Skip to main content

Oscillatory behavior of solutions of odd-order nonlinear delay differential equations

Abstract

The objective of this study is to establish new sufficient criteria for oscillation of solutions of odd-order nonlinear delay differential equations. Based on creating comparison theorems that compare the odd-order equation with a couple of first-order equations, we improve and complement a number of related ones in the literature. To show the importance of our results, we provide an example.

Introduction

In this study, we investigate the oscillatory behavior of solutions of the odd-order delay differential equation (DDE)

$$ \bigl( r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) x^{ \alpha } \bigl( \sigma ( t ) \bigr) =0, $$
(1.1)

where \(t\geq t_{0}\), \(n\in \mathbb{Z} ^{+}\) is odd, α is a ratio of odd positive integers, \(r\in C^{1} ( [ t_{0},\infty ), ( 0,\infty ) ) \), \(r^{\prime } ( t ) \geq 0\), \(\mu _{0,0} ( t,t_{0} ):=\int _{t_{0}}^{t}r^{-1/\alpha } ( s ) \,\mathrm{d}s\rightarrow \infty \) as \(t\rightarrow \infty \), \(q\in C ( [ t_{0},\infty ), [ 0,\infty ) ) \), \(\sigma \in C ( [ t_{0},\infty ), \mathbb{R} ) \), \(\sigma ( t ) < t\), and \(\lim_{t\rightarrow \infty }\sigma ( t ) =\infty \).

Definition 1

Let \(x\in C^{ ( n-1 ) } ( [ t_{x},\infty ) ), t_{x}\geq t_{0}\), and \(r ( x^{ ( n-1 ) } ) ^{\alpha }\in C^{1} ( [ t_{x},\infty ) ) \). The function x is called a solution of (1.1) on \([ t_{x},\infty ) \) if x satisfies (1.1) for all t in \([ t_{x},\infty ) \).

Definition 2

A nontrivial solution x of (1.1) is said to be oscillatory if there exists a sequence of zeros \(\{ t_{n} \} _{n=0}^{\infty }\) (i.e., \(x ( t_{n} ) =0\)) of x such that \(\lim_{n\rightarrow \infty }t_{n}=\infty \); otherwise, it is said to be nonoscillatory.

Although differential equations of even-order have been studied extensively, the study of qualitative behavior of odd-order differential equations has received considerably less attention in the literature, especially the third-order DDEs. However, certain results for third-order equations have been known for a long time and have some applications in mathematical modeling in biology and physics, see [17, 23, 25]. As a matter of fact, equation (1.1) under study is a so-called odd-order half-linear DDE, which has numerous applications in the research area of porous medium, see [13].

Different techniques have been used in studying the asymptotic behavior of DDEs. The articles [1, 39, 1416, 27] were concerned with (in the canonical case and noncanonical case) the oscillation and asymptotic behavior of equation (1.1) and its special cases.

Based on creating comparison theorems that compare the odd-order DDEs with one or a couple of first-order DDEs, Agarwal et al. [1], Baculikova and Dzurina [3, 4] and Chatzarakis et al. [8] studied the oscillatory and asymptotic behavior of special cases of the third-order DDE

$$ \bigl( a ( t ) \bigl( \bigl( b ( t ) x^{ \prime } ( t ) \bigr) ^{\prime } \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) f \bigl( x \bigl( \sigma ( t ) \bigr) \bigr) =0, $$

where \(a,b\in C^{1} ( [ t_{0},\infty ), ( 0,\infty ) ) \). By using the integral averaging technique, Bohner et al. [6] and Moaaz et al. [20] studied the asymptotic behavior of DDE with damping

$$ \bigl( a ( t ) \bigl( b ( t ) \bigl( x^{ \prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime } \bigr) ^{\prime }+p ( t ) \bigl( x^{\prime } ( t ) \bigr) ^{\alpha }+q ( t ) f \bigl( x \bigl( \sigma ( t ) \bigr) \bigr) =0, $$

where \(\alpha \geq 1\) and \(p\in C ( [ t_{0},\infty ), [ 0,\infty ) ) \). On the other hand, [5] used the Riccati transformation to study the asymptotic properties of the odd-order advanced equation

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

where \(g ( t ) >t\). The results concerned with the asymptotic properties and oscillation of the higher-order neutral DDEs were presented in [11, 18, 19, 21, 22, 26].

