1 Introduction and preliminaries

Delay differential equations (DDEs) are of great importance in modeling many phenomena and problems in various applied sciences, see [13]. The mounting interest in studying the qualitative properties of solutions of DDEs is easy to notice, see for example [1–12] and [14–25]. However, the equations with noncanonical operator did not receive the same attention as the equations in the canonical case. One can trace the evolution in the study of the oscillatory properties of higher-order DDEs with noncanonical operator through works of Baculikova et al. [7], Zhang et al. [23–25], and, recently, Moaaz et al. [16, 18].

This study is concerned with finding sufficient oscillation conditions for the solutions of the DDE

$$ \bigl( a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa } \bigr) ^{\prime }+f\bigl(l,v\bigl(g(l)\bigr)\bigr)=0, \quad l\geq l_{0}, $$
(1.1)

in the noncanonical case, that is,

$$ \psi _{0} ( l_{0} ) := \int _{l_{0}}^{\infty } \frac{1}{ ( a(\nu ) ) ^{1/\kappa }}\,\mathrm{d}\nu < \infty . $$
(1.2)

In this study, we suppose that \(\kappa >0\) is a ratio of odd integers, \(a\in C^{1} ( I_{0},\mathbb{R} ^{+} ) \), \(a^{\prime } ( l ) \geq 0\), \(g\in C ( I_{0},\mathbb{R} ^{+} ) \), \(g ( l ) \leq l\), \(g^{\prime } ( l ) >0\), \(\lim_{l\rightarrow \infty }g ( l ) =\infty \), \(I_{\vartheta }:= [ l_{\vartheta },\infty ) \), \(f\in C ( I_{0}\times \mathbb{R} ,\mathbb{R} ) \), and there exists a function \(h\in C ( I_{0}, [ 0,\infty ) ) \) such that \(f ( l,v ) \geq h ( l ) v^{\kappa }\).

By a solution of (1.1), we mean a nontrivial real-valued function \(v\in C ( [ l_{\varkappa },\infty ) ,\mathbb{R} ) \) for some \(l_{\varkappa }\geq l_{0}\), which has the property \(a ( v^{\prime \prime \prime } ) ^{\kappa }\in C^{1} ( [ l_{0},\infty ) ,\mathbb{R} ) \) and satisfies (1.1) on \([ l_{0},\infty ) \). We will consider only those solutions of (1.1) which exist on some half-line \([ l_{\varkappa },\infty ) \) and satisfy the condition

$$ \sup \bigl\{ \bigl\vert v ( l ) \bigr\vert :l_{c}\leq l< \infty \bigr\} >0\quad \text{for any }l_{c}\geq l_{\varkappa }. $$

If v is either positive or negative, eventually, then v is called nonoscillatory; otherwise it is called oscillatory. Equation (1.1) itself is termed oscillatory if all its solutions are oscillatory.

Zhang et al. [25] considered the higher-order DDE

$$ \bigl(a\bigl(v^{(n-1}\bigr)^{\kappa }\bigr)^{\prime } ( l ) +h ( l ) v^{\gamma } \bigl( g ( l ) \bigr) =0, $$
(1.3)

where κ, γ are a ration of odd integers and \(0<\gamma \leq \kappa \). Moreover, Zhang et al. [23] studied the oscillation of solutions for (1.3) and improved the results [25]. For the convenience of the reader, we present some of their results below at \(\kappa =\gamma \) and \(n=4\).

Theorem 1.1

([25, Corollary 2.1])

If

$$ \underset{l\rightarrow \infty }{\lim \inf } \int _{g ( l ) }^{l}h ( s ) \frac{ ( g^{3} ( s ) ) ^{\kappa }}{a ( g ( s ) ) }\,\mathrm{d}s> \frac{ ( 3! ) ^{\kappa }}{\mathrm{e}} $$
(1.4)

and

$$ \underset{l\rightarrow \infty }{\lim \sup } \int _{l_{0}}^{l} \biggl( h ( s ) \biggl( \frac{\varepsilon _{1}\psi _{0} ( s ) g^{2} ( s ) }{2!} \biggr) ^{\kappa }- \frac{\kappa ^{\kappa +1}}{ ( \kappa +1 ) ^{\kappa +1}} \frac{1}{\psi _{0} ( s ) a^{1/\kappa } ( s ) } \biggr) \,\mathrm{d}s=\infty $$
(1.5)

for some \(\varepsilon _{1}\in ( 0,1 ) \), then every nonoscillatory solution of (1.1) tends to zero.

