1 Introduction

The objective of this work is to study the asymptotic behavior of solutions of a class of higher order delay differential equations (DDEs) of the form

$$\begin{aligned} \left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) +\left( h\cdot \left( F\circ \upsilon \circ g\right) \right) \left( t\right) =0, \end{aligned}$$
(1.1)

where \(t\ge t_{0}\), \(n\in \mathbb {Z} ^{+}\) is even, and \(n\ge 4\). We also assume the following:

  1. (H1)

    a,  \(h\in C\left( \left[ t_{0},\infty \right) ,\left[ 0,\infty \right) \right) ,\) \(a\left( t\right) >0\) and \(A_{n-2}\left( t_{0}\right) <\infty \), where

    $$\begin{aligned} A_{0}\left( t\right) :=\int _{t}^{\infty }a^{-1}\left( \varrho \right) \textrm{d}\varrho ,\text { and }A_{k}\left( t\right) :=\int _{t}^{\infty }A_{k-1}\left( \varrho \right) \textrm{d}\varrho , \end{aligned}$$

    for \(k=1,2,\dots ,n-2.\)

  2. (H2)

    \(g\in C\left( \left[ t_{0},\infty \right) , \mathbb {R} ^{+}\right) ,\) \(g\left( t\right) \le t,\) \(g^{\prime }\left( t\right) >0\) and \(\lim _{t\rightarrow \infty }g\left( t\right) =\infty .\)

  3. (H3)

    \(F\in C\left( \mathbb {R},\mathbb {R} \right) ,\) \(F^{\prime }\left( u\right) \ge 0,\) \(uF\left( u\right) >0\) for \( u\ne 0,\) and \(F\left( u\upsilon \right) \ge F\left( u\right) F\left( \upsilon \right) \) for \(u\upsilon >0.\)

By a solution of (1.1), we mean a function \(\upsilon \in C^{n-1}\left( \left[ t_{\upsilon },\infty \right) , \mathbb {R} \right) \), for \(t_{\upsilon }\ge t_{0}\), with \(a\upsilon ^{\left( n-1\right) }\in C^{1}\left( \left[ t_{\upsilon },\infty \right) , \mathbb {R} \right) \), that satisfies (1.1) for all \(t\ge t_{\upsilon }\). We consider only those solutions of (1.1) that do not eventually vanish. A solution \(\upsilon \) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is said to be non-oscillatory [1].

The many applications of DDEs in different sciences were and continue to be the motivation behind the growing interest in studying the qualitative behavior of the solutions of these equations.

In the non-canonical case, it is easy to see how much research has progressed on the oscillatory behavior of the solutions of second-order DDEs. This progress can be traced through the recent results of Baculí ková [2, 3] and Džurina and Jadlovská [4, 5]. They provided improved techniques and sharper criteria for the oscillation of second-order DDE solutions.

In the study of oscillatory behavior, there are two common techniques: Riccati substitution and comparison with first-order equations. In the non-canonical case, Baculíková et al. [6] used the comparison technique to establish the oscillation conditions for the solutions of the DDE

$$\begin{aligned} \left( a\cdot \left( \upsilon ^{(n-1}\right) ^{\alpha }\right) ^{\prime }\left( t\right) +\left( h\cdot \left( F\circ \upsilon \circ g\right) \right) \left( t\right) =0. \end{aligned}$$
(1.2)

On the other hand, Zhang et al. [7] used the Riccati substitution to establish criteria for deciding that all the solutions of the DDE

$$\begin{aligned} \left( a\cdot \left( \upsilon ^{(n-1}\right) ^{\alpha }\right) ^{\prime }\left( t\right) +\left( h\cdot \left( \upsilon ^{\beta }\circ g\right) \right) \left( t\right) =0 \end{aligned}$$
(1.3)

are oscillatory, where \(\alpha \) and \(\beta \) are ratios of odd positive integers. Below, we present two results obtained from the literature, to which we will refer to later.

