Theory and Modern Applications

# New oscillation theorems for a class of even-order neutral delay differential equations

## Abstract

In this work, we study the oscillatory behavior of even-order neutral delay differential equations $$\upsilon ^{n}(l)+b(l)u(\eta (l))=0$$, where $$l\geq l_{0}$$, $$n\geq 4$$ is an even integer and $$\upsilon =u+a ( u\circ \mu )$$. By deducing a new iterative relationship between the solution and the corresponding function, new oscillation criteria are established that improve those reported in (T. Li, Yu.V. Rogovchenko in Appl. Math. Lett. 61:35–41, 2016).

## Introduction

In this paper, we consider the even-order neutral differential equations

$$\upsilon ^{n}(t)+b(t)u\bigl(\eta (t)\bigr)=0,$$
(1.1)

where $$t\geq t_{0}>0$$, $$n\geq 4$$ is an even natural number and $$\upsilon :=u+a\cdot ( u\circ \mu )$$. Moreover, we suppose $$a,b,\eta ,\mu \in C([t_{0},\infty ),\mathbb{R})$$, $$0< a(t)\leq a_{0}$$, $$b(t)\geq 0$$, $$\mu (t)\leq t$$, $$b(t)$$ is not identically zero for large t, and $$\lim _{t\rightarrow \infty }\mu (t)=\lim_{t\rightarrow \infty }\eta (t)=\infty$$.

By a solution of Eq. (1.1), we mean a function $$u\in C ( [ t_{\ast },\infty ) ,\mathbb{R})$$, $$t_{\ast }\geq t_{0}$$, which has the property $$\upsilon \in C^{n} ( [ t_{0},\infty ) , \mathbb{R})$$, and $$u(t)$$ satisfies Eq. (1.1) on $$[t_{\ast },\infty )$$. We only focus on solutions of Eq. (1.1), which exist on $$[ t_{0},\infty )$$ and satisfy

$$\sup \bigl\{ \bigl\vert u(t) \bigr\vert :t_{u}\leq t \bigr\} >0 \quad \text{for every }t\geq t_{u}.$$

As is customary, a solution of Eq. (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative on $$[ t_{0},\infty )$$ and otherwise, it is termed nonoscillatory.

The importance of studying neutral delay differential equations comes from their emergence when modeling many phenomena in different applied sciences, see [2, 3]. The qualitative theory of various classes of neutral differential equations has become an important area of research due to the fact that such equations arise in a variety of real world problems such as in the study of non-Newtonian fluid theory and porous medium problems; see [4].

Very recently, a great development was found in the study of the oscillatory properties of solutions of second order neutral-delay differential equations; see for examples [517]. It would be interesting to extend this development to higher-order differential equations.

In 2016, Li and Rogovchenko [1] studied the oscillatory behavior of solutions of neutral delay equation (1.1). They used an approach similar to that used in [18], and established the relationship between u and υ to have the form

$$u ( t ) \geq \frac{1}{a ( \mu ^{-1} ( t ) ) } \biggl( \upsilon \bigl( \mu ^{-1} ( t ) \bigr) - \frac{\upsilon ( \mu ^{-1} ( \mu ^{-1} ( t ) ) ) }{a ( \mu ^{-1} ( \mu ^{-1} ( t ) ) ) } \biggr) .$$
(1.2)

By using the comparison with the first-order delay equations, they obtained improved criteria over the previous ones in the literature.

In this paper, by improving the relationship (1.2), we establish a new criterion that improves the results in [1]. An example is given to illustrate the importance of our results.

In order to discuss our main results, we need the following auxiliary lemmas.

### Lemma 1.1

([19])

Assume that $$\psi \in C^{n}([t_{0},\infty ),(0,\infty ))$$ and $$\psi ^{(n)}(t)\psi ^{(n-1)}(t)\leq 0$$ for $$t\geq t_{1}$$. If $$\lim_{t\rightarrow \infty }\psi (t)\neq 0$$, then there exists a $$t_{\lambda }\in {}[ t_{1},\infty )$$ such that

$$\psi (t)\geq \frac{\lambda }{ ( n-1 ) !}t^{n-1} \bigl\vert \psi ^{(n-1)}(t) \bigr\vert ,$$
(1.3)

for all $$t\in {}[ t_{\lambda },\infty )$$ and $$\lambda \in (0,1)$$.