Theorem 1.2

([23, Corollary 2.1])

If (1.4), (1.5), and

$$ \underset{l\rightarrow \infty }{\lim \sup } \int _{l_{0}}^{l} \biggl( h ( s ) a^{\kappa } ( s ) - \frac{\kappa ^{\kappa +1}}{ ( \kappa +1 ) ^{\kappa +1}} \frac{ ( a^{\prime } ( s ) ) ^{\kappa +1}}{a ( s ) a_{\ast }^{\kappa } ( s ) } \biggr) \,\mathrm{d}s=\infty $$
(1.6)

for some \(\varepsilon _{1}\in ( 0,1 ) \), where

$$ a ( s ) = \int _{l}^{\infty } ( \eta -l ) \psi _{0} ( \eta ) \,\mathrm{d}\eta $$

and

$$ a_{\ast } ( s ) = \int _{l}^{\infty }\psi _{0} ( \eta ) \,\mathrm{d}\eta , $$

then (1.1) is oscillatory.

Dzurina and Jadlovska [9] considered the second-order DDE

$$ \bigl(a ( l ) \bigl(v^{\prime } ( l ) \bigr)^{\kappa } \bigr)^{ \prime }+h ( l ) v^{\kappa } \bigl( g ( l ) \bigr) =0. $$
(1.7)

Moreover, Dzurina et al. [10] investigated the oscillation of solutions for (1.7) and improved the results [9].

Theorem 1.3

([9, Theorem 3])

Assume that

$$ \underset{l\rightarrow \infty }{\lim \sup } \psi _{0}^{\kappa } ( l ) \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s>1. $$

Then (1.7) is oscillatory.

Theorem 1.4

([10, Theorem 2.3])

Let

$$ \int _{l_{0}}^{\infty }\frac{1}{a^{1/\kappa } ( l ) } \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}l=\infty $$

hold. If

$$ k:=\underset{l\rightarrow \infty }{\lim \inf } \frac{1}{\psi ( l ) }\int _{l}^{\infty }\psi ^{\kappa +1} ( s ) h ( s ) \,\mathrm{d}s>\kappa $$

or

$$ k\leq \kappa \quad \textit{and}\quad K>1-\frac{k}{\kappa }, $$

where

$$ K:=\underset{l\rightarrow \infty }{\lim \sup } \psi ( l ) \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }>1, $$

then (1.7) is oscillatory.

The objective of this paper is to improve and simplify the oscillation criteria of the fourth-order DDE (1.1) in the noncanonical case. In the noncanonical case, it is usual to have oscillation criteria in the form of at least three independent conditions; however, in Sect. 2, we obtain only two independent conditions that guarantee the oscillation of all solutions. In Sect. 3, we take an approach that creates improved criteria for oscillation. Further, the examples provided illustrate the significance of the results.

Lemma 1.1

([5])

Assume that \(F\in C^{m} ( I_{0},\mathbb{R} ) \) and \(F^{ ( m ) } ( l ) \) is eventually of constant sign. Then there are \(l_{u}\geq l_{0}\) and \(\ell \in \mathbb{Z} \), \(0\leq \ell \leq m\), with \(m+\ell \) even for \(F^{ ( m ) } ( l ) \geq 0\) or \(m+\ell \) odd for \(F^{ ( m ) } ( l ) \leq 0\), such that

$$ \ell >0 \textit{ yields}\quad F^{ ( k ) } ( l ) >0 \quad \textit{for } k=0,1,\ldots,\ell -1 $$

and

$$ \ell \leq m-1\textit{ yields}\quad ( -1 ) ^{\ell +k}F^{ ( k ) } ( l ) >0 \quad \textit{for } k=\ell ,\ell +1,\ldots,m-1 $$

for all \(l\in I_{u}\).

2 Simplified criteria for oscillation

Lemma 2.1

Assume that \(v\in C ( [ l_{0},\infty ) , ( 0,\infty ) ) \) is a solution of (1.1). Then \(( a ( l ) ( v^{\prime \prime \prime } ( l ) ) ^{\kappa } ) ^{\prime }\leq 0\), and one of the following cases holds, eventually:

  1. (a)

    \(v^{\prime } ( l )\) and \(v^{\prime \prime \prime } ( l )\) are positive, and \(v^{(4)} ( l )\) is nonpositive;

  2. (b)

    \(v^{\prime } ( l )\) and \(v^{\prime \prime } ( l )\) are positive, and \(v^{\prime \prime \prime } ( l )\) is negative;

  3. (c)

    \(v^{\prime \prime } ( l )\) is positive, and \(v^{\prime } ( l )\) and \(v^{\prime \prime \prime } ( l )\) are negative.

Proof

Assume that \(v\in C ( [ l_{0},\infty ) , ( 0,\infty ) ) \) is a solution of (1.1). From (1.1), we have

$$ \bigl( a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa } \bigr) ^{\prime }\leq -h ( l ) v^{\kappa } ( l ) \leq 0. $$

From (1.1) and Lemma 1.1, there exist three possible cases (a), (b), and (c) for \(l\geq l_{1}\), \(l_{1}\) large enough. The proof is complete. □

Let us define

$$ \psi _{m} ( l ) := \int _{l}^{\infty }\psi _{m-1} ( \nu ) \,\mathrm{d} \nu\quad \text{for }m=1,2. $$

Theorem 2.1

Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If

$$ \underset{l\rightarrow \infty }{\lim \sup } \int _{l_{1}}^{l} \biggl( \frac{1}{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \psi _{2}^{\kappa } \bigl(g ( s ) \bigr)\,\mathrm{d}s \biggr) ^{1/ \kappa } \biggr) \,\mathrm{d}u=\infty , $$
(2.1)

then v satisfies case (b) in Lemma 2.1.

