggr) \,ds \\& \quad \leq - \frac{d ( t ) r ( t ) [ w^{ ( n-1 ) } ( t ) ] ^{\gamma }}{ [ w^{ ( n-2 ) } ( t ) ] ^{\gamma }}+ \frac{d ( t_{1} ) r ( t_{1} ) [ w^{ ( n-1 ) } ( t_{1} ) ] ^{\gamma }}{ [ w^{ ( n-2 ) } ( t_{1} ) ] ^{\gamma }}. \end{aligned}$$

This leads to

$$ \int _{t_{1}}^{t}d ( s ) \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}c_{4}^{ \frac{\delta }{\beta }-\gamma }\sum _{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{r ( s ) ( d^{\prime } ( s ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( s ) }\,ds \leq \frac{d ( t ) }{E^{\gamma } ( t ) }. $$

Hence

$$ \frac{E^{\gamma } ( t ) }{d ( t ) } \int _{t_{1}}^{t}d ( s ) \biggl[ \frac{N}{ ( n-2 ) !}h^{n-2} ( s ) \biggr] ^{\frac{\delta }{\beta }}c_{4}^{ \frac{\delta }{\beta }-\gamma }\sum _{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{r ( s ) ( d^{\prime } ( s ) ) ^{{\gamma +1}}}{ ( \gamma +1 ) ^{{\gamma +1}}d^{\gamma } ( s ) }\,ds \leq 1, $$

which contradicts (3.16), and this completes the proof. □

Theorem 11

Assume that (3.6), (3.8), and (3.9) hold, \(\frac{\delta }{\beta }=\gamma \). If there exists \(\vartheta ( t ) \in C^{1} ( [ t_{0},\infty ] ,R^{+} ) \) such that

$$\begin{aligned}& \int _{t_{0}}^{\infty } \Biggl[ \frac{\vartheta ( s ) }{ ( n-4 ) !} \int _{s}^{\infty } ( s-v ) ^{n-4} \Biggl( \frac{1}{r ( v ) } \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( \chi ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( \chi ) \bigr) \biggl[ \frac{h ( \chi ) }{\chi } \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}d\chi \Biggr) ^{\frac{1}{\gamma }}\,dv \\& \quad{} - \frac{ ( \vartheta ^{\prime } ( s ) ) ^{2}}{4\vartheta ( s ) } \Biggr] \,ds=\infty , \end{aligned}$$
(3.18)

then (1.1) is oscillatory.

Proof

Without loss of generality, we may assume that \(x ( t ) \) is an eventually positive solution of (1.1) such that \(x ( t ) >0\), \(x ( \xi ( t ) ) >0\), \(x ( \eta _{i} ( t ) ) >0\), \(i=1,2,\ldots,m\), for \(t\geq t_{1}\geq t_{0}\). Then, as in Lemma 5, there exist four possible cases. The proofs in the three cases 1, 3, and 4 are the same as in Theorem 9. Now assume that case 2 holds. Define

$$ \varpi ( t ) =\vartheta ( t ) \frac{w^{\prime } ( t ) }{w ( t ) }, $$

then \(\varpi >0\),

$$ \varpi ^{\prime } ( t ) =\vartheta ( t ) \frac{w^{\prime \prime } ( t ) }{w ( t ) }- \frac{\varpi ^{2} ( t ) }{\vartheta ( t ) }+ \frac{\vartheta ^{\prime } ( t ) \varpi ( t ) }{\vartheta ( t ) }. $$
(3.19)

Now by integrating (3.4) from t to v

$$ r ( v ) \bigl( w^{{ ( n-1 ) }} ( v ) \bigr) ^{\gamma }-r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma }\leq - \int _{t}^{v} \sum_{i=1}^{m}q_{i} ( s ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) w^{\frac{\delta }{\beta }} \bigl( h ( s ) \bigr) \,ds. $$
(3.20)

As in the proof of Lemma 7, \(t^{3-n}\) \(w ( t ) \) is nonincreasing since \(h ( t ) < t\), then

$$ w \bigl( h ( t ) \bigr) \geq \biggl[ \frac{h ( t ) }{t} \biggr] ^{n-3}w ( t ) . $$

By substituting in (3.20) and letting \(v\rightarrow \infty \), we have

$$ r ( t ) \bigl( w^{{ ( n-1 ) }} ( t ) \bigr) ^{\gamma }\geq w^{\frac{\delta }{\beta }} ( t ) \int _{t}^{\infty }\sum_{i=1}^{m}q_{i} ( s ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}\,ds. $$

Integrating again from t to ∞, we get

$$ w^{{ ( n-2 ) }} ( t ) \leq -w^{ \frac{\delta }{\gamma \beta }} ( t ) \int _{t}^{\infty } \Biggl( \frac{1}{r ( v ) }\int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( s ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}\,ds \Biggr) ^{\frac{1}{\gamma }} \,dv. $$
(3.21)