### Lemma 1.2

([20])

Assume that the $$\psi \in C^{ ( k+1 ) }([t_{0},\infty ))$$ with $$\psi ^{(i)}(t)>0$$ for $$i=0,1,2,\ldots,k$$ and $$\psi ^{(k+1)}(t)\leq 0$$ for all $$t\geq t_{1}$$. Then there exists a $$t_{\lambda }\in {}[ t_{1},\infty )$$ such that

$$\frac{h(t)}{h^{\prime }(t)}\geq \frac{\lambda t}{k},$$

for all $$t\in {}[ t_{\lambda },\infty )$$ and $$\lambda \in (0,1)$$.

### Lemma 1.3

([21])

If $$\psi \in C^{n}([t_{0},\infty ),(0,\infty ))$$, $$\psi ^{(n)}(t)$$ is eventually of one sign for large t, then there exist a $$t_{u}\geq t_{0}$$ and an integer $$t\in [ 0,n ]$$ with $$( -1 ) ^{n+t}\psi ^{(n)}(t) \geq 0$$, such that $$t>0$$ yields

$$\psi ^{(k)}(t)>0, \quad t\geq t_{u},k=0,1,\ldots,t-1,$$

and

$$t\leq n-1\quad \textit{yields}\quad (-1)^{t+k}\psi ^{(k)}(t)>0, \quad t \geq t_{u}, k=t, t+1,\ldots,n-1.$$

## Main results

Through this section, we will be using the next notation: $$\mu ^{ [ -1 ] }:=\mu ^{-1}$$, $$\mu ^{- [ h+1 ] }:=\mu ^{-1}\circ \mu ^{- [ h ] }$$ for $$h=1,2,\ldots$$ ,

\begin{aligned}& \widetilde{a} ( t ) :=\sum_{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) } \biggl( 1- \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] }(t)}{\mu ^{- [ 2k-1 ] }(t)} \biggr) ^{(n-1)/\lambda _{0}} \biggr) , \\& \widehat{a} ( t ) :=\sum_{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) } \biggl( 1- \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] } ( t ) }{\mu ^{- [ 2k-1 ] } ( t ) } \biggr) ^{1/\lambda _{2}} \biggr) , \end{aligned}

and

$$B ( t ) =\min \bigl\{ b \bigl( \eta ^{-1} ( t ) \bigr) ,b \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \bigr\} ,$$

where n is an even positive integer and $$\lambda _{1},\lambda _{2}\in ( 0,1 )$$.

### Lemma 2.1

Assume that u is an eventually positive solution. Then, we have two cases for the derivatives of υ as

\begin{aligned}& ( 1 ) \quad \upsilon ( t ) >0, \qquad \upsilon ^{\prime }(t)>0,\qquad \upsilon ^{\prime \prime }(t)>0, \qquad \upsilon ^{(n-1)}(t)>0; \\& ( 2 ) \quad \upsilon ( t ) >0, \qquad ( -1 ) ^{k+1}\upsilon ^{(k)}(t)>0 \quad \textit{for all }k\in \{1,2,\ldots,n-3 \} . \end{aligned}

### Proof

From the definition of υ, we get that $$\upsilon ( t ) >0$$ for large t. From Eq. (1.1), $$\upsilon ^{(n)}(t)\leq 0$$. Based on the facts that n is even and $$\upsilon ^{n}(t)\leq 0$$, cases $$( 1 )$$ and $$( 2 )$$ are deduced directly from Lemma 1.3. □

### Theorem 2.1

Assume that $$\mu ^{\prime }(t)>0$$ and there exists an even integer m such that

$$\frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] }(t)}{\mu ^{- [ 2k-1 ] }(t)} \biggr) ^{(n-1)/\lambda _{0}}\leq 1,$$
(2.1)

for all $$k=1,2,\ldots,n/2$$. Suppose that there exist functions $$\chi \in C^{1}([t_{0},\infty ),\mathbb{R})$$ and $$\varkappa \in C^{1}([t_{0},\infty ),\mathbb{R})$$ satisfying

$$\varkappa ( t ) \leq \eta (t), \qquad \varkappa ( t ) < \mu (t), \qquad \lim _{t\rightarrow \infty }\varkappa (t)= \infty$$
(2.2)

and

$$\chi ( t ) \leq \eta (t), \qquad \chi ( t ) < \mu (t), \qquad \chi ^{\prime } ( t ) \geq 0, \qquad \lim_{t \rightarrow \infty }\chi (t)=\infty .$$
(2.3)

If there exist $$\lambda _{i}\in (0,1)$$, $$i=0,1,2$$, such that the first-order delay equations

$$G^{\prime } ( t ) + \frac{\lambda _{1}}{ ( n-1 ) !}b(t)\widetilde{a} \bigl( \eta ( t ) \bigr) \bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr) ^{n-1}G\bigl(\mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr)=0$$
(2.4)

and

$$\phi ^{\prime } ( t ) +\frac{\lambda _{2}}{(n-3)!}\mu ^{-1}( \chi ( t ) \phi ( \mu ^{-1}\bigl(\chi ( t ) \bigr) \int _{t}^{\infty } ( s-3 ) ^{n-3}b ( s ) \widehat{a} \bigl( \eta (s) \bigr) \,\mathrm{d}s=0$$
(2.5)

are oscillatory, every solution of Eq. (1.1) is oscillatory.