Proof

Assume on the contrary that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution (1.1) and satisfies either case (a) or case (c).

First, we suppose that (c) holds on \(I_{1}\). Since \(( a ( l ) ( v^{\prime \prime \prime } ( l ) ) ^{\kappa } ) ^{\prime }\leq 0\), we have

$$ a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa }\leq a ( l_{1} ) \bigl( v^{\prime \prime \prime } ( l_{1} ) \bigr) ^{\kappa }:=-L< 0, $$
(2.2)

which is

$$ a^{1/\kappa } ( l ) v^{\prime \prime \prime } ( l ) \leq -L^{1/\kappa }. $$
(2.3)

If we divide (2.3) by \(a^{1/\kappa }\) and then integrate from lto ϱ, we find

$$ v^{\prime \prime } ( \varrho ) \leq v^{\prime \prime } ( l ) -L^{1/\kappa } \int _{l}^{\varrho } \frac{1}{a^{1/\kappa } ( s ) }\,\mathrm{d}s. $$

Letting \(\varrho \rightarrow \infty \), we get

$$ 0\leq v^{\prime \prime } ( l ) -L^{1/\kappa }\psi _{0}(l). $$
(2.4)

Integrating (2.4) from l to ∞, we obtain

$$ -v^{\prime } ( l ) \geq L^{1/\kappa }\psi _{1}(l). $$
(2.5)

Integrating (2.5) from l to ∞ implies that

$$ v ( l ) \geq L^{1/\kappa }\psi _{2}(l). $$
(2.6)

From (1.1) and (2.6), we have

$$ \bigl(a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa } \bigr)^{\prime }\leq -h ( l ) L\psi _{2}^{\kappa }\bigl(g ( l ) \bigr). $$
(2.7)

Integrating (2.7) from \(l_{1}\) to l, we obtain

$$\begin{aligned} a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa } \leq &a ( l_{1} ) \bigl(v^{\prime \prime \prime } ( l_{1} ) \bigr)^{\kappa }-L \int _{l_{1}}^{l}h ( s ) \psi _{2}^{\kappa } \bigl(g ( s ) \bigr)\,\mathrm{d}s \\ \leq &-L \int _{l_{1}}^{l}h ( s ) \psi _{2}^{\kappa } \bigl(g ( s ) \bigr)\,\mathrm{d}s. \end{aligned}$$
(2.8)

Integrating (2.8) from \(l_{1}\) to l, we get

$$ v^{\prime \prime } ( l ) \leq v^{\prime \prime } ( l_{1} ) -L^{1/\kappa } \int _{l_{1}}^{l} \biggl( \frac{1}{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \psi _{2}^{\kappa } \bigl(g ( s ) \bigr)\,\mathrm{d}s \biggr) ^{1/\kappa } \biggr) \,\mathrm{d}u. $$

At \(l\rightarrow \infty \), we arrive at a contradiction with (2.1).

Finally, let case (a) hold on \(I_{1}\). On the other hand, it follows from (2.1) and (1.2) that \(\int _{l_{1}}^{l}h ( s ) \psi _{2}^{\kappa }(s)\,\mathrm{d}s\) must be unbounded. Further, since \(\psi _{2}^{\prime }(s)<0\), it is easy to see that

$$ \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s\rightarrow \infty \quad \text{as }l\rightarrow \infty . $$
(2.9)

Integrating (1.1) from \(l_{2}\) to l, we get

$$\begin{aligned} a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa } \leq &a ( l_{2} ) \bigl(v^{\prime \prime \prime } ( l_{2} ) \bigr)^{\kappa }- \int _{l_{2}}^{l}h ( s ) v^{ \kappa }\bigl(g ( s ) \bigr)\,\mathrm{d}s \\ \leq &a ( l_{2} ) \bigl(v^{\prime \prime \prime } ( l_{2} ) \bigr)^{\kappa }-v^{\kappa }\bigl(g ( l_{2} ) \bigr) \int _{l_{2}}^{l}h ( s ) \,\mathrm{d}s. \end{aligned}$$
(2.10)

From (2.9) and (2.10), we get a contradiction with the positivity of \(a ( l ) (v^{\prime \prime \prime } ( l ) )^{ \kappa }\). This completes the proof. □

Theorem 2.2

Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If

$$ \underset{l\rightarrow \infty }{\lim \sup } \psi _{2}^{\kappa } ( l ) \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s>1, $$
(2.11)

then v satisfies case (b) in Lemma 2.1.

