On the Asymptotic Behavior of Advanced Differential Equations with a Non-Canonical Operator

Abstract

In this paper, we aim to study the oscillatory behavior of a class of even-order advanced differential equations with a non-canonical operator. In addition, we present results on the asymptotic behavior of this type of equations and provide an example that illustrates our main results.

1. Introduction

In recent decades, many authors have studied problems of a number of different classes of advanced differential equations including the asymptotic and oscillatory behavior of their solutions, see [1,2,3,4,5,6,7,8] and the references cited therein. For some more recent oscillation results, see [9,10,11,12,13,14,15,16,17,18,19,20]. The interest in studying advanced differential equations is also caused by the fact that they appear in models of several areas in science. In [21,22,23], singular systems of differential equations are used to study the dynamics and stability properties of electrical power systems. Some additional mathematical background on this can be found in [24]. Systems of differential equations with delays are used to study additional properties of electrical power systems in [25,26]. Non-linear advanced differential equations can be used to describe complex dynamical networks, see [27,28,29], and bring new insight to their stability. Furthermore, this type of equations can be also used in the modeling of dynamical networks of interacting free-bodies, see [30]. Finally, properties of advanced differential equations are used in the study of singular differential equations of fractional order, see [31,32]. Several other examples in Physics can be found in [33]. In this paper, we consider an even-order non-linear advanced differential equation with a non-canonical operator of the following type:

$$L_y+q\left(\upsilon \right)g\left(y\left(\eta \left(\upsilon \right)\right)\right)=0,L_y:={\left(a\left(\upsilon \right){\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime },$$

(1)

where $\upsilon \ge {\upsilon }_0,$ $\kappa$ is even and $\beta$ is a quotient of odd positive integers. The operator $L_y$ is said to be in canonical form if ${\int }_{{\upsilon }_0}^{\infty }{a}^{-1/\beta }\left(s\right)\mathrm{d}s=\infty$; otherwise, it is called noncanonical. Throughout this work, we suppose that:

C1:

$a\in C^1\left([{\upsilon }_0,\infty ),\mathbb{R}\right),$$a\left(\upsilon \right)>0,a^{\prime }\left(\upsilon \right)\ge 0,$

C2:

$q,\eta \in C\left([{\upsilon }_0,\infty ),\mathbb{R}\right),$$q\left(\upsilon \right)\ge 0,\eta \left(\upsilon \right)\ge \upsilon ,$${\displaystyle \underset{\upsilon \to \infty }{lim}}\eta \left(\upsilon \right)=\infty ,$

C3:

$g\in C\left(\mathbb{R},\mathbb{R}\right)$ such that $g\left(x\right)/x^{\beta }\ge k>0,$for $x\ne 0$ and under the condition

$$\zeta \left(\upsilon \right)={\int }_{{\upsilon }_0}^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}\mathrm{d}s<\infty .$$

(2)

Definition 1.

The function y$\in C^{\kappa -1}[{\upsilon }_y,\infty ),{\upsilon }_y\ge {\upsilon }_0,$is called a solution of (1), if ${\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\in C^1[{\upsilon }_y,\infty ),$for $a\in C^1\left([{\upsilon }_0,\infty ),\mathbb{R}\right),a\left(\upsilon \right)>0$ and $y\left(\upsilon \right)$ satisfies (1) on $[{\upsilon }_y,\infty )$.

Definition 2.

Let

$$D=\{\left(\upsilon ,s\right)\in {\mathbb{R}}^2:\upsilon \ge s\ge {\upsilon }_0\}\mathrm{and}{D}_0=\{\left(\upsilon ,s\right)\in {\mathbb{R}}^2:\upsilon >s\ge {\upsilon }_0\}.$$

A kernel function $H_i\in C\left(D,\mathbb{R}\right)$ is said to belong to the function class ℑ, written by $H\in \Im$, if, for $i=1,2$,

(i) 

$H_i\left(\upsilon ,s\right)>0,$on $D_0$and $H_i\left(\upsilon ,s\right)=0$ for $\upsilon \ge {\upsilon }_0$with $\left(\upsilon ,s\right)\notin D_0;$

(ii) 