### Proof

Assume that Eq. (1.1) has an eventually positive solution u. It follows from (1.1) that $$\upsilon ^{n}(t)=-b(t)u(\eta (t))\leq 0$$. Thus, using Lemma 2.1, we see that there are two cases for the derivatives of υ for large t, either $$( 1 )$$ or $$( 2 )$$.

Assume that $$( 1 )$$ holds. Since υ is an increasing positive function, we obtain $$\lim_{t\rightarrow \infty }\upsilon (t)\neq 0$$. Therefore, by virtue of Lemma 1.1, we get

$$\upsilon (t)\geq \frac{\lambda }{ ( n-1 ) !}t^{{(n-1)}} \upsilon ^{(n-1)}(t),$$
(2.6)

for every $$\lambda \in (0,1)$$ and for all large t. It follows from the definition of $$\upsilon (t)$$ that

$$u(t)=\frac{1}{a(\mu ^{-1}(t))} \bigl( \upsilon \bigl(\mu ^{-1}(t)\bigr)-u \bigl(\mu ^{-1}(t)\bigr) \bigr)$$

and

$$u ( t ) =\frac{\upsilon (\mu ^{-1}(t)}{a(\mu ^{-1}(t))}- \frac{1}{a(\mu ^{-1}(t))} \biggl( \frac{\upsilon (\mu ^{-1}(\mu ^{-1}(t)))}{a(\mu ^{-1}(\mu ^{-1}(t)))}- \frac{u(\mu ^{-1}(\mu ^{-1}(t)))}{a(\mu ^{-1}(\mu ^{-1}(t)))} \biggr) .$$

If we repeat the previous procedure, then there exists an even positive integer n such that

\begin{aligned} u ( t ) =&\sum_{k=1}^{n} \frac{ ( -1 ) ^{k+1}}{\sum_{m=1}^{k}a ( \mu ^{- [ m ] } ( t ) ) }\upsilon \bigl(\mu ^{- [ k ] }(t)\bigr)+ \frac{1}{\sum_{m=1}^{n}a ( \tau ^{- [ m ] } ( t ) ) }u\bigl(\mu ^{- [ n ] }(t)\bigr) \\ \geq &\sum_{k=1}^{n} \frac{ ( -1 ) ^{k+1}}{\sum_{m=1}^{k}a ( \mu ^{- [ m ] } ( t ) ) } \upsilon \bigl(\mu ^{- [ k ] }(t)\bigr) \\ \geq &\sum_{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) } \biggl( \upsilon \bigl( \mu ^{- [ 2k-1 ] } ( t ) \bigr) - \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \upsilon \bigl( \mu ^{- [ 2k ] } ( t ) \bigr) \biggr) . \end{aligned}
(2.7)

Now, using Lemma 1.2, we obtain

$$\frac{\upsilon (t)}{\upsilon ^{\prime }(t)}\geq \frac{\lambda _{0}t}{n-1},$$

for all $$\lambda _{0}\in ( 0,1 )$$ and $$t\geq t_{1}$$, and so

\begin{aligned} \biggl( \frac{\upsilon (t)}{t^{(n-1)/\lambda _{0}}} \biggr) ^{\prime } =&\frac{\upsilon ^{\prime } ( t ) }{t^{(n-1)/\lambda _{0}}}- \frac{n-1}{\lambda _{0}} \frac{\upsilon ( t ) }{t^{(n-1)/\lambda _{0}+1}} \\ =&\frac{t\upsilon ^{\prime } ( t ) -\frac{n-1}{\lambda _{0}}\upsilon ( t ) }{t^{(n-1)/\lambda _{0}+1}}\leq 0. \end{aligned}
(2.8)