Proof

Assume on the contrary that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution (1.1) and satisfies case (a) or case (c).

First, we suppose that (c) holds on \(I_{1}\). Then

$$ v^{\prime \prime } ( l ) \geq - \int _{l}^{\infty }a^{-1/ \kappa } ( s ) a^{1/\kappa } ( s ) v^{\prime \prime \prime } ( s ) \,\mathrm{d}s\geq -a^{1/\kappa } ( l ) v^{\prime \prime \prime } ( l ) \psi _{0} ( l ) . $$
(2.12)

Integrating (2.12) twice from l to ∞, we arrive at

$$ v^{\prime } ( l ) \leq \int _{l}^{\infty }a^{1/\kappa } ( s ) v^{\prime \prime \prime } ( s ) \psi ( s ) \,\mathrm{d}s\leq a^{1/\kappa } ( l ) v^{ \prime \prime \prime } ( l ) \psi _{1} ( l ) $$
(2.13)

and

$$ v ( l ) \geq - \int _{l}^{\infty }a^{1/\kappa } ( s ) v^{\prime \prime \prime } ( s ) \psi _{1} ( s ) \,\mathrm{d}s\geq -a^{1/\kappa } ( l ) v^{\prime \prime \prime } ( l ) \psi _{2} ( l ) . $$
(2.14)

Integrating (1.1) from \(l_{1}\) to l, we get

$$ a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa }\leq a ( l_{1} ) \bigl(v^{\prime \prime \prime } ( l_{1} ) \bigr)^{\kappa }- \int _{l_{1}}^{l}h ( s ) v^{\kappa }\bigl(g ( s ) \bigr)\,\mathrm{d}s, $$

since \(g^{\prime } ( l ) >0\) and \(s\leq l\), we obtain

$$ a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa }\leq -v^{\kappa } \bigl(g ( l ) \bigr) \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s. $$
(2.15)

Since \(g ( l ) \leq l\), we have

$$ a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa }\leq -v^{\kappa } ( l ) \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s. $$
(2.16)

From (2.14) and (2.16), we find

$$ a ( l ) \bigl(v^{\prime \prime \prime } ( l ) \bigr)^{ \kappa }\leq a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa }\psi _{2}^{\kappa } ( l ) \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s. $$
(2.17)

Dividing both sides of inequality (2.17) by \(a ( l ) (v^{\prime \prime \prime } ( l ) )^{ \kappa }\) and taking the limsup, we arrive at

$$ \underset{l\rightarrow \infty }{\lim \sup }\psi _{2}^{\kappa } ( l ) \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s\leq 1, $$

we arrive at a contradiction with (2.11).

Next, we suppose that case (a) holds on \(I_{1}\). From (2.11) and the fact that \(\psi _{2} ( l ) <\infty \), we get that (2.9) holds. Then, this part of the proof is similar to that of Theorem 2.1. This completes the proof. □

Theorem 2.3

Assume that (2.1) or (2.11) holds. If there is \(\rho \in C^{1} ( I_{0},\mathbb{R} ^{+} ) \) such that

$$ \lim_{l\rightarrow \infty }\sup \frac{\psi _{0}^{\kappa } ( l ) }{\rho (l)} \int _{l_{0}}^{l} \biggl( \rho (s)h(s) \biggl( \frac{\lambda }{2!}g^{2}(s) \biggr) ^{\kappa }- \frac{a ( s ) ( \rho ^{\prime }(s) ) ^{\kappa +1}}{ ( \kappa +1 ) ^{\kappa +1}\rho ^{\kappa }(s)} \biggr) \,\mathrm{d}s>1 $$
(2.18)

holds for some \(\lambda _{1}\in ( 0,1 ) \), then all solutions of (1.1) are oscillatory.

Proof

Suppose that (1.1) has a nonoscillatory solution v in \(I_{0}\). Then we assume that v is eventually positive. From Lemma 2.1, we have three cases for v and its derivatives. Using Theorems 2.1 and 2.2, we have that condition (2.1) or (2.11) ensures that solution v satisfies case (b). On the other hand, using Theorem 2.2 in [18], we find that condition (2.18) contrasts with case (b). This completes the proof. □

Example 2.1

Consider the DDE

$$ \bigl(l^{3\kappa +1}\bigl(v^{\prime \prime \prime }(l)\bigr)^{\kappa } \bigr)^{\prime }+h_{0}v^{ \kappa } ( \epsilon l ) =0, $$
(2.19)

where \(h_{0}>0\) and \(\epsilon \in ( 0,1 ] \). Note that \(a ( l ) :=l^{3\kappa +1}\), \(g ( l ) :=\epsilon l\), \(f ( v ) :=v^{\kappa }\), and \(h ( l ) :=h_{0}\). Thus, we have that

$$ \psi _{0} ( l ) = \frac{\kappa }{ ( 2\kappa +1 ) l^{ ( 2\kappa +1 ) /\kappa }},\qquad \psi _{1} ( l ) = \frac{\kappa ^{2}}{ ( 2\kappa +1 ) ( \kappa +1 ) l^{ ( \kappa +1 ) /\kappa }} $$

and

$$ \psi _{2} ( l ) = \frac{\kappa ^{3}}{ ( 2\kappa +1 ) ( \kappa +1 ) l^{1/\kappa }}. $$