$H_i\left(\upsilon ,s\right)$ has a continuous and nonpositive partial derivative $\partial H_i/\partial s$ on $D_0$ and there exist functions $\tau ,\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$ and $h_i\in C\left(D_0,\mathbb{R}\right)$ such that

$$\frac{\partial }{\partial s}{H}_1\left(\upsilon ,s\right)+\frac{{\tau }^{\prime }\left(s\right)}{\tau \left(s\right)}{H}_1\left(\upsilon ,s\right)=h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)$$

(3)

and

$$\frac{\partial }{\partial s}{H}_2\left(\upsilon ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right)=h_2\left(\upsilon ,s\right)\sqrt{H_2\left(\upsilon ,s\right)}.$$

(4)

Next we will discuss the results in [34,35,36]. Actually, our purpose in this article is to complement and improve these results. Agarwal et al. in [34,35] studied the even-order nonlinear advanced differential equations

$${\left({\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+q\left(\upsilon \right)y^{\beta }\left(\eta \left(\upsilon \right)\right)=0.$$

(5)

By means of the Riccati transformation technique, the authors established some oscillation criteria of (5). Grace and Lalli [36] investigated the second-order neutral Emden–Fowler delay dynamic equations

$$y^{\left(\kappa \right)}\left(\upsilon \right)+q\left(\upsilon \right)y\left(\eta \left(\upsilon \right)\right)=0,$$

(6)

and established some new oscillation for (5) under the condition

$${\int }_{{\upsilon }_0}^{\infty }\frac{1}{a^{1/\beta }\left(s\right)}\mathrm{d}s=\infty .$$

(7)

To prove this, we apply the previous results to the equation

$$y^{\left(\kappa \right)}\left(\upsilon \right)+\frac{q_0}{{\upsilon }^{\kappa }}y\left(\lambda \upsilon \right)=0,\upsilon \ge 1.$$

(8)

if we set $\kappa =4$and $\lambda =2,$then by applying conditions in [34,35,36] on Equation (8), we find the results in [35] improves those in [36]. Moreover, the those in [34] improves results in [35,36]. Thus, the motivation in our paper is to complement and improve results in [34,35,36]. We will use the following methods:

• Integral averaging technique.
• Riccati transformations technique.
• Method of comparison with second-order differential equations.

We will also use the following lemmas from (1):

Lemma 1

([3]). If $y^{\left(i\right)}\left(\upsilon \right)>0,$ $i=0,1,\dots ,\kappa ,$ and $y^{\left(\kappa +1\right)}\left(\upsilon \right)<0,$ then

$$\frac{y\left(\upsilon \right)}{{\upsilon }^{\kappa }/\kappa !}\ge \frac{y^{\prime }\left(\upsilon \right)}{{\upsilon }^{\kappa -1}/\left(\kappa -1\right)!}.$$

Lemma 2

([19]). Suppose that $y\in C^{\kappa }\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right),$ $y^{\left(\kappa \right)}$ is of a fixed sign on $\left[{\upsilon }_0,\infty \right),$ $y^{\left(\kappa \right)}$ not identically zero and there exists a ${\upsilon }_1\ge {\upsilon }_0$ such that

$$y^{\left(\kappa -1\right)}\left(\upsilon \right)y^{\left(\kappa \right)}\left(\upsilon \right)\le 0,$$

for all $\upsilon \ge {\upsilon }_1$. If we have ${lim}_{\upsilon \to \infty }y\left(\upsilon \right)\ne 0$, then there exists ${\upsilon }_{\theta }\ge {\upsilon }_1$ such that

$$y\left(\upsilon \right)\ge \frac{\theta }{\left(\kappa -1\right)!}{\upsilon }^{\kappa -1}\left|y^{\left(\kappa -1\right)}\left(\upsilon \right)\right|,$$

for every $\theta \in \left(0,1\right)$ and $\upsilon \ge {\upsilon }_{\theta }$.

Lemma 3

([2]). Let $\beta$bea ratio of two odd numbers, $V>0$ and U are constants. Then

$$Ux-V{x}^{\left(\beta +1\right)/\beta }\le \frac{{\beta }^{\beta }}{{(\beta +1)}^{\beta +1}}\frac{U^{\beta +1}}{V^{\beta }},V>0.$$

Lemma 4.