Taking into account that $$\mu ( t ) \leq t$$, we get $$\mu ^{- [ 2k-1 ] }(t)\leq \mu ^{- [ 2k ] }(t)$$. Thus, from (2.8), we find

$$\upsilon \bigl( \mu ^{- [ 2k ] }(t) \bigr) \leq \upsilon \bigl( \mu ^{- [ 2k-1 ] }(t) \bigr) \biggl( \frac{\mu ^{- [ 2k ] }(t)}{\mu ^{- [ 2k-1 ] }(t)} \biggr) ^{(n-1)/\lambda _{0}},$$

which with (2.7) gives

\begin{aligned}[b] u(t)&\geq \sum_{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) } \\ &\quad {}\times\biggl( 1- \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] }(t)}{\mu ^{- [ 2k-1 ] }(t)} \biggr) ^{(n-1)/ \lambda _{0}} \biggr) \upsilon \bigl( \mu ^{- [ 2k-1 ] } ( t ) \bigr) . \end{aligned}
(2.9)

Since $$\upsilon ^{\prime } ( t ) >0$$ and $$\mu ^{- [ 2k-1 ] } ( t ) >\mu ^{-1} ( t )$$, we have that $$\upsilon ( \mu ^{- [ 2k-1 ] } ( t ) ) >\upsilon ( \mu ^{-1} ( t ) )$$ for all $$k=1,2,\ldots,n/2$$. Therefore, (2.9) becomes

\begin{aligned} u(t) \geq &\upsilon \bigl( \mu ^{-1} ( t ) \bigr) \sum _{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) } \biggl( 1- \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] }(t)}{\mu ^{- [ 2k-1 ] }(t)} \biggr) ^{(n-1)/ \lambda _{0}} \biggr) \\ =&\widetilde{a} ( t ) \upsilon \bigl( \mu ^{-1} ( t ) \bigr) , \end{aligned}

which, with the facts that $$\varkappa ( t ) \leq \eta ( t )$$ and $$\mu ^{\prime } ( t ) >0$$, gives

$$u \bigl( \eta ( t ) \bigr) \geq \widetilde{a} \bigl( \eta ( t ) \bigr) \upsilon \bigl( \mu ^{-1} \bigl( \eta ( t ) \bigr) \bigr) \geq \widetilde{a} \bigl( \eta ( t ) \bigr) \upsilon \bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr) .$$

Then, Eq. (1.1) will become

$$\upsilon ^{n}(t)+b(t)\widetilde{a} \bigl( \eta ( t ) \bigr) \upsilon \bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr) \leq 0.$$
(2.10)

Now, using Lemma 1.1, we arrive at

$$\upsilon (t)\geq \frac{\lambda _{1}}{ ( n-1 ) !}t^{n-1} \upsilon ^{(n-1)}(t),$$
(2.11)

for all $$\lambda _{1}\in ( 0,1 )$$. It follows from Eqs. (2.10) and (2.11) that

$$\upsilon ^{n} ( t ) + \frac{\lambda _{1}}{ ( n-1 ) !}b(t)\widetilde{a} \bigl( \eta ( t ) \bigr) \bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr) ^{n-1}\upsilon ^{(n-1)}\bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr)\leq 0.$$

Clearly, $$G ( t ) :=\upsilon ^{n-1} ( t )$$ is a positive solution of the first-order delay differential inequality

$$G^{\prime } ( t ) + \frac{\lambda _{1}}{ ( n-1 ) !}b(t)\widetilde{a} \bigl( \eta ( t ) \bigr) \bigl( \mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr) ^{n-1}G\bigl(\mu ^{-1} \bigl( \varkappa ( t ) \bigr) \bigr)\leq 0.$$
(2.12)

It follows from [22] that Eq. (2.6) also has a positive solution for all $$\lambda _{0},\lambda _{1}\in (0,1)$$, but this contradicts our assumption.

Assume that $$( 2 )$$ holds. It follows from Lemma 1.2 that

$$\upsilon ( t ) \geq \lambda _{2}t\upsilon ^{\prime } ( t ) ,$$
(2.13)

for all $$\lambda _{3}\in ( 0,1 )$$ and $$t\geq t_{1}\geq t_{0}$$, and so

$$\biggl( \frac{\upsilon (t)}{t^{1/\lambda _{2}}} \biggr) ^{\prime }= \frac{\upsilon ^{\prime } ( t ) }{t^{1/\lambda _{2}}}- \frac{1}{\lambda _{2}} \frac{\upsilon ( t ) }{t^{ ( 1+\lambda _{2} ) /\lambda _{2}}}=\frac{1}{\lambda _{2}} \frac{\lambda _{2}t\upsilon ^{\prime } ( t ) -\upsilon ( t ) }{t^{ ( 1+\lambda _{2} ) /\lambda _{2}}}\leq 0.$$