Now, condition (2.11) reduces to

$$ \frac{\kappa ^{3\kappa }h_{0}}{ ( ( 2\kappa +1 ) ( \kappa +1 ) ) ^{\kappa }}>1. $$
(2.20)

Furthermore, if \(\rho ( l ) :=1/l^{2\kappa +1}\), then condition (2.18) becomes

$$ h_{0} \biggl( \frac{\lambda }{2!}\epsilon ^{2} \biggr) ^{\kappa }> \frac{(2\kappa +1)^{\kappa +1}}{(\kappa +1)^{\kappa +1}}. $$
(2.21)

Using Theorem 2.3, we have that (2.19) is oscillatory if (2.20) and (2.21) hold.

Remark 2.4

Note that, we used two conditions only for testing the oscillation of the fourth-order DDEs. Moreover, our results can also be applied to ordinary DEs when \(g ( l ) =l\).

3 Improved criteria for oscillation

Theorem 3.1

Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If the DE

$$ v^{\prime } ( l ) + \frac{1}{\psi _{2} ( g ( l ) ) } \biggl( \int _{l}^{\infty } \int _{\varsigma }^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}u\,\mathrm{d}\varsigma \biggr) v \bigl( g ( l ) \bigr) =0 $$
(3.1)

is oscillatory, then the solution v does not satisfy case (c).

Proof

Suppose the contrary that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.12) and (2.15) hold. From (2.12), we have

$$ \biggl( \frac{v^{\prime \prime } ( l ) }{\psi ( l ) } \biggr) ^{\prime }= \frac{\psi ( l ) v^{\prime \prime \prime } ( l ) +v^{\prime \prime } ( l ) a^{-1/\kappa } ( l ) }{\psi ^{2} ( l ) } \geq 0. $$

Thus, we get that

$$ -v^{\prime } ( l ) \geq \int _{l}^{\infty } \frac{v^{\prime \prime } ( s ) }{\psi ( s ) } \psi ( s ) \,\mathrm{d}s\geq \frac{v^{\prime \prime } ( l ) }{\psi ( l ) }\int _{l}^{\infty }\psi ( s ) \,\mathrm{d}s, $$

that is, \(-v^{\prime } ( l ) \psi ( l ) \geq v^{\prime \prime } ( l ) \psi _{1} ( l ) \). Therefore,

$$ \biggl( \frac{v^{\prime } ( l ) }{\psi _{1} ( l ) } \biggr) ^{\prime }= \frac{\psi _{1} ( l ) v^{\prime \prime } ( l ) +v^{\prime } ( l ) \psi ( l ) }{\psi _{1}^{2} ( l ) }\leq 0. $$
(3.2)

Using (3.2), we obtain that

$$ -v ( l ) \leq \int _{l}^{\infty } \frac{v^{\prime } ( s ) }{\psi _{1} ( s ) }\psi _{1} ( s ) \,\mathrm{d}s \leq \frac{v^{\prime } ( l ) }{\psi _{1} ( l ) } \int _{l}^{ \infty }\psi _{1} ( s ) \,\mathrm{d}s, $$

that is, \(-\psi _{1} ( l ) v ( l ) \leq v^{\prime } ( l ) \psi _{2} ( l ) \). Hence,

$$ \biggl( \frac{v ( l ) }{\psi _{2} ( l ) } \biggr) ^{\prime }=\frac{\psi _{2} ( l ) v^{\prime } ( l ) +v ( l ) \psi _{1} ( l ) }{\psi _{2}^{2} ( l ) } \geq 0. $$
(3.3)

Now, integrating (2.15) from l to ∞ and using (3.3), we get

$$\begin{aligned} -v^{\prime \prime } ( l ) \leq &- \int _{l}^{\infty } \frac{v ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa } \,\mathrm{d}u \\ \leq &- \int _{l}^{\infty } \frac{v ( g ( u ) ) }{\psi _{2} ( g ( u ) ) } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }\,\mathrm{d}u \\ \leq &- \frac{v ( g ( l ) ) }{\psi _{2} ( g ( l ) ) } \int _{l}^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }\,\mathrm{d}u. \end{aligned}$$
(3.4)

Integrating (3.4) from l to ∞, we find

$$\begin{aligned} v^{\prime } ( l ) \leq &- \int _{l}^{\infty } \frac{v ( g ( \varsigma ) ) }{\psi _{2} ( g ( \varsigma ) ) } \int _{\varsigma }^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }\,\mathrm{d}u \,\mathrm{d}\varsigma \\ \leq &- \frac{v ( g ( l ) ) }{\psi _{2} ( g ( l ) ) } \int _{l}^{\infty } \int _{\varsigma }^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}u\,\mathrm{d}\varsigma . \end{aligned}$$