Theorem 1.1

[6, Theorem 4 with \(F\left( \upsilon \right) =\upsilon ^{\alpha }\)] All solutions of (1.3) are oscillatory if the first-order DDEs

$$\begin{aligned} w^{\prime }\left( t\right) +\left( h\cdot \left( \frac{\epsilon _{1}g^{n-1}}{ \left( n-1\right) !\left( a^{1/\alpha }\circ g\right) }\right) ^{\alpha }\cdot \left( w\circ g\right) \right) \left( t\right) =0, \end{aligned}$$
(1.4)
$$\begin{aligned} w^{\prime }\left( t\right) +\frac{1}{a^{1/\alpha }\left( t\right) }\cdot \left( \int _{t_{0}}^{t}h\left( \varrho \right) \left( \frac{\epsilon _{2}g^{n-2}\left( \varrho \right) }{\left( n-2\right) !}\right) ^{\alpha } \textrm{d}\varrho \right) ^{1/\alpha }\cdot \left( w\circ g\right) \left( t\right) =0\qquad \qquad \end{aligned}$$

are oscillatory, and there is a \(\eta \in C^{1}\left( \left[ t_{0},\infty \right) \right) \) with \(\eta \left( t\right) >t\), \(\eta ^{\prime }\left( t\right) \ge 0\) and \(\left( \eta _{n-2}\circ g\right) \left( t\right) <t\), such that

$$\begin{aligned} w^{\prime }\left( t\right) +\frac{1}{a^{1/\alpha }\left( t\right) }\cdot \left( \int _{t_{0}}^{t}h\left( \varrho \right) \textrm{d}\varrho \right) ^{1/\alpha }\cdot \left( J_{n-2}\circ g\right) \left( t\right) \cdot \left( w\circ \eta _{n-2}\circ g\right) \left( t\right) =0\qquad \qquad \end{aligned}$$

is oscillatory for some \(\epsilon _{1},\epsilon _{2}\in \left( 0,1\right) \), where

$$\begin{aligned} \eta _{1}=\eta ,\text { }\eta _{i+1}=\eta _{i}\circ \eta ,\text { }J_{1}\left( t\right) =\eta -t\text { and }J_{i+1}\left( t\right) =\int _{t}^{\eta }J_{i}\left( \varrho \right) \textrm{d}\varrho , \end{aligned}$$

for \(i=1,2,\dots ,n-3\).

Theorem 1.2

[7, Theorem 2.1] All solutions of (1.3) are oscillatory if the first-order DDE (1.4) is oscillatory and the following conditions hold:

$$\begin{aligned} \underset{t\rightarrow \infty }{\lim \sup }\int _{t_{0}}^{t}\left( \left( \frac{\epsilon _{0}g^{n-2}\left( \varrho \right) }{\left( n-2\right) !} \right) ^{\alpha }h\left( \varrho \right) A_{0}^{\alpha }\left( \varrho \right) -\frac{\alpha ^{*}}{a^{1/\alpha }\left( \varrho \right) A_{0}\left( \varrho \right) }\right) \textrm{d}\varrho =\infty , \end{aligned}$$

and

$$\begin{aligned} \underset{t\rightarrow \infty }{\lim \sup }\int _{t_{0}}^{t}\left( h\left( \varrho \right) B^{\alpha }\left( \varrho \right) -\frac{\alpha ^{*}\left( B^{\prime }\left( \varrho \right) \right) ^{\alpha +1}}{B\left( \varrho \right) B_{1}^{\alpha }\left( \varrho \right) }\right) \textrm{d} \varrho =\infty , \end{aligned}$$

where \(\epsilon _{0}\in \left( 0,1\right) ,\) \(\alpha ^{*}=\left( \alpha /\left( \alpha +1\right) \right) ^{\alpha +1}\), \(A_{0}\left( t\right) =\int _{t}^{\infty }a^{-1/\alpha }\left( \varrho \right) \textrm{d}\varrho \),

$$\begin{aligned} B\left( t\right) =\frac{1}{\left( n-3\right) !}\int _{t}^{\infty }\left( \varrho -t\right) ^{n-3}A_{0}\left( \varrho \right) \textrm{d}\varrho \end{aligned}$$

and

$$\begin{aligned}B_{1}\left( t\right) =\frac{1}{\left( n-4\right) !}\int _{t}^{\infty }\left( \varrho -t\right) ^{n-4}A_{0}\left( \varrho \right) \textrm{d} \varrho . \end{aligned}$$

We note here that the linear delay equation

$$\begin{aligned} \upsilon ^{\left( n\right) }\left( t\right) +\left( h\cdot \left( \upsilon \circ g\right) \right) \left( t\right) =0 \end{aligned}$$

has been studied by Koplatadze et al. [8]. They took into account the odd- and even-order cases of this equation.