Similarly, integrating (3.21) from t to ∞ a total \(( n-4 ) \) times, we find

$$ w^{{\prime \prime }} ( t ) \leq - \frac{w^{\frac{\delta }{\gamma \beta }} ( t ) }{ ( n-4 ) !} \int _{t}^{\infty } ( t-v ) ^{n-4} \Biggl( \frac{1}{r ( v ) } \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( s ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}\,ds \Biggr) ^{ \frac{1}{\gamma }} \,dv. $$

Thus, by substituting in (3.19), we have

$$\begin{aligned} \varpi ^{\prime } ( t ) & \leq -\vartheta ( t ) \frac{w^{\frac{\delta }{\gamma \beta }} ( t ) }{ ( n-4 ) !w ( t ) } \\ &\quad{}\times \int _{t}^{\infty } ( t-v ) ^{n-4} \Biggl( \frac{1}{r ( v ) } \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( s ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}\,ds \Biggr) ^{\frac{1}{\gamma }}\,dv \\ &\quad{} - \frac{\varpi ^{2} ( t ) }{\vartheta ( t ) } + \frac{\vartheta ^{\prime } ( t ) \varpi ( t ) }{\vartheta ( t ) }. \end{aligned}$$

By completing the squares, we have

$$\begin{aligned} \varpi ^{\prime } ( t ) \leq &-\vartheta ( t ) \frac{w^{\frac{\delta }{\gamma \beta }} ( t ) }{ ( n-4 ) !w (t ) } \\ &{}\times \int _{t}^{\infty } ( t-v ) ^{n-4} \Biggl( \frac{1}{r (v ) } \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( s ) P_{2}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}\,ds \Biggr) ^{\frac{1}{\gamma }} \,dv \\ &{}+ \frac{ ( \vartheta ^{\prime } ( t ) ) ^{2}}{4\vartheta ( t ) }. \end{aligned}$$

This yields

$$\begin{aligned}& \int _{t_{1}}^{t} \Biggl[ \frac{\vartheta ( s ) }{ ( n-4 )!} \int _{s}^{\infty } ( s-v ) ^{n-4} \Biggl( \frac{1}{r ( v )} \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( \chi ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( \chi ) \bigr) \biggl[ \frac{h ( \chi ) }{\chi } \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}d\chi \Biggr) ^{\frac{1}{\gamma }}\,dv \\& \qquad{} - \frac{ ( \vartheta ^{\prime } ( s ) ) ^{2}}{4\vartheta ( s ) } \Biggr] \,ds \\& \quad \leq \varpi ( t_{1} ) , \end{aligned}$$

which contradicts (3.18), and this completes the proof. □

4 Examples and conclusion

Example 12

Consider the differential equation

$$ \biggl( t^{2} \biggl( x ( t ) +tx^{3} \biggl( \frac{3t}{4} \biggr) \biggr) ^{\prime \prime \prime } \biggr) ^{\prime }+ \frac{q_{0}}{t}x^{3} ( t ) +\frac{q_{0}}{t^{2}}x^{3} ( 2t ) =0, \quad q_{0}>0, t\geq 1. $$
(4.1)

Here \(\delta =\beta =3\), \(\gamma =1\), \(n=4\), \(r ( t ) =t^{2}\), \(p ( t ) =t\), \(\xi ( t ) =\frac{3t}{4}\), \(q_{1} ( t ) =\frac{q_{0}}{t}\), \(q_{2} ( t ) =\frac{q_{0}}{t^{2}}\), \(\eta _{1} ( t ) =t\), \(\eta _{2} ( t ) =2t\), \(\xi ^{-1} ( t ) =\frac{4}{3}t\), \(\xi ^{-1} ( \xi ^{-1} ( t ) ) = \frac{16}{9}t\). Taking \(\ \eta ( t ) =\frac{t}{2}\), then \(\eta ( t ) <\eta _{i} ( t ) \), \(\lim_{t \rightarrow \infty }\eta ( t ) = \infty \), \(\eta ( t ) <\xi ( t ) \), \(h ( t ) =\frac{2}{3}t\),

$$\begin{aligned}& P_{1} ( t ) =\frac{3}{4t} \biggl[ 1- \frac{4}{3k_{1}^{\frac{2}{3}}{}t^{\frac{1}{3}}} \sqrt[3]{\frac{9}{16}} \biggr] ,\qquad P_{2} ( t ) =\frac{3}{4t} \biggl[ 1- \biggl( \sqrt[3]{\frac{3}{4}} \biggr) \frac{1}{k_{2}^{\frac{2}{3}}{}t^{\frac{1}{3}}} \biggr] ,\qquad \\& P_{3} ( t ) =\frac{3}{4t} \biggl[1-\frac{1}{k_{3}^{\frac{2}{3}}{}t^{\frac{1}{3}}} \biggr] . \end{aligned}$$