Thus, it is easy to see that v is a positive solution of the first-order delay differential inequality

$$ v^{\prime } ( l ) + \frac{1}{\psi _{2} ( g ( l ) ) } \biggl( \int _{l}^{\infty } \int _{\varsigma }^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}u\,\mathrm{d}\varsigma \biggr) v \bigl( g ( l ) \bigr) \leq 0. $$

Using [22], we have that (3.1) has also a positive solution, a contradiction. This completes the proof. □

Corollary 3.1

Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If

$$ \underset{l\rightarrow \infty }{\lim \inf } \int _{g ( l ) }^{l} \frac{1}{\psi _{2} ( g ( \vartheta ) ) } \biggl( \int _{\vartheta }^{\infty } \int _{\varsigma }^{\infty } \frac{\psi _{2} ( g ( u ) ) }{a^{1/\kappa } ( u ) } \biggl( \int _{l_{1}}^{u}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }\,\mathrm{d}u \,\mathrm{d}\varsigma \biggr) \,\mathrm{d}\vartheta > \frac{1}{\mathrm{e}}, $$
(3.5)

then the solution v does not satisfy case (c).

Proof

Using [22], we note that condition (3.5) ensures the oscillation of (3.1). This completes the proof. □

Lemma 3.1

Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1) and case (c) holds. If

$$ \int _{l_{0}}^{\infty } \biggl( \frac{1}{a ( \varsigma ) }\int _{l_{1}}^{\varsigma }h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}\varsigma =\infty , $$
(3.6)

then \(\lim_{l\rightarrow \infty }v ( l ) =0\).

Proof

Suppose that v satisfies case (c). Then we obtain that \(\lim_{l\rightarrow \infty }v ( l ) =c\geq 0\). We claim that \(\lim_{l\rightarrow \infty }v ( l ) =0\). Suppose the contrary that \(c>0\). Thus, there exists \(l_{1}\geq l_{0}\) such that \(v ( g ( l ) ) \geq c\) for \(l\geq l_{1}\), and hence

$$ - \bigl( a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa } \bigr) ^{\prime }\geq h ( l ) v^{\kappa } \bigl( g ( l ) \bigr) \geq c^{\kappa }h ( l ) $$
(3.7)

for \(l\geq l_{1}\). Integrating (3.7) twice from \(l_{1}\) to l, we obtain

$$ v^{\prime \prime \prime } ( l ) \leq -c \biggl( \frac{1}{a ( l ) } \int _{l_{1}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa } $$

and

$$ v^{\prime \prime } ( l ) \leq v^{\prime \prime } ( l_{1} ) -c \int _{l_{1}}^{l} \biggl( \frac{1}{a ( \varsigma ) }\int _{l_{1}}^{\varsigma }h ( s ) \,\mathrm{d}s \biggr) ^{1/ \kappa }\,\mathrm{d}\varsigma . $$

Letting \(l\rightarrow \infty \) and using (3.6), we obtain that \(\lim_{l\rightarrow \infty }v^{\prime \prime }(l)=-\infty \), which contradicts \(v^{\prime \prime }(l)>0\). Thus, the proof is complete. □

Lemma 3.2

Assume that (3.6) holds, \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1), and case (c) holds. If there exists a constant \(\mu \geq 0\) such that

$$ \psi _{2} ( l ) \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }\geq \mu , $$
(3.8)

then

$$ \frac{\mathrm{d}}{\mathrm{d}l} \biggl( \frac{v ( l ) }{\psi _{2}^{\mu } ( l ) } \biggr) \leq 0. $$
(3.9)

Proof

Suppose that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.13) holds. Integrating (1.1) from \(l_{1}\) to l and using \(v^{\prime } ( l ) <0\), we find

$$\begin{aligned} a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa } \leq &a ( l_{1} ) \bigl( v^{\prime \prime \prime } ( l_{1} ) \bigr) ^{\kappa }- \int _{l_{1}}^{l}h ( s ) v^{\kappa } \bigl( g ( s ) \bigr) \,\mathrm{d}s \\ \leq &a ( l_{1} ) \bigl( v^{\prime \prime \prime } ( l_{1} ) \bigr) ^{\kappa }-v^{\kappa } \bigl( g ( l ) \bigr) \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s+v^{ \kappa } \bigl( g ( l ) \bigr) \int _{l_{0}}^{l_{1}}h ( s ) \,\mathrm{d}s. \end{aligned}$$
(3.10)