In this paper, by using an iterative method, we create sharper estimates for increasing and decreasing positive solutions of (1.1). Thus, we create sharper criteria for oscillation of (1.1). Moreover, iterative technique allows us to test the oscillation, even when the related results fail to apply. The results reported in this paper generalize, complement, and improve those in [79, 1416, 27]. To show the importance of our results, we provide an example.

Remark 1.1

We restrict our discussion to those solutions x of (1.1) which satisfy \(\sup \{ \vert x ( t ) \vert : t\geq T \} >0\) for every \(T\in [ t_{0},\infty ) \).

Remark 1.2

All functional inequalities and properties, such as increasing, decreasing, positive, and so on, are assumed to hold eventually, that is, they are satisfied for all t large enough.

Main results

Lemma 2.1

([2, Lemma 2.2.3])

Let\(F\in C^{n} ( [ t_{0},\infty ), ( 0,\infty ) ), F^{ ( n-1 ) } ( t ) F^{ ( n ) } ( t ) \leq 0\)for\(t\geq t_{F} \), and\(\lim_{t\rightarrow \infty }F ( t ) \neq 0\). Then, for every\(\delta \in ( 0,1 )\), there exists\(t_{\delta }\in [ t_{F},\infty ) \)such that

$$ F ( t ) \geq \frac{\delta }{ ( n-1 ) !}t^{n-1} \bigl\vert F^{ ( n-1 ) } ( t ) \bigr\vert \quad\textit{for all }t\in [ t_{\delta },\infty ). $$

Lemma 2.2

([5, Lemma 2])

Ifxis a positive solution of (1.1), then all derivatives\(x^{ ( k ) } ( t ), 1\leq k\leq n-1 \), are of constant signs, \(r ( t ) ( x^{ ( n-1 ) } ( t ) ) ^{\alpha }\)is nonincreasing, andxsatisfies either

$$ x^{\prime } ( t ) >0,\qquad x^{\prime \prime } ( t ) >0, \qquad x^{ ( n-1 ) } ( t ) >0,\qquad x^{ ( n ) } ( t ) < 0 $$
(2.1)

or

$$ ( -1 ) ^{m}x^{ ( m ) }>0,\quad m=1,2,\ldots,n. $$
(2.2)

Definition 3

The set of all positive solutions of (1.1) with property (2.1) or (2.2) is denoted by \(X_{I}^{+}\) or \(X_{D}^{+}\), respectively.

Lemma 2.3

Assume that\(x\in X_{I}^{+}\). Then

$$ x \bigl( \sigma ( t ) \bigr) \geq \eta _{k} \bigl( \sigma ( t ) \bigr) x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr), $$
(2.3)

where

$$ \eta _{0} ( t ):= \frac{\delta _{0}}{ ( n-1 ) !}t^{n-1}, $$

and

$$ \eta _{k+1} ( t ):= \frac{\delta _{k}}{ ( n-2 ) !}r^{1/\alpha } ( t ) \int _{t_{1}}^{t}s^{n-2} \biggl( \frac{1}{r ( s ) }\exp \biggl( \int _{s}^{t}\frac{1}{r ( u ) }q ( u ) \eta _{k}^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u \biggr) \biggr) ^{1/\alpha }\,\mathrm{d}s $$

for all\(\delta _{k}\in ( 0,1 ) \)and\(k=0,1,\ldots \) .

Proof

Let \(x\in X_{I}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). Next, we will prove (2.3) using induction. For \(k=0\), using Lemma 2.1, we see that

$$ x \bigl( \sigma ( t ) \bigr) \geq \frac{\delta _{0}}{ ( n-1 ) !}\sigma ^{n-1} ( t ) x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr) \geq \eta _{0} \bigl( \sigma ( t ) \bigr) x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr). $$

Now, we assume that \(x ( \sigma ( t ) ) \geq \eta _{k} ( \sigma ( t ) ) x^{ ( n-1 ) } ( \sigma ( t ) ) \) for \(k>0\). Since \(x^{ ( n ) }<0\) and \(\sigma ( t ) < t\), we have that

$$ x \bigl( \sigma ( t ) \bigr) \geq \eta _{k} \bigl( \sigma ( t ) \bigr) x^{ ( n-1 ) } ( t ). $$
(2.4)

Then, from (1.1) and (2.4), we get

$$ \bigl( r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) \eta _{k}^{\alpha } \bigl( \sigma ( t ) \bigr) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha }\leq 0. $$
(2.5)