Suppose that $y$is an eventually positive solution of (1). Then, there exist three possible cases:

$$\begin{gathered} \begin{array}{ll}\left({\mathbf{S}}_1\right)& y\left(\upsilon \right)>0,y^{\prime }\left(\upsilon \right)>0,y^{″}\left(\upsilon \right)>0,y^{\left(\kappa -1\right)}\left(\upsilon \right)>0,y^{\left(\kappa \right)}\left(\upsilon \right)<0,\\ \left({\mathbf{S}}_2\right)& y\left(\upsilon \right)>0,y^{\left(r\right)}\left(\upsilon \right)>0,y^{(r+1)}\left(\upsilon \right)< 0 for all odd integer \\ r\in \{1,3,\dots ,\kappa -3\},y^{(\kappa -1)}\left(\upsilon \right)>0,y^{\left(\kappa \right)}\left(\upsilon \right)<0,\\ \left({\mathbf{S}}_3\right)& y\left(\upsilon \right)>0,y^{\left(\kappa -2\right)}\left(\upsilon \right)>0,y^{\left(\kappa -1\right)}\left(\upsilon \right)<0,L_y\le 0,\end{array} \end{gathered}$$

for$\upsilon \ge {\upsilon }_1,$where${\upsilon }_1\ge {\upsilon }_0$is sufficiently large.

2. Oscillation Criteria

Theorem 1.

Assume that (2) holds. If the differential equations

$${\left(\frac{\left(\kappa -2\right)!a^{\frac{1}{\beta }}\left(\upsilon \right)}{{\left(\theta {\upsilon }^{\kappa -2}\right)}^{\beta }}{\left(y^{\prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+kq\left(\upsilon \right)y^{\beta }\left(\upsilon \right)=0,\forall \theta \in (0,1),$$

(9)

$$y^{″}\left(\upsilon \right)+y\left(\upsilon \right)\frac{1}{\left(\kappa -4\right)!}{\int }_{\upsilon }^{\infty }{\left(\varsigma -\upsilon \right)}^{\kappa -4}{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma =0,$$

(10)

and

$${\left(a\left(\upsilon \right){\left(y^{\prime }\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+y^{\beta }\left(\upsilon \right)kq\left(\upsilon \right){\left(\frac{\zeta \left(\eta \left(\upsilon \right)\right)}{\zeta \left(\upsilon \right)}\right)}^{\beta }{\left(\frac{{\theta }_1}{\left(\kappa -2\right)!}{\eta }^{\kappa -2}\left(\upsilon \right)\right)}^{\beta }=0,{\theta }_1\in (0,1)$$

(11)

are oscillatory for every constant $\theta ,{\theta }_1\in (0,1)$, then every solution of (1) is either oscillatory or satisfies $\underset{\upsilon \to \infty }{lim}y\left(\upsilon \right)=0$.

Proof. 

Assume to the contrary that y is a positive solution of (1). Then, we can suppose that $y\left(\upsilon \right)$and $y\left(\eta \left(\upsilon \right)\right)$ are positive for all $\upsilon \ge {\upsilon }_1$ sufficiently large. From Lemma 4, we have three possible cases $\left({\mathbf{S}}_1\right)$, $\left({\mathbf{S}}_2\right)$ and $\left({\mathbf{S}}_3\right)$. Let case $\left({\mathbf{S}}_1\right)$ hold. Using Lemma 2, we find

$$y^{\prime }\left(\upsilon \right)\ge \frac{\theta }{\left(\kappa -2\right)!}{\upsilon }^{\kappa -2}{y}^{(\kappa -1)}\left(\upsilon \right),$$

(12)

for every $\theta \in (0,1)$ and for all large $\upsilon$. We set

$$\phi \left(\upsilon \right):=\tau \left(\upsilon \right)\left(\frac{a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }}{y^{\beta }\left(\upsilon \right)}\right),$$

(13)

and observe that $\phi \left(\upsilon \right)>0$for$\upsilon \ge {\upsilon }_1,$where$\tau \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$and

$$\begin{array}{lll}{\phi }^{\prime }\left(\upsilon \right)& =& {\tau }^{\prime }\left(\upsilon \right)\frac{a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }}{y^{\beta }\left(\upsilon \right)}+\tau \left(\upsilon \right)\frac{{\left(a{\left(y^{(\kappa -1)}\right)}^{\beta }\right)}^{\prime }\left(\upsilon \right)}{y^{\beta }\left(\upsilon \right)}\\ & & -\beta \tau \left(\upsilon \right)\frac{y^{\beta -1}\left(\upsilon \right)y^{\prime }\left(\upsilon \right)a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }}{y^{2\beta }\left(\upsilon \right)}.\end{array}$$