Very recently, Moaaz et al. [9] extended the results about the second-order equations to even-order equations in the non-canonical case. They adopted a strategy that involved new monotonic properties for positive decreasing solutions and used those properties to iteratively develop new oscillation criteria.

This study aims to establish a new criterion to determine the oscillation of all solutions of Eq. (1.1) in the non-canonical case. The approach followed is an extension of the approach used by Koplatadze et al. [8] and later by Baculíková [2] to obtain an effective oscillation criterion for second-order equations. The new criterion ensures that Eq. (1.1) is oscillatory without the need to check the additional condition (1.4), which has traditionally been imposed on all previous related results. In addition, the new criterion also introduces a measure of oscillation that is sharper than previous results in the literature.

2 Main Results

Using Lemma 2.2.1 in [10], we can classify the positive solutions of (1.1) as follows:

Lemma 2.1

[11, Lemma 3] Suppose that \(\upsilon \) is an eventually positive solution of (1.1). Then, \(\left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) \le 0\), and there are eventually the following three cases:

\(\left( C_{1}\right) \):

\(\upsilon ^{\prime }\) and \(\upsilon ^{\left( n-1\right) }\ \)are positive, and \(\upsilon ^{\left( n\right) }\ \)is negative;

\(\left( C_{2}\right) \):

\(\upsilon ^{\prime }\) and \(\upsilon ^{\left( n-2\right) }\ \)are positive, and \(\upsilon ^{\left( n-1\right) }\ \)is negative;

\(\left( C_{3}\right) \):

\(\left( -1\right) ^{m}\upsilon ^{\left( m\right) }\) are positive, for \(m=1,\dots ,n-1.\)

Lemma 2.2

Suppose that \(\upsilon \) is an eventually positive solution of (1.1) and satisfies \(\left( C_{3}\right) \). Then, there is a \(t_{1}\ge t_{0}\) such that

$$\begin{aligned} \frac{\upsilon \left( g\left( t\right) \right) }{F\left( \upsilon \left( g\left( t\right) \right) \right) }\ge & {} A_{n-2}\left( g\left( t\right) \right) \int _{t_{1}}^{g\left( t\right) }h\left( \varrho \right) \textrm{d} \varrho +\int _{g\left( t\right) }^{t}A_{n-2}\left( \varrho \right) h\left( \varrho \right) \textrm{d}\varrho \nonumber \\{} & {} +F\left( A_{n-2}^{-1}\left( g\left( t\right) \right) \right) \int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( A_{n-2}\left( g\left( \varrho \right) \right) \right) \textrm{d} \varrho . \nonumber \\ \end{aligned}$$
(2.1)

Proof

Since \(\upsilon \) is an eventually positive solution of (1.1) and satisfies \(\left( C_{3}\right) \), there is a \(t_{1}\ge t_{0}\) such that \( \upsilon \left( t\right) >0\) and \(\upsilon \left( g\left( t\right) \right) >0 \) for all \(t\ge t_{1}\). We also have

$$\begin{aligned} \int _{t}^{\infty }\upsilon ^{\left( n-1\right) }\left( \varrho \right) \textrm{d}\varrho =\lim _{s\rightarrow \infty }\upsilon ^{\left( n-2\right) }\left( s\right) -\upsilon ^{\left( n-2\right) }\left( t\right) \ge -\upsilon ^{\left( n-2\right) }\left( t\right) . \end{aligned}$$