If we set \(w:=r ( t ) ( x^{ ( n-1 ) } ( t ) ) ^{\alpha }\), then (2.5) becomes

$$ w^{\prime } ( t ) \leq -\frac{1}{r ( t ) }q ( t ) \eta _{k}^{\alpha } \bigl( \sigma ( t ) \bigr) w ( t ). $$

Applying the Grönwall inequality, we find

$$ w ( s ) \geq w ( t ) \exp \biggl( \int _{s}^{t} \frac{1}{r ( u ) }q ( u ) \eta _{k}^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u \biggr) $$

or

$$ x^{ ( n-1 ) } ( s ) \geq r^{1/\alpha } ( t ) x^{ ( n-1 ) } ( t ) \biggl( \frac{1}{r ( s ) }\exp \biggl( \int _{s}^{t} \frac{1}{r ( u ) }q ( u ) \eta _{k}^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u \biggr) \biggr) ^{1/\alpha }. $$
(2.6)

Using Lemma 2.1 with \(F:=x^{\prime }>0\), we see that

$$ x^{\prime } ( t ) \geq \frac{\delta _{k}t^{n-2}}{ ( n-2 ) !}x^{ ( n-1 ) } ( t ) \quad\text{for all }\delta _{k} \in ( 0,1 ). $$

By integrating this inequality from \(t_{1}\) to t and taking into account (2.6), we see that

$$\begin{aligned} x ( t ) &\geq \frac{\delta _{k}}{ ( n-2 ) !}\int _{t_{1}}^{t}s^{n-2}x^{ ( n-1 ) } ( s ) \,\mathrm{d}s \\ &\geq x^{ ( n-1 ) } ( t ) \frac{\delta _{k}}{ ( n-2 ) !}r^{1/\alpha } ( t ) \int _{t_{1}}^{t}s^{n-2} \biggl( \frac{1}{r ( s ) }\exp \biggl( \int _{s}^{t} \frac{1}{r ( u ) }q ( u ) \eta _{k}^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u \biggr) \biggr) ^{1/\alpha }\,\mathrm{d}s \\ &\geq \eta _{k+1} ( t ) x^{ ( n-1 ) } ( t ). \end{aligned}$$

Therefore, we have that

$$ x \bigl( \sigma ( t ) \bigr) \geq \eta _{k+1} \bigl( \sigma ( t ) \bigr) x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr). $$

The proof is complete. □

Lemma 2.4

Assume that\(x\in X_{D}^{+}\). Then

$$ x ( u ) \geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{l,n-2} ( v,u ), $$
(2.7)

where

$$ \mu _{l,k+1} ( v,u ):= \int _{u}^{v}\mu _{l,k} ( v,s ) \, \mathrm{d}s $$

and

$$ \mu _{l+1,0} ( v,u ):= \int _{u}^{v} \frac{1}{r^{1/\alpha } ( s ) }\exp \biggl( \int _{s}^{v}q ( u ) \mu _{l,n-2}^{ \alpha } \bigl( u,\sigma ( u ) \bigr) \,\mathrm{d}u \biggr) ^{1/\alpha }\, \mathrm{d}s $$

for\(k=0,1,\ldots,n-3\), and\(l=0,1,2,\ldots \) .

Proof

Let \(x\in X_{D}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). Next, we will prove (2.7) using induction. For \(l=0\), since \(( r(z^{ ( n-1 ) }) ) ^{\prime }\leq 0\), we get that

$$\begin{aligned} -x^{ ( n-2 ) } ( u ) &\geq x^{ ( n-2 ) } ( v ) -x^{ ( n-2 ) } ( u ) = \int _{u}^{v}\frac{1}{r^{1/\alpha } ( s ) }r^{1/\alpha } ( s ) x^{ ( n-1 ) } ( s ) \,\mathrm{d}s \\ &\geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{0,0} ( v,u ). \end{aligned}$$
(2.8)

Integrating (2.8) from u to v, we have

$$ -x^{ ( n-3 ) } ( u ) \leq x^{ ( n-3 ) } ( v ) -x^{ ( n-3 ) } ( u ) =r^{1/ \alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{0,1} ( v,u ). $$
(2.9)

Integrating (2.9) \(n-3\) times from u to v, we get

$$ x ( u ) \geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{0,n-2} ( v,u ). $$