Using (12) and (13), we obtain

$$\begin{array}{lll}{\phi }^{\prime }\left(\upsilon \right)& \le & \frac{{\tau }_{+}^{\prime }\left(\upsilon \right)}{\tau \left(\upsilon \right)}\phi \left(\upsilon \right)+\tau \left(\upsilon \right)\frac{{\left(a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{y^{\beta }\left(\upsilon \right)}\\ & & -\beta \tau \left(\upsilon \right)\frac{\theta }{\left(\kappa -2\right)!}{\upsilon }^{\kappa -2}\frac{a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta +1}}{y^{\beta +1}\left(\upsilon \right)}\\ & \le & \frac{{\tau }^{\prime }\left(\upsilon \right)}{\tau \left(\upsilon \right)}\phi \left(\upsilon \right)+\tau \left(\upsilon \right)\frac{{\left(a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{y^{\beta }\left(\upsilon \right)}\\ & & -\frac{\beta \theta {\upsilon }^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.\end{array}$$

(14)

From (1) and (14), we obtain

$${\phi }^{\prime }\left(\upsilon \right)\le \frac{{\tau }^{\prime }\left(\upsilon \right)}{\tau \left(\upsilon \right)}\phi \left(\upsilon \right)-k\tau \left(\upsilon \right)\frac{q\left(\upsilon \right)y^{\beta }\left(\eta \left(\upsilon \right)\right)}{y^{\beta }\left(\upsilon \right)}-\frac{\beta \theta {\upsilon }^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.$$

Note that $y^{\prime }\left(\upsilon \right)>0$ and $\eta \left(\upsilon \right)\ge \upsilon ,$ thus, we find

$${\phi }^{\prime }\left(\upsilon \right)\le \frac{{\tau }^{\prime }\left(\upsilon \right)}{\tau \left(\upsilon \right)}\phi \left(\upsilon \right)-k\tau \left(\upsilon \right)q\left(\upsilon \right)-\frac{\beta \theta {\upsilon }^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(\upsilon \right)a\left(\upsilon \right)\right)}^{\frac{1}{\beta }}}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}.$$

(15)

If we set$\tau \left(\upsilon \right)=k=1$ in (15), then we find

$${\phi }^{\prime }\left(\upsilon \right)+\frac{\beta \theta {\upsilon }^{\kappa -2}}{\left(\kappa -2\right)!a^{\frac{1}{\beta }}\left(\upsilon \right)}\phi {\left(\upsilon \right)}^{\frac{\beta +1}{\beta }}+q\left(\upsilon \right)\le 0.$$

From [37], we can see that Equation (9) is non-oscillatory, which is a contradiction.

Let case $\left({\mathbf{S}}_2\right)$ hold. If we set

$$\psi \left(\upsilon \right):=\vartheta \left(\upsilon \right)\frac{y^{\prime }\left(\upsilon \right)}{y\left(\upsilon \right)},$$

we see that $\psi \left(\upsilon \right)>0$for$\upsilon \ge {\upsilon }_1,$where$\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\left(0,\infty \right)\right)$. By differentiating $\psi \left(\upsilon \right)$, we find

$${\psi }^{\prime }\left(\upsilon \right)=\frac{{\vartheta }^{\prime }\left(\upsilon \right)}{\vartheta \left(\upsilon \right)}\psi \left(\upsilon \right)+\vartheta \left(\upsilon \right)\frac{y^{″}\left(\upsilon \right)}{y\left(\upsilon \right)}-\frac{1}{\vartheta \left(\upsilon \right)}\psi {\left(\upsilon \right)}^2.$$

(16)

Now, by integrating (1) from $\upsilon$ to m and using $y^{\prime }\left(\upsilon \right)>0$, we get

$$a\left(m\right){\left(y^{(\kappa -1)}\left(m\right)\right)}^{\beta }-a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }=-{\int }_{\upsilon }^{m}q\left(s\right)g\left(y\left(\eta \left(s\right)\right)\right)ds.$$

By virtue of $y^{\prime }\left(\upsilon \right)>0$ and $\eta \left(\upsilon \right)\ge \upsilon ,$ we get

$$a\left(m\right){\left(y^{(\kappa -1)}\left(m\right)\right)}^{\beta }-a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }\le -k{y}^{\beta }\left(\upsilon \right){\int }_{\upsilon }^{u}q\left(s\right)ds.$$