Thus, from the fact that $$\mu ^{- [ 2k ] } ( t ) \leq \mu ^{- [ 2k-1 ] } ( t )$$, we conclude that

$$\upsilon \bigl( \mu ^{- [ 2k ] } ( t ) \bigr) \leq \biggl( \frac{\mu ^{- [ 2k ] } ( t ) }{\mu ^{- [ 2k-1 ] } ( t ) } \biggr) ^{1/\lambda _{2}}\upsilon \bigl( \mu ^{- [ 2k-1 ] } ( t ) \bigr) .$$
(2.14)

Combining (2.7) and (2.14), we obtain

\begin{aligned}[b] u ( t ) &\geq \sum_{k=1}^{n/2} \frac{1}{\sum_{m=1}^{2k-1}a ( \mu ^{- [ m ] } ( t ) ) }\\ &\quad {}\times \biggl( 1- \frac{1}{a ( \mu ^{- [ 2k ] } ( t ) ) } \biggl( \frac{\mu ^{- [ 2k ] } ( t ) }{\mu ^{- [ 2k-1 ] } ( t ) } \biggr) ^{1/\lambda _{2}} \biggr) \upsilon \bigl( \mu ^{- [ 2k-1 ] } ( t ) \bigr) . \end{aligned}
(2.15)

Since $$\mu ^{-[2\kappa -1]}(t)\geq \mu ^{-1}(t)$$ for all $$k=1,2,\ldots,n/2$$, (2.15) becomes

$$u(t)\geq \widehat{a} ( t ) \upsilon \bigl(\mu ^{-1}(t)\bigr).$$
(2.16)

Therefore, (1.1) will be

$$\upsilon ^{n}(t)+b(t)\widehat{a} \bigl( \eta (t) \bigr) \upsilon \bigl( \mu ^{-1}\bigl(\eta (t)\bigr)\bigr)\leq 0,$$

which, with $$\chi ( t ) \leq \eta (t)$$ and $$\upsilon ^{\prime } ( t ) >0$$, give

$$\upsilon ^{n}(t)+b(t)\widehat{a} \bigl( \eta (t) \bigr) \upsilon \bigl( \mu ^{-1}\bigl(\chi ( t ) \bigr)\bigr)\leq 0.$$
(2.17)

Integrating (2.17) from t to ∞ consecutively $$n-2$$ times and using the properties of derivatives in case $$( 2 )$$, we get

$$\upsilon ^{\prime \prime } ( t ) +\frac{1}{(n-3)!} \upsilon (\mu ^{-1} \bigl(\chi ( t ) \bigr) \int _{t}^{\infty } ( s-3 ) ^{n-3}b ( s ) \widehat{a} \bigl( \eta (s) \bigr) \,\mathrm{d}s\leq 0.$$
(2.18)

By setting $$\phi ( t ) =\upsilon ^{\prime } ( t )$$ and using (2.13), we conclude that $$\phi ( t )$$ is a positive solution of the first-order delay differential inequality,

$$\phi ^{\prime } ( t ) +\frac{\lambda _{2}}{(n-3)!}\mu ^{-1}( \chi ( t ) \phi ( \mu ^{-1}\bigl(\chi ( t ) \bigr) \int _{t}^{\infty } ( s-3 ) ^{n-3}b ( s ) \widehat{a} \bigl( \eta (s) \bigr) \,\mathrm{d}s\leq 0.$$
(2.19)

It follows from [22] that the Eq. (2.5) also has a positive solution, which contradicts our assumption. Therefore, the proof of this theorem is complete. □

### Corollary 2.1

Assume that there exist an even integer m and functions $$\varkappa \in C^{1}([t_{0},\infty ),\mathbb{R})$$, $$\chi \in C^{1}([t_{0},\infty ),\mathbb{R})$$ such that (2.1)(2.3) hold. If

$$\mathop{\lim \inf }_{t\rightarrow \infty } \int _{\mu ^{-1} ( \varkappa ( t ) ) }^{t}b(s)\widetilde{a} \bigl( \eta ( s ) \bigr) \bigl( \mu ^{-1} \bigl( \varkappa ( s ) \bigr) \bigr) ^{n-1}\,\mathrm{d}s> \frac{ ( n-1 ) !}{\lambda _{1}\mathrm{e}}$$
(2.20)

and

$$\mathop{\lim \inf }_{t\rightarrow \infty } \int _{\mu ^{-1}(\chi ( t ) }^{t}\mu ^{-1} \bigl( \chi ( \varrho ) \bigr) \biggl( \int _{\varrho }^{\infty } ( s-3 ) ^{n-3}b ( s ) \widehat{a} \bigl( \eta (s) \bigr) \,\mathrm{d}s \biggr) \,\mathrm{d} \varrho > \frac{ ( n-3 ) !}{\lambda _{2}\mathrm{e}},$$
(2.21)

for some $$\lambda _{i}\in ( 0,1 )$$, $$i=0,1,2$$, every solution of Eq. (1.1) is oscillatory.