Now, we assume that \(x ( u ) \geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{l,n-2} ( v,u ) \) for \(l>0\). Thus, we find

$$ x \bigl( \sigma ( t ) \bigr) \geq r^{1/\alpha } ( t ) x^{ ( n-1 ) } ( t ) \mu _{l,n-2} \bigl( t,\sigma ( t ) \bigr), $$

which, with (1.1), gives

$$ \bigl( r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \mu _{l,n-2}^{\alpha } \bigl( t,\sigma ( t ) \bigr) \leq 0. $$
(2.10)

If we set \(\psi:=r ( t ) ( x^{ ( n-1 ) } ( t ) ) ^{\alpha }\), then (2.10) becomes

$$ \psi ^{\prime } ( t ) \leq -q ( t ) \mu _{l,n-2}^{ \alpha } \bigl( t,\sigma ( t ) \bigr) \psi ( t ). $$

Applying the Grönwall inequality, we find

$$ \psi ( s ) \geq \psi ( v ) \exp \biggl( \int _{s}^{v}q ( u ) \mu _{l,n-2}^{\alpha } \bigl( u,\sigma ( u ) \bigr) \,\mathrm{d}u \biggr) $$

or

$$ x^{ ( n-1 ) } ( s ) \geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \biggl( \frac{1}{r ( s ) }\exp \biggl( \int _{s}^{v}q ( u ) \mu _{l,n-2}^{\alpha } \bigl( u,\sigma ( u ) \bigr) \,\mathrm{d}u \biggr) \biggr) ^{1/\alpha }. $$

Thus, from (2.8), we see that

$$\begin{aligned} -x^{ ( n-2 ) } ( u ) &\geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \int _{u}^{v} \frac{1}{r^{1/\alpha } ( s ) }\exp \biggl( \int _{s}^{v}q ( u ) \mu _{l,n-2}^{\alpha } \bigl( u,\sigma ( u ) \bigr) \,\mathrm{d}u \biggr) ^{1/\alpha }\,\mathrm{d}s \\ &\geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{l+1,0} ( v,u ). \end{aligned}$$

Integrating this inequality \(n-2\) times from u to v, we get

$$ x ( u ) \geq r^{1/\alpha } ( v ) x^{ ( n-1 ) } ( v ) \mu _{l+1,n-2} ( v,u ). $$

Thus, the proof is complete. □

Theorem 2.1

Assume thatxis a positive solution of (1.1) and\(\eta _{k}\)is defined as in Lemma 2.3. If the delay differential equation

$$ w^{\prime } ( t ) + \frac{1}{r ( \sigma ( t ) ) }q ( t ) \eta _{k}^{\alpha } \bigl( \sigma ( t ) \bigr) w \bigl( \sigma ( t ) \bigr) =0 $$
(2.11)

is oscillatory for some\(\delta _{k}\in ( 0,1 ) \)and some\(k\in \mathbb{N} \), then\(X_{I}^{+}\)is empty.

Proof

Assume to the contrary that \(x\in X_{I}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). From Lemma 2.3, we have that (2.3) holds. Combining (1.1) and (2.3), we obtain

$$ \bigl( r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) \eta _{k}^{\alpha } \bigl( \sigma ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr) \bigr) ^{\alpha } \leq 0. $$
(2.12)

If we set \(w:=r ( x^{ ( n-1 ) } ) ^{\alpha }\), then (2.12) becomes

$$ w^{\prime } ( t ) + \frac{1}{r ( \sigma ( t ) ) }q ( t ) \eta _{k}^{\alpha } \bigl( \sigma ( t ) \bigr) w \bigl( \sigma ( t ) \bigr) \leq 0. $$

In view of [24, Theorem 1], we have that (2.11) also has a positive solution, a contradiction. Thus, the proof is complete. □

Corollary 2.1

Assume thatxis a positive solution of (1.1) and\(\eta _{k}\)is defined as in Lemma 2.3. If

$$ \underset{t\rightarrow \infty }{\lim \inf } \int _{\sigma ( t ) }^{t}\frac{1}{r ( \sigma ( u ) ) }q ( u ) \eta _{k}^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u> \frac{1}{\mathrm{e}} $$
(2.13)

for some\(\delta _{k}\in ( 0,1 ) \)and some\(k\in \mathbb{N} \), then\(X_{I}^{+}\)is empty.