Using Lemma 3.1, we get that \(\lim_{l\rightarrow \infty }v ( l ) =0\). Thus, there is \(l_{2}\geq l_{1}\) such that

$$ a ( l_{1} ) \bigl( v^{\prime \prime \prime } ( l_{1} ) \bigr) ^{\kappa }+v^{\kappa } \bigl( g ( l ) \bigr) \int _{l_{0}}^{l_{1}}h ( s ) \,\mathrm{d}s< 0 \quad \text{for every }l\geq l_{2}, $$

which, with (3.10), gives

$$ a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa }\leq -v^{\kappa } \bigl( g ( l ) \bigr) \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s\leq -v^{ \kappa } ( l ) \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s. $$
(3.11)

Next, we have that

$$ \frac{\mathrm{d}}{\mathrm{d}l} \biggl( \frac{v ( l ) }{\psi _{2}^{\mu } ( l ) } \biggr) = \frac{\psi _{2}^{\mu } ( l ) v^{\prime } ( l ) +\mu \psi _{2}^{\mu -1} ( l ) \psi _{1} ( l ) v ( l ) }{\psi _{2}^{2\mu } ( l ) }. $$
(3.12)

Combining (2.13) and (3.11), we get

$$ v^{\prime } ( l ) \leq -v ( l ) \psi _{1} ( l ) \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }. $$

This implies

$$\begin{aligned} \psi _{2}^{\mu } ( l ) v^{\prime } ( l ) +\mu \psi _{2}^{\mu -1} ( l ) \psi _{1} ( l ) v ( l ) \leq &- \psi _{2}^{\mu } ( l ) \psi _{1} ( l ) v ( l ) \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }+\mu \psi _{2}^{\mu -1} ( l ) \psi _{1} ( l ) v ( l ) \\ =& \biggl( -\psi _{2} ( l ) \biggl( \int _{l_{0}}^{l}h ( s ) \,\mathrm{d}s \biggr) ^{1/\kappa }+\mu \biggr) \psi _{2}^{ \mu -1} ( l ) \psi _{1} ( l ) v ( l ) . \end{aligned}$$

It follows from (3.8) that \(\psi _{2}^{\mu } ( l ) v^{\prime } ( l ) +\mu \psi _{2}^{\mu -1} ( l ) \psi _{1} ( l ) v ( l ) \leq 0\), which, with (3.12), implies that the function \(v ( l ) /\psi _{2}^{\mu } ( l ) \) is nonincreasing. This completes the proof. □

Theorem 3.2

Assume that (3.6) holds. If there exists a constant \(\mu \geq 0\) such that (3.8) holds, and the equation

$$ \biggl( \frac{1}{\psi _{1}^{\kappa } ( l ) } \bigl( v^{ \prime } ( l ) \bigr) ^{\kappa } \biggr) ^{\prime }+h ( l ) \biggl( \frac{\psi _{2} ( g ( l ) ) }{\psi _{2} ( l ) } \biggr) ^{\mu \kappa }v^{\kappa } ( l ) =0 $$
(3.13)

is oscillatory, then the solution v does not satisfy case (c).

Proof

Assume on the contrary that (1.1) has a positive solution v which satisfies case (c). Using Theorem 2.2 and Lemma 3.2, we get that (2.13) and (3.9) hold, respectively. Integrating (3.9) from \(g ( l ) \) to l, we obtain

$$ v \bigl( g ( l ) \bigr) \geq \biggl( \frac{\psi _{2} ( g ( l ) ) }{\psi _{2} ( l ) } \biggr) ^{\mu }v ( l ) , $$

which with (1.1) gives

$$ \bigl( a ( l ) \bigl( v^{\prime \prime \prime } ( l ) \bigr) ^{\kappa } \bigr) ^{\prime }\leq -h ( l ) \biggl( \frac{\psi _{2} ( g ( l ) ) }{\psi _{2} ( l ) } \biggr) ^{\mu \kappa }v^{\kappa } ( l ) . $$
(3.14)

Integrating (2.13) from l to ∞ provides

$$ v ( l ) \geq -a^{1/\kappa } ( l ) v^{\prime \prime \prime } ( l ) \psi _{2} ( l ) . $$
(3.15)

Next, we define

$$ w ( l ) :=a ( l ) \biggl( \frac{v^{\prime \prime \prime } ( l ) }{v ( l ) } \biggr) ^{\kappa }< 0. $$
(3.16)

From (3.14) and (3.16), we conclude that

$$ w^{\prime } ( l ) \leq -h ( l ) \biggl( \frac{\psi _{2} ( g ( l ) ) }{\psi _{2} ( l ) } \biggr) ^{\mu \kappa }-\kappa \frac{a ( l ) ( v^{\prime \prime \prime } ( l ) ) ^{\kappa }}{v^{\kappa +1} ( l ) }v^{ \prime } ( l ) , $$

which, in view of (2.13), gives

$$ w^{\prime } ( l ) +h ( l ) \biggl( \frac{\psi _{2} ( g ( l ) ) }{\psi _{2} ( l ) } \biggr) ^{ \mu \kappa }+\kappa \psi _{1} ( l ) w^{ ( \kappa +1 ) /\kappa } ( l ) \leq 0. $$
(3.17)

In view of [6], differential equation (3.13) is nonoscillatory if and only if there exists a function \(w\in C ( [ l_{1},\infty ) ,\mathbb{R} ) \) satisfying inequality (3.17) for \(l\geq l_{1}\), \(l_{1}\) large enough, which is a contradiction. This completes the proof. □

Using Theorems 3.2, 1.3, and 1.4, we establish the following oscillation criteria for (1.1) under the assumption \(\psi _{2} ( l_{0} ) <\infty \).