### Proof

Applying a well-known criterion [23, Theorem 2] for first-order delay differential equations (2.4) and (2.5) to be oscillatory, we obtain immediately the criteria (2.20) and (2.21), respectively. □

### Remark 2.2

Combining Theorem 2.1 and the results reported in [2426] for the oscillation of Eqs. (2.4) and (2.5), one can derive various oscillation criteria for Eq. (1.1).

By using a Riccati transformation, we obtain the following criterion.

### Theorem 2.3

Assume that

$$\bigl( \eta ^{-1} ( t ) \bigr) ^{\prime }\geq \eta _{0}>0,\qquad \eta ^{-1} \bigl( \mu ( t ) \bigr) \geq t, \qquad \mu ^{ \prime } ( t ) \geq \mu _{0}>0,$$
(2.22)

and there exist an even integer m and a function $$\chi \in C^{1}([t_{0},\infty ),\mathbb{R})$$ such that (2.1), (2.3) and (2.21) hold. If there exist $$\lambda _{0}\in (0,1)$$ and a function $$\rho \in C^{1}([t_{0},\infty ), ( 0,\infty ) )$$ such that

$$\mathop{\lim \sup }_{t\rightarrow \infty } \int _{t_{1}}^{t} \biggl( \eta _{0}\rho ( s ) B ( s ) - \frac{ ( n-2 ) !}{4\lambda _{0}} \biggl( 1+\frac{a_{0}}{\mu _{0}} \biggr) \frac{ ( \rho ^{\prime } ( s ) ) ^{2}}{s^{{(n-2)}}\rho ( s ) } \biggr) \,\mathrm{d}s=\infty ,$$
(2.23)

then every solution of Eq. (1.1) is oscillatory.

### Proof

Assume that Eq. (1.1) has an eventually positive solution u. It follows from (1.1) that $$\upsilon ^{n}(t)=-b(t)u(\eta (t))\leq 0$$. Thus, using Lemma 2.1, we have that there are two cases for the derivatives of υ for large t, either $$( 1 )$$ or $$( 2 )$$.

Assume that $$( 1 )$$ holds. From Eq. (1.1), we obtain

$$\frac{1}{ ( \eta ^{-1} ( t ) ) ^{\prime }} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} ( t ) \bigr) \bigr) ^{\prime }+b \bigl( \eta ^{-1} ( t ) \bigr) u ( t ) =0$$
(2.24)

and

$$\frac{1}{ ( \eta ^{-1} ( \mu ( t ) ) ) ^{\prime }} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \bigr) ^{\prime }+b \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) u \bigl( \mu ( t ) \bigr) =0.$$
(2.25)

Combining (2.24) and (2.25), and using (2.22), we find

\begin{aligned} &\frac{1}{\eta _{0}} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} ( t ) \bigr) \bigr) ^{\prime }+ \frac{a_{0}}{\eta _{0}\mu _{0}} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \bigr) ^{\prime }\\ &\quad {}+b \bigl( \eta ^{-1} ( t ) \bigr) u ( t ) +a_{0}b \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) u \bigl( \mu ( t ) \bigr) \leq 0, \end{aligned}

and so

$$\frac{1}{\eta _{0}} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} ( t ) \bigr) \bigr) ^{\prime }+ \frac{a_{0}}{\eta _{0}\mu _{0}} \bigl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \bigr) ^{\prime }+B ( t ) \bigl( u ( t ) +a_{0}u \bigl( \mu ( t ) \bigr) \bigr) \leq 0.$$

Then,

$$\frac{1}{\eta _{0}} \biggl( \upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} ( t ) \bigr) +\frac{a_{0}}{\mu _{0}}\upsilon ^{ ( n-1 ) } \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \biggr) ^{\prime }+B ( t ) \upsilon ( t ) \leq 0.$$
(2.26)

Now, we define the Riccati transformation as

$$\varpi ( t ) :=\rho ( t ) \frac{\upsilon ^{(n-1)} ( \eta ^{-1} ( t ) ) }{\upsilon (t)}.$$

Thus, $$\varpi ( t ) >0$$ for $$t\geq t_{1}\geq t_{0}$$, and

$$\varpi ^{\prime } ( t ) = \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }\varpi ( t ) -\rho ( t ) \biggl( \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( t ) ) ) \upsilon ^{\prime }(t)}{\upsilon ^{2}(t)}- \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( t ) ) ) ^{\prime }}{\upsilon (t)} \biggr) .$$
(2.27)