Proof

In view of [12, Theorem 2], condition (2.13) guarantees that the delay equation (2.11) is oscillatory. □

Theorem 2.2

Assume thatxis a positive solution of (1.1), \(\sigma ^{\prime } ( t ) >0\), and\(\mu _{l,k}\)is defined as in Lemma 2.4. If

$$ \underset{t\rightarrow \infty }{\lim \sup } \int _{\sigma ( t ) }^{t}q ( u ) \mu _{l,n-2}^{\alpha } \bigl( \sigma ( t ),\sigma ( u ) \bigr) \,\mathrm{d}u>1 $$
(2.14)

for some\(l\in \mathbb{N} \), then\(X_{D}^{+}\)is empty.

Proof

Assume to the contrary that \(x\in X_{D}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). From Lemma 2.4, we have that (2.7) holds. Integrating (1.1) from \(\sigma ( t ) \) to t, we obtain

$$ r \bigl( \sigma ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr) \bigr) ^{ \alpha }-r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha }= \int _{\sigma ( t ) }^{t}q ( u ) x^{\alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u, $$

and so

$$ r \bigl( \sigma ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr) \bigr) ^{ \alpha }\geq \int _{\sigma ( t ) }^{t}q ( u ) x^{ \alpha } \bigl( \sigma ( u ) \bigr) \,\mathrm{d}u. $$
(2.15)

Using (2.7) with \(u=\sigma ( u ) \) and \(v=\sigma ( t ) \), we get that

$$ x \bigl( \sigma ( u ) \bigr) \geq r^{1/\alpha } \bigl( \sigma ( t ) \bigr) x^{ ( n-1 ) } \bigl( \sigma ( t ) \bigr) \mu _{l,n-2} \bigl( \sigma ( t ),\sigma ( u ) \bigr), $$

with (2.15), gives

$$ \int _{\sigma ( t ) }^{t}q ( u ) \mu _{l,n-2}^{ \alpha } \bigl( \sigma ( t ),\sigma ( u ) \bigr) \,\mathrm{d}u\leq 1, $$

which contradicts condition (2.14). This completes the proof. □

Theorem 2.3

Assume thatxis a positive solution of (1.1) and\(\mu _{l,k}\)is defined as in Lemma 2.4. If there exists a function\(\theta \in C ( [ t_{0},\infty ), ( 0,\infty ) ) \)satisfying\(\theta ( t ) < t\)and\(\sigma ( t ) <\theta ( t ) \)such that the delay differential equation

$$ \varphi ^{\prime } ( t ) +q ( t ) \mu _{l,n-2}^{ \alpha } \bigl( \theta ( t ),\sigma ( t ) \bigr) \varphi \bigl( \theta ( t ) \bigr) =0 $$
(2.16)

is oscillatory for some\(l\in \mathbb{N} \), then\(X_{D}^{+}\)is empty.

Proof

Assume to the contrary that \(x\in X_{D}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). From Lemma 2.4, we have that (2.7) holds. Using (2.7) with \(u=\sigma ( t ) \) and \(v=\theta ( t ) \), we get that

$$ x \bigl( \sigma ( t ) \bigr) \geq r^{1/\alpha } \bigl( \theta ( t ) \bigr) x^{ ( n-1 ) } \bigl( \theta ( t ) \bigr) \mu _{l,n-2} \bigl( \theta ( t ),\sigma ( t ) \bigr). $$

Thus, from (1.1), we obtain

$$ \bigl( r ( t ) \bigl( x^{ ( n-1 ) } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }+q ( t ) \mu _{l,n-2}^{\alpha } \bigl( \theta ( t ), \sigma ( t ) \bigr) r \bigl( \theta ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \theta ( t ) \bigr) \bigr) ^{\alpha }\leq 0. $$
(2.17)

If we set \(\varphi:=r ( x^{ ( n-1 ) } ) ^{\alpha }\), then (2.17) becomes

$$ \varphi ^{\prime } ( t ) +q ( t ) \mu _{l,n-2}^{ \alpha } \bigl( \theta ( t ),\sigma ( t ) \bigr) \varphi \bigl( \theta ( t ) \bigr) \leq 0. $$

In view of [24, Theorem 1], we have that (2.16) also has a positive solution, a contradiction. Thus, the proof is complete. □

Theorem 2.4

Assume thatxis a positive solution of (1.1), \(( \sigma ^{-1} ( t ) ) ^{\prime }>0\)and\(\mu _{l,k}\)is defined as in Lemma 2.4. If there exists a function\(\vartheta \in C ( [ t_{0},\infty ), ( 0, \infty ) ) \)satisfying\(\vartheta ( t ) >t\)and\(\sigma ( \vartheta ( t ) ) < t\)such that the delay differential equation

$$ \varphi ^{\prime } ( t ) + \bigl( \sigma ^{-1} ( t ) \bigr) ^{\prime }q \bigl( \sigma ^{-1} ( t ) \bigr) \mu _{l,n-2}^{\alpha } \bigl( \vartheta ( t ),t \bigr) \varphi \bigl( \sigma \bigl( \vartheta ( t ) \bigr) \bigr) =0 $$
(2.18)

is oscillatory for some\(l\in \mathbb{N} \), then\(X_{D}^{+}\)is empty.