Furthermore, according to Lemma 2.1 it is \(\left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) \le 0\), and thus we have

$$\begin{aligned} -\upsilon ^{\left( n-2\right) }\left( t\right)\le & {} \int _{t}^{\infty } \frac{1}{a\left( \varrho \right) }a\left( \varrho \right) \upsilon ^{\left( n-1\right) }\left( \varrho \right) \textrm{d}\varrho \le a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right) \int _{t}^{\infty }\frac{1}{ a\left( \varrho \right) }\textrm{d}\varrho \nonumber \\= & {} a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right) A_{0}\left( t\right) . \end{aligned}$$
(2.2)

Integrating this inequality over \(\left[ t,\infty \right) \), we obtain

$$\begin{aligned} \upsilon ^{\left( n-3\right) }\left( t\right)\le & {} \int _{t}^{\infty }a\left( \varrho \right) \upsilon ^{\left( n-1\right) }\left( \varrho \right) A_{0}\left( \varrho \right) \textrm{d}\varrho \\\le & {} a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right) \int _{t}^{\infty }A_{0}\left( \varrho \right) \textrm{d}\varrho \\= & {} \left( a\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) \cdot A_{1}\left( t\right) . \end{aligned}$$

Integrating this inequality \(n-3\) times over \(\left[ t,\infty \right) \), and taking into account the behavior of the derivatives of \(\upsilon \) in \( \left( C_{3}\right) \), we conclude that

$$\begin{aligned} \upsilon ^{\prime }\left( t\right) \le \left( a\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) \cdot A_{n-3}\left( t\right) \end{aligned}$$
(2.3)

and

$$\begin{aligned} \upsilon \left( t\right) \ge -\left( a\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) \cdot A_{n-2}\left( t\right) . \end{aligned}$$
(2.4)

Next, we define

$$\begin{aligned} H\left( t\right) :=\left( a\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) \cdot A_{n-2}\left( t\right) +\upsilon \left( t\right) . \end{aligned}$$

From (2.4), we have that \(H\left( t\right) \ge 0\) for \(t\ge t_{1}\). Using (2.3) and (1.1), we have

$$\begin{aligned} H^{\prime }\left( t\right)= & {} \left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) \cdot A_{n-2}\left( t\right) -a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right) A_{n-3}\left( t\right) +\upsilon ^{\prime }\left( t\right) \\\le & {} \left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) \cdot A_{n-2}\left( t\right) \\= & {} -A_{n-2}\left( t\right) \cdot h\left( t\right) \cdot \left( F\circ \upsilon \circ g\right) \left( t\right) \le 0. \end{aligned}$$

Integrating this inequality over \(\left[ t,\infty \right) \), we arrive at

$$\begin{aligned} H\left( t\right) \ge \int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho . \end{aligned}$$
(2.5)

Integrating (1.1) from \(t_{1}\) to t, we find that

$$\begin{aligned} a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right)= & {} a\left( t_{1}\right) \upsilon ^{\left( n-1\right) }\left( t_{1}\right) -\int _{t_{1}}^{t}h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\< & {} -\int _{t_{1}}^{t}h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho . \end{aligned}$$
(2.6)

From (2.5) and (2.6), we obtain

$$\begin{aligned} \upsilon \left( t\right)\ge & {} -a\left( t\right) \upsilon ^{\left( n-1\right) }\left( t\right) A_{n-2}\left( t\right) +\int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \\> & {} A_{n-2}\left( t\right) \int _{t_{1}}^{t}h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\{} & {} +\int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d} \varrho . \end{aligned}$$

Thus, we have

$$\begin{aligned} \upsilon \left( g\left( t\right) \right)> & {} A_{n-2}\left( g\left( t\right) \right) \int _{t_{1}}^{g\left( t\right) }h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\{} & {} +\int _{g\left( t\right) }^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\= & {} A_{n-2}\left( g\left( t\right) \right) \int _{t_{1}}^{g\left( t\right) }h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\{} & {} +\int _{g\left( t\right) }^{t}A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \nonumber \\{} & {} +\int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d} \varrho . \end{aligned}$$
(2.7)