Using Lemma 1.1 and the fact that $$\upsilon ^{ ( n ) }\leq 0$$, we arrive at

\begin{aligned}[b] \upsilon ^{\prime } ( t ) &\geq \frac{\lambda _{0}}{ ( n-2 ) !}t^{n-2}\upsilon ^{(n-1)} ( t ) \geq \frac{\lambda _{0}}{ ( n-2 ) !}t^{n-2}\upsilon ^{(n-1)} \bigl( \eta ^{-1} \bigl( \mu ( t ) \bigr) \bigr) \\ & \geq \frac{\lambda _{0}}{ ( n-2 ) !}t^{n-2}\upsilon ^{(n-1)} \bigl( \eta ^{-1}(t) \bigr) . \end{aligned}
(2.28)

Hence, (2.27) yields

$$\varpi ^{\prime } ( t ) \leq \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }\varpi ( t ) - \frac{\lambda _{0}}{ ( n-2 ) !}\frac{t^{{(n-2)}}}{\rho ( t ) }\varpi ^{2} ( t ) +\rho ( t ) \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( t ) ) ) ^{\prime }}{\upsilon (t)}.$$
(2.29)

Next, we define function

$$\omega ( t ) :=\rho ( t ) \frac{\upsilon ^{(n-1)} ( \eta ^{-1} ( \mu ( t ) ) ) }{\upsilon (t)}.$$

Then $$\omega ( t ) >0$$ for $$t\geq t_{1}\geq t_{0}$$ and

\begin{aligned} \omega ^{\prime } ( t ) =&\frac{\rho ^{\prime } ( t ) }{\rho ( t ) }\omega ( t ) -\rho ( t ) \biggl( \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( \mu ( t ) ) ) ) \upsilon ^{\prime }(t)}{\upsilon ^{2}(t)}- \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( \mu ( t ) ) ) ) ^{\prime }}{\upsilon (t)} \biggr) \\ \leq &\frac{\rho ^{\prime } ( t ) }{\rho ( t ) } \omega ( t ) -\frac{\lambda _{0}}{ ( n-2 ) !} \frac{t^{{(n-2)}}}{\rho ( t ) } \omega ^{2} ( t ) +\rho ( t ) \frac{ ( \upsilon ^{(n-1)} ( \eta ^{-1} ( \mu ( t ) ) ) ) ^{\prime }}{\upsilon (t)}. \end{aligned}
(2.30)

Combining (2.29) and (2.30), we get

\begin{aligned} \varpi ^{\prime } ( t ) +\frac{a_{0}}{\mu _{0}}\omega ^{ \prime } ( t ) \leq &\frac{\rho ^{\prime } ( t ) }{\rho ( t ) }\varpi ( t ) - \frac{\lambda _{0}}{ ( n-2 ) !}\frac{t^{{(n-2)}}}{\rho ( t ) }\varpi ^{2} ( t ) +\frac{a_{0}}{\mu _{0}} \biggl[ \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }\omega ( t ) \\ &{} -\frac{\lambda _{0}}{ ( n-2 ) !} \frac{t^{{(n-2)}}}{\rho ( t ) }\omega ^{2} ( t ) \biggr] -\eta _{0}\rho ( t ) B ( t ) . \end{aligned}

Using the fact that

$$Hy-Ky^{2}\leq \frac{1}{4}\frac{H^{2}}{K}, \quad K>0,$$

we obtain

$$\varpi ^{\prime } ( t ) +\frac{a_{0}}{\mu _{0}}\omega ^{ \prime } ( t ) \leq \frac{ ( n-2 ) !}{4\lambda _{0}} \biggl( 1+\frac{a_{0}}{\mu _{0}} \biggr) \frac{ ( \rho ^{\prime } ( t ) ) ^{2}}{t^{{(n-2)}}\rho ( t ) }-\eta _{0}\rho ( t ) B ( t ) .$$

Integrating the above inequality from $$t_{1}$$ to t, we have

$$\int _{t_{1}}^{t} \biggl( \eta _{0}\rho ( s ) B ( s ) -\frac{ ( n-2 ) !}{4\lambda _{0}} \biggl( 1+\frac{a_{0}}{\mu _{0}} \biggr) \frac{ ( \rho ^{\prime } ( s ) ) ^{2}}{s^{{(n-2)}}\rho ( s ) } \biggr) \,\mathrm{d}s\leq \varpi ( t_{1} ) + \frac{a_{0}}{\mu _{0}}\omega ( t_{1} ) ,$$

Assume that case $$( 2 )$$ holds. If we are back to the proof of Corollary 2.1, then we get a contradiction with (2.21). Hence, the proof is complete. □

Next, we give an example to illustrate our main results.