Proof

Assume to the contrary that \(x\in X_{D}^{+}\). Then there exists \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\) and \(x ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). From Lemma 2.4, we have that (2.7) holds. From (1.1), we get

$$ \bigl( r \bigl( \sigma ^{-1} ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \sigma ^{-1} ( t ) \bigr) \bigr) ^{\alpha } \bigr) ^{\prime }+ \bigl( \sigma ^{-1} ( t ) \bigr) ^{\prime }q \bigl( \sigma ^{-1} ( t ) \bigr) x^{\alpha } ( t ) =0. $$
(2.19)

Using (2.7) with \(u=t\) and \(v=\vartheta ( t ) \), we have

$$ x ( t ) \geq r^{1/\alpha } \bigl( \vartheta ( t ) \bigr) x^{ ( n-1 ) } \bigl( \vartheta ( t ) \bigr) \mu _{l,n-2} \bigl( \vartheta ( t ),t \bigr), $$

which with (2.19) gives

$$\begin{aligned} 0 \geq {}& \bigl( r \bigl( \sigma ^{-1} ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \sigma ^{-1} ( t ) \bigr) \bigr) ^{\alpha } \bigr) ^{\prime } \\ &{}+ \bigl( \sigma ^{-1} ( t ) \bigr) ^{\prime }q \bigl( \sigma ^{-1} ( t ) \bigr) \mu _{l,n-2}^{\alpha } \bigl( \vartheta ( t ),t \bigr) r \bigl( \vartheta ( t ) \bigr) \bigl( x^{ ( n-1 ) } \bigl( \vartheta ( t ) \bigr) \bigr) ^{\alpha }. \end{aligned}$$
(2.20)

If we set \(\varphi ( t ):=r ( x^{ ( n-1 ) } ) ^{\alpha } ( \sigma ^{-1} ( t ) ) \), then (2.20) becomes

$$ \varphi ^{\prime } ( t ) + \bigl( \sigma ^{-1} ( t ) \bigr) ^{\prime }q \bigl( \sigma ^{-1} ( t ) \bigr) \mu _{l,n-2}^{\alpha } \bigl( \vartheta ( t ),t \bigr) \varphi \bigl( \sigma \bigl( \vartheta ( t ) \bigr) \bigr) \leq 0. $$

In view of [24, Theorem 1], we have that (2.18) also has a positive solution, a contradiction. Thus, the proof is complete. □

Applying a well-known criterion [12, Theorem 2] for delay equations (2.16) and (2.18) to be oscillatory, we obtain the following two corollaries.

Corollary 2.2

Assume thatxis a positive solution of (1.1) and\(\mu _{l,k}\)is defined as in Lemma 2.4. If there exists a function\(\theta \in C ( [ t_{0},\infty ), ( 0,\infty ) ) \)satisfying\(\theta ( t ) < t\)and\(\sigma ( t ) <\theta ( t ) \)such that

$$ \underset{t\rightarrow \infty }{\lim \inf } \int _{\theta ( t ) }^{t}q ( u ) \mu _{l,n-2}^{\alpha } \bigl( \theta ( u ),\sigma ( u ) \bigr) \,\mathrm{d}u>\frac{1}{\mathrm{e}} $$
(2.21)

for some\(l\in \mathbb{N} \), then\(X_{D}^{+}\)is empty.

Corollary 2.3

Assume thatxis a positive solution of (1.1), \(( \sigma ^{-1} ( t ) ) ^{\prime }>0\)and\(\mu _{l,k}\)is defined as in Lemma 2.4. If there exists a function\(\vartheta \in C ( [ t_{0},\infty ), ( 0, \infty ) ) \)satisfying\(\vartheta ( t ) >t\)and\(\sigma ( \vartheta ( t ) ) < t\)such that

$$ \underset{t\rightarrow \infty }{\lim \inf } \int _{\sigma ( \vartheta ( t ) ) }^{t} \bigl( \sigma ^{-1} ( u ) \bigr) ^{\prime }q \bigl( \sigma ^{-1} ( u ) \bigr) \mu _{l,n-2}^{\alpha } \bigl( \vartheta ( u ),u \bigr) \,\mathrm{d}u> \frac{1}{\mathrm{e}} $$
(2.22)

for some\(l\in \mathbb{N} \), then\(X_{D}^{+}\)is empty.