Now let \(\upsilon ( t ) =t\), \(\vartheta ( t ) =t\), then

$$\begin{aligned}& \int _{t_{0}}^{\infty } \Biggl[ \upsilon ( s ) \biggl[ \frac{h ( s ) }{s} \biggr] ^{ ( n-1 ) \delta /\beta }\sum_{i=1}^{m}q_{i} ( s ) P_{1}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{L [ \upsilon ^{\prime } ( s ) ] ^{2}r ( s ) }{s^{ ( n-1 ) \delta /\beta -1}\upsilon ( s ) } \Biggr] \,ds\\& \quad= \int _{1}^{\infty } \biggl( s \biggl( \frac{2}{3} \biggr) ^{3} \biggl[ \frac{3q_{0}}{4s^{2}} \biggl( 1-\frac{4}{3k_{1}^{\frac{2}{3}}{}s^{\frac{1}{3}}} \sqrt[3]{\frac{9}{16}} \biggr) +\frac{3q_{0}}{8s^{3}} \biggl( 1- \frac{4}{3k_{1}^{\frac{2}{3}}{}s^{\frac{1}{3}}}\sqrt[3]{\frac{9}{32}} \biggr) \biggr] - \frac{L}{s} \biggr) \,ds=\infty . \end{aligned}$$

If \(q_{0}>\frac{9L}{2}\), \(L= [ \frac{ ( n-1 ) !}{k} ] ^{ \delta /\beta } \frac{\beta }{4\delta ( n-1 ) c_{1}^{\delta /\beta -\gamma }}= \frac{1}{2k}\), i.e., \(q_{0}>\frac{9}{4k}\), \(k\in ( 0,1 ) \)

$$\begin{aligned}& \int _{t_{0}}^{\infty } \Biggl[ E^{\gamma } ( s ) c_{2}^{ \frac{\delta }{\beta }-\gamma } \biggl[ \frac{Nh^{n-2} ( s ) }{ ( n-2 ) !} \biggr] ^{\delta /\beta }\sum_{i=1}^{m}q_{i} ( s ) P_{3}^{ \frac{\delta }{\beta }} \bigl( \eta _{i} ( s ) \bigr) - \frac{\gamma ^{\gamma +1}}{ ( \gamma +1 ) ^{\gamma +1}E ( s ) r^{1/\gamma } ( s ) } \Biggr] \,ds\\& \quad= \int _{t_{0}}^{\infty } \biggl[ \frac{2}{9}Ns \biggl[ \frac{3q_{0}}{4s^{2}} \biggl( 1- \frac{1}{k_{3}^{\frac{2}{3}}{}s^{\frac{1}{3}}} \biggr) + \frac{3q_{0}}{8s^{3}} \biggl( 1-\frac{1}{k_{3}^{\frac{2}{3}}{}s^{\frac{1}{3}}} \sqrt[3]{\frac{1}{2}} \biggr) \biggr] -\frac{1}{4s} \biggr] \,ds=\infty . \end{aligned}$$

If \(q_{0}>\frac{3}{2N}\),

$$ \int _{t}^{\infty }E ( s ) \,ds= \int _{t}^{\infty } \frac{1}{s}\,ds=\infty . $$

Moreover,

$$\begin{aligned}& \int _{t_{0}}^{\infty } \Biggl[ \frac{\vartheta ( s ) }{ ( n-4 ) !} \int _{s}^{\infty } ( s-v ) ^{n-4} \Biggl( \frac{1}{r ( v ) } \int _{v}^{\infty }\sum_{i=1}^{m}q_{i} ( \chi ) P_{2}^{\frac{\delta }{\beta }} \bigl( \eta _{i} ( \chi ) \bigr) \biggl[ \frac{h ( \chi ) }{\chi } \biggr] ^{ ( n-3 ) \frac{\delta }{\beta }}d\chi \Biggr) ^{\frac{1}{\gamma }}\,dv \\& \qquad{} - \frac{ ( \vartheta ^{\prime } ( s ) ) ^{2}}{4\vartheta ( s ) } \Biggr] \,ds\\& \quad= \int _{1}^{\infty } \biggl( q_{0}s \biggl[ \frac{1}{4s^{2}}- \frac{9}{56}\frac{1}{k_{2}^{\frac{2}{3}}{}s^{\frac{7}{3}}}\sqrt[3]{ \frac{3}{4}}+ \frac{1}{24s^{3}}-\frac{9}{280} \frac{1}{k_{2}^{\frac{2}{3}}{}s^{\frac{10}{3}}} \sqrt[3]{\frac{3}{8}} \biggr] -\frac{1}{4s} \biggr) \,ds=\infty . \end{aligned}$$

If \(q_{0}>1\), then by Theorem 11, every solution of (4.1)oscillates for \(q_{0}>\max [ 1,\frac{3}{2N},\frac{9}{4k} ] \), \(N,k\in ( 0,1 ) \).

Conclusion 13

In this work, we use techniques of the Riccati and generalized Riccati transformations, integral averaging, and the method of comparison to establish some new oscillation criteria for the even-order differential equation with superlinear neutral term (1.1) in noncanonical case. The obtained results improve and complement some previous criteria in the literature. An example is provided to support the theoretical findings.

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El-Gaber, A.A. Oscillatory criteria of noncanonical even-order differential equations with a superlinear neutral term. Bound Value Probl 2024, 67 (2024). https://doi.org/10.1186/s13661-024-01873-z

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