On the other hand, from (2.2), we get

$$\begin{aligned} \left( \frac{\upsilon ^{\left( n-2\right) }}{A_{0}}\right) ^{\prime }=\frac{1 }{A_{0}^{2}\left( t\right) }\left( \left( A_{0}\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) +\left( a^{-1}\cdot \upsilon ^{\left( n-2\right) }\right) \left( t\right) \right) \ge 0, \end{aligned}$$
(2.8)

which leads to

$$\begin{aligned} -\upsilon ^{\left( n-3\right) }\left( t\right) \ge \int _{t}^{\infty }A_{0}\left( \varrho \right) \frac{\upsilon ^{\left( n-2\right) }\left( \varrho \right) }{A_{0}\left( \varrho \right) }\textrm{d}\varrho \ge \frac{ \upsilon ^{\left( n-2\right) }\left( t\right) }{A_{0}\left( t\right) } A_{1}\left( t\right) . \end{aligned}$$

This implies

$$\begin{aligned} \left( \frac{\upsilon ^{\left( n-3\right) }}{A_{1}}\right) ^{\prime }\left( t\right) =\frac{1}{A_{1}^{2}\left( t\right) }\left( \left( A_{1}\cdot \upsilon ^{\left( n-2\right) }\right) \left( t\right) +\left( A_{0}\cdot \upsilon ^{\left( n-3\right) }\right) \left( t\right) \right) \le 0. \end{aligned}$$

Repeating the same procedure \(n-3\) times, we obtain that \(\left( \frac{ \upsilon }{A_{n-2}}\right) ^{\prime }\left( t\right) \ge 0\). Hence, \( \upsilon \left( g\left( \varrho \right) \right) \ge \frac{A_{n-2}\left( g\left( \varrho \right) \right) }{A_{n-2}\left( g\left( t\right) \right) } \upsilon \left( g\left( t\right) \right) ,\) for \(t\le \varrho \). Then, from (H3), we have

$$\begin{aligned} F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \ge F\left( A_{n-2}^{-1}\left( g\left( t\right) \right) \right) F\left( A_{n-2}\left( g\left( \varrho \right) \right) \right) F\left( \upsilon \left( g\left( t\right) \right) \right) ,\text { for }t\le \varrho . \nonumber \\ \end{aligned}$$
(2.9)

Using the fact that \(\upsilon ^{\prime }\left( t\right) <0\) and (2.9), the inequality (2.7) becomes

$$\begin{aligned} \frac{\upsilon \left( g\left( t\right) \right) }{F\left( \upsilon \left( g\left( t\right) \right) \right) }\ge & {} A_{n-2}\left( g\left( t\right) \right) \int _{t_{1}}^{g\left( t\right) }h\left( \varrho \right) \textrm{d} \varrho +\int _{g\left( t\right) }^{t}A_{n-2}\left( \varrho \right) h\left( \varrho \right) \textrm{d}\varrho \\{} & {} +F\left( A_{n-2}^{-1}\left( g\left( t\right) \right) \right) \int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( A_{n-2}\left( g\left( \varrho \right) \right) \right) \textrm{d} \varrho , \end{aligned}$$

and the proof is complete. \(\square \)

Theorem 2.3

Suppose that \(\lim _{\omega \rightarrow 0}\frac{\omega }{F\left( \omega \right) }=L<\infty \) and

$$\begin{aligned}{} & {} \underset{t\rightarrow \infty }{\lim \sup }\left[ A_{n-2}\left( g\left( t\right) \right) \int _{t_{0}}^{g\left( t\right) }h\left( \varrho \right) \textrm{d}\varrho +\int _{g\left( t\right) }^{t}A_{n-2}\left( \varrho \right) h\left( \varrho \right) \textrm{d}\varrho \right. \nonumber \\{} & {} \left. \left. +F\left( A_{n-2}^{-1}\left( g\left( t\right) \right) \right) \int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) F\left( A_{n-2}\left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho \right] >L\right. . \qquad \qquad \end{aligned}$$
(2.10)

If for some \(\epsilon _{0}\in \left( 0,1\right) \), the DDE

$$\begin{aligned}{} & {} \left( a\left( t\right) A_{0}^{2}\left( t\right) w^{\prime }\left( t\right) \right) ^{\prime }\nonumber \\ {}{} & {} \quad +A_{0}\left( t\right) h\left( t\right) F\left( \frac{ \epsilon _{0}}{\left( n-2\right) !}g^{n-2}\left( t\right) \right) F\left( A_{0}\left( g\left( t\right) \right) w\left( g\left( t\right) \right) \right) =0 \qquad \qquad \end{aligned}$$
(2.11)

is oscillatory, then all solutions of (1.1) are oscillatory.