### Example 2.1

Consider a fourth-order neutral delay differential equation

$$\bigl( u(t)+a_{0}u ( \beta t ) \bigr) ^{ ( 4 ) }+ \frac{b_{0}}{t^{4}}u ( \delta t ) =0, \quad t\geq 1,$$
(2.31)

where $$a_{0},b_{0}>0$$ and $$0<\delta \leq \beta <1$$. It is easy to see that $$B ( t ) = ( b_{0}\delta ^{4} ) /t^{4}$$,

$$\widetilde{a} ( t ) = \biggl( 1- \frac{1}{\beta ^{3/\lambda _{0}}a_{0}} \biggr) \sum _{k=1}^{n/2}\frac{1}{a_{0}^{2k-1}}:=A_{0}$$

and

$$\widehat{a} ( t ) := \biggl( 1- \frac{1}{\beta ^{1/\lambda _{2}}a_{0}} \biggr) \sum _{k=1}^{n/2}\frac{1}{a_{0}^{2k-1}}:=A_{1}.$$

By choosing $$\varkappa ( t ) =\chi ( t ) =\delta t$$, we see that (2.2) and (2.3) hold, and conditions (2.20) and (2.21) reduce to

$$b_{0}\ln \frac{\beta }{\delta }> \frac{6\beta ^{3}}{\delta ^{3}A_{0}\mathrm{e}}$$

and

$$b_{0}\ln \frac{\beta }{\delta }> \frac{12\beta }{\delta A_{1}\mathrm{e}},$$

respectively. Thus, from Corollary 2.1, we see that every solution of Eq. (2.31) is oscillatory if

$$b_{0}>\max \biggl\{ \frac{6\beta ^{3}}{\delta ^{3}A_{0}\mathrm{e}\ln \beta /\delta }, \frac{12\beta }{\delta A_{1}\mathrm{e}\ln \beta /\delta } \biggr\} .$$
(2.32)

Moreover, the condition (2.23) reduces to

$$\mathop{\lim \sup }_{t\rightarrow \infty } \int _{t_{1}}^{t} \biggl( \delta ^{4}\eta _{0}b_{0}-\frac{9}{2\lambda _{0}} \biggl( 1+ \frac{a_{0}}{\mu _{0}} \biggr) \biggr) \frac{1}{s}\,\mathrm{d}s=\infty ,$$

when

$$b_{0}>\frac{9}{2} \biggl( 1+\frac{a_{0}}{\mu _{0}} \biggr) \frac{1}{\delta ^{4}\eta _{0}}.$$

Thus, from Theorem 2.3, we see that every solution of Eq. (2.31) is oscillatory if

$$b_{0}>\max \biggl\{ \frac{9}{2} \biggl( 1+\frac{a_{0}}{\mu _{0}} \biggr) \frac{1}{\delta ^{4}\eta _{0}}, \frac{6\beta ^{3}}{\delta ^{3}A_{0}\mathrm{e}\ln \beta /\delta } \biggr\} .$$

### Remark 2.4

Although the results of Li and Rogovchenko in [1] improved their previous results, they used Lemma 1.2 with $$\lambda =1$$ (and this is inaccurate); see Remark 12 in [20]. Theorem 2.1, with $$n=2$$, is a correction of Theorem 2.1 in [1]. Moreover, our results improve the results in [1], since the iterative nature of the two functions $$\widetilde{a} ( t )$$ and $$\widehat{a} ( t )$$ enables us to test for oscillations, even when the previously known results fail to apply. Let us consider a special case of (2.31), namely,

$$\bigl( u(t)+10u ( 0.9t ) \bigr) ^{ ( 4 ) }+ \frac{110}{t^{4}}u ( 0.5t ) =0.$$
(2.33)

We note that the condition (2.32) fail to apply on (2.33) when $$n=2,4$$ (consequently, the results in [1] also fail). But, at $$n=6$$, the condition (2.32) is satisfied. Therefore, our results improve the previous results in the literature.

### Remark 2.5

It would be of interest to further investigate Eq. (1.1) with different neutral coefficients; see [27] and [28] for more details. It would also be interesting to extend this development to higher-order nonlinear neutral differential equations.

Not applicable.

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## Acknowledgements

The author is grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out some inaccuracies.

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Anis, M., Moaaz, O. New oscillation theorems for a class of even-order neutral delay differential equations. Adv Differ Equ 2021, 258 (2021). https://doi.org/10.1186/s13662-021-03421-w

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• DOI: https://doi.org/10.1186/s13662-021-03421-w

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### Keywords

• Even-order differential equations
• Kneser solutions
• Oscillatory solutions