1 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, 3–9, 14–16, 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 [7–9, 14–16, 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.

2 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 \(q_{0}>16.988\).