Proof

We proceed by contradiction. Let us assume that \(\upsilon \) is an eventually positive solution of (1.1). Then, there is a \(t_{1}\ge t_{0}\), such that \(\upsilon \left( t\right) >0\) and \(\upsilon \left( g\left( t\right) \right) >0\) for all \(t\ge t_{1}\). It follows from Lemma 2.1 that \(\upsilon \) satisfies one of the cases \(\left( C_{1}\right) -\left( C_{3}\right) \).

Assume that case \(\left( C_{1}\right) \) holds. Proceeding similarly as in the proof of Theorem 1 in [2], we can prove that (2.10) implies that

$$\begin{aligned} \int _{t_{1}}^{\infty }h\left( \varrho \right) A_{2}\left( \varrho \right) \textrm{d}\varrho =\infty . \end{aligned}$$
(2.12)

Integrating (1.1) from \(t_{1}\) to \(\infty \) and using the fact that \( \upsilon ^{\left( n-1\right) }\) is a decreasing positive function, we find that

$$\begin{aligned} a\left( t_{1}\right) \upsilon ^{\left( n-1\right) }\left( t_{1}\right) \ge \int _{t_{1}}^{\infty }h\left( \varrho \right) F\left( \upsilon \left( g\left( \varrho \right) \right) \right) \textrm{d}\varrho . \nonumber \end{aligned}$$

Since \(\upsilon \left( t\right) >0\) and \(\upsilon ^{\prime }\left( t\right) >0\), there is a \(t_{2}\ge t_{1}\), such that \(\upsilon \left( g\left( t\right) \right) >l\) for \(t\ge t_{1}\). Then, from (H3), we arrive at

$$\begin{aligned} a\left( t_{1}\right) \upsilon ^{\left( n-1\right) }\left( t_{1}\right) \ge F\left( l\right) \int _{t_{1}}^{\infty }h\left( \varrho \right) \textrm{d} \varrho \ge F\left( l\right) \int _{t_{1}}^{\infty }h\left( \varrho \right) A_{2}\left( \varrho \right) \textrm{d}\varrho , \end{aligned}$$

which contradicts (2.12).

Assume that case \(\left( C_{2}\right) \) holds. We have

$$\begin{aligned}{} & {} \left( a\left( t\right) \cdot A_{0}^{2}\left( t\right) \cdot \left( \frac{ \upsilon ^{\left( n-2\right) }}{A_{0}}\right) ^{\prime }\left( t\right) \right) ^{\prime } \nonumber \\{} & {} \quad = \left( a\left( t\right) \cdot A_{0}^{2}\left( t\right) \cdot \left( \frac{\left( A_{0}\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) +\left( a^{-1}\cdot \upsilon ^{\left( n-2\right) }\right) \left( t\right) }{A_{0}^{2}\left( t\right) }\right) \right) ^{\prime } \nonumber \\{} & {} \quad = \left( A_{0}\left( t\right) \cdot \left( a\cdot \upsilon ^{\left( n-1\right) }\right) \left( t\right) +\upsilon ^{\left( n-2\right) }\left( t\right) \right) ^{\prime } \nonumber \\{} & {} \quad = A_{0}\left( t\right) \cdot \left( a\cdot \upsilon ^{\left( n-1\right) }\right) ^{\prime }\left( t\right) , \end{aligned}$$
(2.13)

which, by virtue of (1.1), yields that

$$\begin{aligned} \left( a\left( t\right) \cdot A_{0}^{2}\left( t\right) \cdot \left( \frac{ \upsilon ^{\left( n-2\right) }}{A_{0}}\right) ^{\prime }\left( t\right) \right) ^{\prime }+A_{0}\left( t\right) \cdot h\left( t\right) \cdot \left( F\circ \upsilon \circ g\right) \left( t\right) =0. \nonumber \\ \end{aligned}$$
(2.14)