Theorem 2.5

Assume that\(\eta _{k}\)and\(\mu _{l,k}\)are defined as in Lemmas2.3and2.4, respectively. Then every solution of (1.1) is oscillatory if one of the following conditions is satisfied for some\(\delta _{k}\in ( 0,1 ) \)and some\(k,l\in \mathbb{N} \):

  1. (a)

    There exists a function\(\theta \in C ( [ t_{0},\infty ), ( 0,\infty ) ) \)satisfying\(\theta ( t ) < t\)and\(\sigma ( t ) <\theta ( t ) \)such that the delay differential equations (2.11) and (2.16) are oscillatory;

  2. (b)

    There exists a function\(\vartheta \in C ( [ t_{0},\infty ), ( 0, \infty ) ) \)satisfying\(\vartheta ( t ) >t\), \(( \sigma ^{-1} ( t ) ) ^{\prime }>0\)and\(\sigma ( \vartheta ( t ) ) < t\)such that the delay differential equations (2.11) and (2.18) are oscillatory.

Corollary 2.4

Assume that\(\eta _{k}\)and\(\mu _{l,k}\)are defined as in Lemmas2.3and2.4, respectively. Then every solution of (1.1) is oscillatory if one of the following conditions is satisfied for some\(\delta _{k}\in ( 0,1 ) \)and some\(k,l\in \mathbb{N} \):

  1. (a)

    Conditions (2.13) and (2.14) hold;

  2. (b)

    Conditions (2.13) and (2.21) hold;

  3. (c)

    Conditions (2.13) and (2.22) hold.

Remark 2.1

The article [10] was concerned with the oscillation of equations (2.11), (2.16), and (2.18). Thus, one can obtain a number of oscillation criteria for (1.1) by using related results reported in [10].

Example 2.1

Consider the third-order differential equation

$$ x^{\prime \prime \prime }+\frac{q_{0}}{t^{3}}x ( \lambda t ) =0, $$
(2.23)

where \(t\geq 1\), \(q_{0}>0\), and \(\lambda \in ( 0,2/3 ) \). It is easy to verify that \(\eta _{0} ( t ):=\frac{\delta _{0}}{2}\lambda ^{2}t^{2}\), \(\mu _{0,0} ( v,u ) =v-u\), \(\mu _{0,1} ( v,u ) =\frac{1}{2} ( v-u ) ^{2}\),

$$ \mu _{1,0} ( v,u ) =q_{0} \frac{ ( 1-\lambda ) ^{2}}{2}v\ln \frac{v}{u} $$

and

$$ \mu _{1,1} ( v,u ) =q_{0} \frac{ ( 1-\lambda ) ^{2}}{2}v \biggl( v-u \biggl( 1+\ln \frac{v}{u} \biggr) \biggr). $$

Thus, by choosing \(k=0\), \(l=1\) and \(\theta ( t ):=\frac{3}{2}\lambda t\), conditions (2.13) and (2.21) reduce to

$$ q_{0}\lambda ^{2}\ln \frac{1}{\lambda }>\frac{2}{\mathrm{e}} $$
(2.24)

and

$$ q_{0}^{2}\frac{3}{4}\lambda ^{2} ( \lambda -1 ) ^{2} \biggl( \frac{1}{2}-\ln \frac{3}{2} \biggr) \ln \frac{2}{3\lambda }> \frac{1}{\mathrm{e}}, $$
(2.25)

respectively. Using Corollary 2.4(b), we see that every solution of (2.23) is oscillatory if (2.24) and (2.25) hold.

Remark 2.2

Apparently, Corollary 2.4(a) and Theorem 2 in [8] are the same for \(n=3\). Consider a particular case of (2.23), namely \(x^{\prime \prime \prime }+q_{0}t^{-3}x ( 0.5t ) =0\). By using the results in Example 2.1, this equation is oscillatory if \(q_{0}>16.988\).

References

  1. 1.

    Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of third-order nonlinear delay differential equations. Taiwan. J. Math. 17(2), 545–558 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

    Baculikova, B., Dzurina, J.: Oscillation of third-order functional differential equations. Electron. J. Qual. Theory Differ. Equ. 43, 1 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Baculikova, B., Dzurina, J.: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 24(4), 466–470 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Baculikova, B., Dzurina, J.: On the oscillation of odd order advanced differential equations. Bound. Value Probl. 2014, 214 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bohner, M., Grace, S.R., Sager, I., Tunc, E.: Oscillation of third-order nonlinear damped delay differential equations. Appl. Math. Comput. 278, 21–32 (2016)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Chatzarakis, G.E., Dzurina, J., Jadlovska, I.: Oscillatory and asymptotic properties of third-order quasilinear delay differential equations. J. Inequal. Appl. 2019, Article ID 23 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chatzarakis, G.E., Grace, S.R., Jadlovska, I.: Oscillation criteria for third-order delay differential equations. Adv. Differ. Equ. 2017, 330 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chatzarakis, G.E., Grace, S.R., Jadlovska, I., Li, T., Tunc, T.: Oscillation criteria for third-order Emden–Fowler differential equations with unbounded neutral coefficients. Complexity 2019, Article ID 5691758 (2019)

    Article  Google Scholar 

  10. 10.

    Chatzarakis, G.E., Li, T.: Oscillation criteria for delay and advanced differential equations with nonmonotone arguments. Complexity 2018, Article ID 8237634 (2018)

    Article  Google Scholar 

  11. 11.

    Dzurina, J., Grace, S.R., Jadlovska, I.: On nonexistence of Kneser solutions of third-order neutral delay differential equations. Appl. Math. Lett. 88, 193–200 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kitamura, Y., Kusano, T.: Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Am. Math. Soc. 78(1), 64–68 (1980)

    MathSciNet  Article  Google Scholar 

  13. 13.

    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 

  14. 14.

    Li, T., Rogovchenko, Y.V.: Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstr. Appl. Anal. 2014, Article ID 395368 (2014)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    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 

  16. 16.

    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 

  17. 17.

    McKean, H.P.: Nagumo’s equation. Adv. Math. 4(3), 209–223 (1970)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Moaaz, O., Baleanu, D., Muhib, A.: New aspects for non-existence of Kneser solutions of neutral differential equations with odd-order. Mathematics 8(4), 494 (2020)

    Article  Google Scholar 

  19. 19.

    Moaaz, O., Chalishajar, D., Bazighifan, O.: Asymptotic behavior of solutions of the third order nonlinear mixed type neutral differential equations. Mathematics 8(4), 485 (2020)

    Article  Google Scholar 

  20. 20.

    Moaaz, O., Elabbasy, E.M., Shaaban, E.: Oscillation criteria for a class of third order damped differential equations. Arab J. Math. Sci. 24(1), 16–30 (2018)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Moaaz, O., Muhib, A.: New oscillation criteria for nonlinear delay differential equations of fourth-order. Appl. Math. Comput. 377, 125192 (2020)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Moaaz, O., Qaraad, B., El-Nabulsi, R.A., Bazighifan, O.: New results for Kneser solutions of third-order nonlinear neutral differential equations. Mathematics 8(5), 686 (2020)

    Article  Google Scholar 

  23. 23.

    Padhi, S., Pati, S.: Theory of Third-Order Differential Equations. Springer, New Delhi (2014)

    Google Scholar 

  24. 24.

    Philos, C.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. (Basel) 36(2), 168–178 (1981)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Vreeke, S.A., Sandquist, G.M.: Phase space analysis of reactor kinetics. Nucl. Sci. Eng. 42(3), 295–305 (1970)

    Article  Google Scholar 

  26. 26.

    Xing, G., Li, T., Zhang, C.: Oscillation of higher-order quasi-linear neutral differential equations. Adv. Differ. Equ. 2011, 45 (2011)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Zhang, C., Li, T., Sun, B., Thandapani, E.: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 24(9), 1618–1621 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The author presents their sincere thanks to the editors.

Availability of data and materials

No data sharing (where no datasets are produced).

Funding

For this paper, no direct funding was received.

Author information

Affiliations

Authors

Contributions

All authors read and approved the final manuscript.

Corresponding author

Correspondence to Osama Moaaz.

Ethics declarations

Competing interests

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

Moaaz, O. Oscillatory behavior of solutions of odd-order nonlinear delay differential equations. Adv Differ Equ 2020, 357 (2020). https://doi.org/10.1186/s13662-020-02821-8

Download citation

Keywords

  • Odd-order
  • Delay differential equations
  • Oscillatory behavior