Corollary 3.2

Assume that (3.6) holds and there exists a constant \(\mu \geq 0\) such that (3.8) holds. If \(\psi _{2} ( l_{0} ) <\infty \) and

$$ \underset{l\rightarrow \infty }{\lim \sup }\psi _{2}^{\kappa } ( l ) \int _{l_{0}}^{l}h ( s ) \biggl( \frac{\psi _{2} ( g ( s ) ) }{\psi _{2} ( s ) } \biggr) ^{ \mu \kappa }\,\mathrm{d}s>1 $$
(3.18)

or

$$ \underset{l\rightarrow \infty }{\lim \inf } \frac{1}{\psi _{2} ( l ) } \int _{l}^{\infty }\psi _{2}^{\kappa +1} ( s ) h ( s ) \biggl( \frac{\psi _{2} ( g ( s ) ) }{\psi _{2} ( s ) } \biggr) ^{\mu \kappa }\,\mathrm{d}s> \biggl( \frac{\kappa }{\kappa +1} \biggr) ^{\kappa +1} $$
(3.19)

hold, then the solution v does not satisfy case (c).

Theorem 3.3

Assume that (1.4), (1.5), and (3.5) hold, then all solutions of equation (1.1) are oscillatory.

Proof

Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists \(l_{1}\in [ l_{0},\infty ) \) such that \(v ( l ) >0\) and \(v ( g ( l ) ) >0\) for \(l\geq l_{1}\). Using Lemma 2.1, there exist three possible cases (a)–(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.1, we get a contradiction with condition (3.5). This completes the proof. □

Theorem 3.4

Assume that (3.6), (1.4), and (1.5hold and there exists a constant \(\mu \geq 0\) such that (3.8) holds. If \(\psi _{2} ( l_{0} ) <\infty \) and (3.19) hold, then all solutions of equation (1.1) are oscillatory.

Proof

Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists \(l_{1}\in [ l_{0},\infty ) \) such that \(v ( l ) >0\) and \(v ( g ( l ) ) >0\) for \(l\geq l_{1}\). Using Lemma 2.1, there exist three possible cases (a)–(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.2, we get a contradiction with condition (3.19). This completes the proof. □

Example 3.1

Consider the delay differential equation

$$ \bigl(\mathrm{e}^{3l}\bigl(v^{\prime \prime \prime }(l)\bigr)^{3} \bigr)^{\prime }+h_{0} \mathrm{e}^{3l}v^{3} ( l-1 ) =0, $$
(3.20)

where \(h_{0}>0\). We note that \(a ( l ) :=\mathrm{e}^{3l}\), \(h ( l ) :=h_{0}\mathrm{e}^{3l}\), \(f ( v ) :=v^{3}\), and \(g ( l ) :=l-1\). Thus, we have that

$$ \psi _{i} ( l ) =\mathrm{e}^{-l} \quad \text{for }i=0,1,2. $$

It is easy to verify that \(\psi _{2} ( l_{0} ) <\infty \), (3.6), (1.4), and (1.5) are satisfied. Now, (3.5) holds if \(h_{0}>0.149 36\). Moreover, if we choose \(\mu := ( h_{0}/3 ) ^{1/3} \), then we see that (3.8) is satisfied and (3.19) holds if \(h_{0}>0.115 05\).

Hence, by Theorem 3.3, every solution of (3.20) is oscillatory if \(h_{0}>0.149 36\). Further, by Theorem 3.4, every solution of (3.20) is oscillatory if \(h_{0}>0.115 05\).

Remark 3.5

By using [23, Corollary 2.1], equation (3.20) is oscillatory when \(h_{0}>0.316 41\). Thus, we note that Theorem 3.4 provides a better criterion for the oscillation of (3.20). Moreover, our oscillation criteria take into account the influence of \(g ( l ) \), which has not been taken care of in the related results [18, 25].

4 Conclusion

In this work, we simplified and improved the oscillation criteria for a class of even-order delay differential equations. In the noncanonical case, it always sets three conditions to check the oscillation of even-order DDEs. First, we obtained a criterion with only two conditions to check the oscillation. Furthermore, we improved the three-condition oscillation criteria by creating a better estimate of the ratio \(v ( g ( l ) ) /v ( l ) \). Through the example, we compared our results with the previous results and explained the importance of our new oscillation criteria. It will be interesting to extend our results of this study to the neutral and mixed case.