From Lemma [10, Lemma 2.2.3], we get

$$\begin{aligned} \left( \upsilon \circ g\right) \left( t\right) \ge \frac{\epsilon _{0}}{ \left( n-2\right) !}g^{n-2}\left( t\right) \cdot \left( \upsilon ^{\left( n-2\right) }\circ g\right) \left( t\right) , \end{aligned}$$
(2.15)

for all \(\epsilon _{0}\in \left( 0,1\right) \). Combining (2.15) and (2.14) and using the transformation \(\upsilon ^{\left( n-2\right) }\left( t\right) =A_{0}\left( t\right) w\left( t\right) \), we find that

$$\begin{aligned}{} & {} \left( a\left( t\right) A_{0}^{2}\left( t\right) w^{\prime }\left( t\right) \right) ^{\prime }\nonumber \\ {}{} & {} \quad +A_{0}\left( t\right) h\left( t\right) F\left( \frac{ \epsilon _{0}}{\left( n-2\right) !}g^{n-2}\left( t\right) \right) F\left( A_{0}\left( g\left( t\right) \right) w\left( g\left( t\right) \right) \right) \le 0. \end{aligned}$$

It follows from Corollary 1 in [12] that the DDE (2.11) also has a positive solution, which is a contradiction.

Assume that case \(\left( C_{3}\right) \) holds. From Lemma 2.2, we have that (2.1) holds, which contradicts the hypothesis (2.10). The proof is complete. \(\square \)

Corollary 2.4

Suppose that \(F\left( u\right) =u\). Then, conditions

$$\begin{aligned}{} & {} \underset{t\rightarrow \infty }{\lim \sup }\left[ A_{n-2}\left( g\left( t\right) \right) \int _{t_{0}}^{g\left( t\right) }h\left( \varrho \right) \textrm{d}\varrho +\int _{g\left( t\right) }^{t}A_{n-2}\left( \varrho \right) h\left( \varrho \right) \textrm{d}\varrho \right. \nonumber \\{} & {} \quad \left. \left. +A_{n-2}^{-1}\left( g\left( t\right) \right) \int _{t}^{\infty }A_{n-2}\left( \varrho \right) h\left( \varrho \right) A_{n-2}\left( g\left( \varrho \right) \right) \textrm{d}\varrho \right] >1\right. , \end{aligned}$$
(2.16)

and for some \(\epsilon _{0}\in \left( 0,1\right) \)

$$\begin{aligned}{} & {} \underset{t\rightarrow \infty }{\lim \inf }\int _{g\left( t\right) }^{t}g^{n-2}\left( s\right) h\left( s\right) A_{0}\left( s\right) A_{0}\left( g\left( s\right) \right) \left( \int _{t_{0}}^{g\left( s\right) } \frac{\textrm{d}\varrho }{a\left( \varrho \right) A_{0}^{2}\left( \varrho \right) }\right) \textrm{d}s \nonumber \\ {}{} & {} \quad >\frac{\left( n-2\right) !}{\epsilon _{0}\textrm{ e}}, \end{aligned}$$
(2.17)

guarantee that all solutions of (1.1) are oscillatory.

Proof

The proof is the same as the proof of Theorem 2.3. It is enough just to know from Theorem 4 in [13] that the criterion (2.17) guarantees that (2.11) is oscillatory. \(\square \)

3 Examples

We present two examples taken from the literature to compare different oscillation criteria, showing that Theorem 2.1 provides the sharpest results.

Example 3.1

Consider the DDE

$$\begin{aligned} \left( \textrm{e}^{t}\upsilon ^{\prime \prime \prime }\left( t\right) \right) ^{\prime }+h_{0}\textrm{e}^{t}\upsilon \left( t-\lambda \right) =0, \end{aligned}$$
(3.1)

where \(\lambda >0\) and \(h_{0}>0\). It is easy to check that \(A_{i}\left( t\right) =\textrm{e}^{-t},\) for \(i=0,1,2\). Furthermore, conditions (2.17 ) and (2.16) reduce to

$$\begin{aligned} \underset{t\rightarrow \infty }{\lim \inf }\int _{t-\lambda }^{t}h_{0}\left( s-\lambda \right) ^{2}\left( \textrm{1}-e^{t_{0}}\textrm{e}^{-s+\lambda }\right) \textrm{d}s=\infty \end{aligned}$$

and

$$\begin{aligned} h_{0}>\frac{1}{2+\lambda }, \end{aligned}$$
(3.2)

respectively. Then, according to Corollary 2.4, every solution of (3.1) is oscillatory if (3.2) holds.

Remark 3.1

From Example 3 in [6], Eq. (3.1), when \(\lambda =1\), is oscillatory if \(h_{0}>2^{5}/\textrm{e}\). However, from Example 3.1, the finer bound \(h_{0}>1/3\) guarantees that for \(\lambda =1\), Eq. (3.1) is oscillatory. Moreover, for all \(\lambda <2\), the results in [7] give the most efficient condition as \(h_{0}<0.25\), while our results support the most efficient condition for all \(\lambda >2\). On the other hand, the results in [6, 7] do not take into account the impact of \( \lambda \).

Example 3.2

Consider the DDE of the Euler type

$$\begin{aligned} \left( t^{4}\upsilon ^{\prime \prime \prime }\left( t\right) \right) ^{\prime }+h_{0}\upsilon \left( \lambda t\right) =0, \end{aligned}$$
(3.3)

where \(\lambda \in \left( 0,1\right) \) and \(h_{0}>0\). It is easy to check that \(A_{0}\left( t\right) =\frac{1}{3}t^{-3},\) \(A_{1}\left( t\right) =\frac{ 1}{6}t^{-2},\) \(A_{2}\left( t\right) =\frac{1}{6}t^{-1}\). Conditions (2.16 ) and (2.17) reduce to \(\frac{1}{6}h_{0}\left( 2+\ln \frac{1}{\lambda } \right) >1\), and \(h_{0}\lambda ^{2}\ln \frac{1}{\lambda }>\frac{6}{\textrm{e} }\), respectively. Then, from Corollary 2.4, every solution of (3.3) is oscillatory if

$$\begin{aligned} h_{0}>\frac{6}{\textrm{e}\lambda ^{2}\ln \frac{1}{\lambda }}. \end{aligned}$$
(3.4)

Remark 3.2

By choosing \(\eta \left( t\right) =ct\), where \(c=\left( 1+\lambda ^{-1/2}\right) /2\), we can apply Theorem 1.1 to Example 3.2. Then, Eq. (3.3) is oscillatory if

$$\begin{aligned} h_{0}>\max \left\{ \frac{6\lambda }{\textrm{e}\ln \frac{1}{\lambda }},\frac{6 }{\textrm{e}\lambda ^{2}\ln \frac{1}{\lambda }},\frac{2}{\textrm{e}\lambda ^{2}\left( c-1\right) \left( c^{2}-1\right) \ln \frac{1}{c^{2}\lambda }} \right\} . \end{aligned}$$
(3.5)

We note the difficulty of obtaining an unknown function \(\eta \) that satisfies the conditions of Theorem 1.1. On the other hand, from Theorem 1.2, Eq. (3.3) is oscillatory if

$$\begin{aligned} h_{0}>\max \left\{ \frac{6\lambda }{\textrm{e}\ln \frac{1}{\lambda }},\frac{9 }{2\lambda ^{2}},\frac{3}{2}\right\} . \end{aligned}$$
(3.6)

If we set \(\lambda =1/2\), then conditions (3.5) and (3.6) reduce to \(h_{0}>98.162\) and \(h_{0}>18\), respectively, while condition (3.4) gives \(h_{0}>12.738\).

4 Conclusion

The study of the oscillatory behavior of solutions of higher order differential equations depends on finding conditions that exclude positive solutions. By extending the results in [8] to higher order equations, we have established a new condition that excludes positive decreasing solutions. Our results involve two conditions to ensure that all solutions of (1.1) are oscillatory, while all previous results need three assumptions. Through some examples, we show that our results improve other results in the literature. As future work, it would be interesting to extend the obtained result to Eq. (1.3), as well as to the neutral case. In addition, as an open problem, it is proposed to extend our results to equations of odd order in the non-canonical or non-linear cases.