Letting $m\to \infty$, we see that

$$a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }\ge k{y}^{\beta }\left(\upsilon \right){\int }_{\upsilon }^{\infty }q\left(s\right)\mathrm{d}s$$

and so

$$y^{(\kappa -1)}\left(\upsilon \right)\ge y\left(\upsilon \right){\left(\frac{k}{a\left(\upsilon \right)}{\int }_{\upsilon }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }.$$

Integrating again from $\upsilon$ to ∞, $\kappa -4$ times, we get

$$y^{″}\left(\upsilon \right)+\frac{y\left(\upsilon \right)}{\left(\kappa -4\right)!}{\int }_{\upsilon }^{\infty }{\left(\varsigma -\upsilon \right)}^{\kappa -4}{\left(\frac{k}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma \le 0.$$

(17)

From (16) and (17), we obtain

$${\psi }^{\prime }\left(\upsilon \right)\le \frac{{\vartheta }^{\prime }\left(\upsilon \right)}{\vartheta \left(\upsilon \right)}\psi \left(\upsilon \right)-\frac{\vartheta \left(\upsilon \right)}{\left(\kappa -4\right)!}\varpi \left(s\right)-\frac{1}{\vartheta \left(\upsilon \right)}\psi {\left(\upsilon \right)}^2,$$

(18)

where

$$\varpi \left(s\right)={\int }_{\upsilon }^{\infty }{\left(\varsigma -\upsilon \right)}^{\kappa -4}{\left(\frac{k}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\infty }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma .$$

If we now set $\vartheta \left(\upsilon \right)=k=1$ in (18), then we obtain

$${\psi }^{\prime }\left(\upsilon \right)+{\psi }^2\left(\upsilon \right)+\frac{1}{\left(\kappa -4\right)!}\varpi \left(s\right)\varsigma \le 0.$$

From [37], we see Equation (10) is non-oscillatory, which is a contradiction.

Let case $\left({\mathbf{S}}_3\right)$ hold. By recalling that $a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }$ is non-increasing, we obtain

$$a^{1/\beta }\left(s\right)y^{(\kappa -1)}\left(s\right)\le a^{1/\beta }\left(\upsilon \right)y^{(\kappa -1)}\left(\upsilon \right),s\ge \upsilon \ge {\upsilon }_1.$$

Dividing the latter inequality by $a^{1/\beta }\left(s\right)$ and integrating the resulting inequality from $\upsilon$ to u, we get

$$y^{(\kappa -2)}\left(u\right)\le y^{(\kappa -2)}\left(\upsilon \right)+a^{1/\beta }\left(\upsilon \right)y^{(\kappa -1)}\left(\upsilon \right){\int }_{\upsilon }^u{a}^{-1/\beta }\left(s\right)\mathrm{ds}.$$

Letting $u\to \infty ,$we obtain

$$0\le y^{(\kappa -2)}\left(\upsilon \right)+a^{1/\beta }\left(\upsilon \right)y^{(\kappa -1)}\left(\upsilon \right)\zeta \left(\upsilon \right).$$

Thus,

$$\frac{-a^{1/\beta }\left(\upsilon \right)y^{(\kappa -1)}\left(\upsilon \right)\zeta \left(\upsilon \right)}{y^{(\kappa -2)}\left(\upsilon \right)}\le 1.$$

(19)

Furthermore, we get

$${\left(\frac{y^{(\kappa -2)}\left(\upsilon \right)}{\zeta \left(\upsilon \right)}\right)}^{\prime }\ge 0,$$

(20)

due to (19). Now define

$$\varphi \left(\upsilon \right)=\frac{a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }}{{\left(y^{(\kappa -2)}\left(\upsilon \right)\right)}^{\beta }},$$

(21)

we see that $\varphi \left(\upsilon \right)<0$for$\upsilon \ge {\upsilon }_1,$and

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{{\left(a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }}{{\left(y^{(\kappa -2)}\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta a\left(\upsilon \right){\left(y^{(\kappa -1)}\left(\upsilon \right)\right)}^{\beta +1}}{{\left(y^{(\kappa -2)}\left(\upsilon \right)\right)}^{\beta +1}}.$$

It follows from (1) and (19) that

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{-kq\left(\upsilon \right)y^{\beta }\left(\eta \left(\upsilon \right)\right)}{{\left(y^{(\kappa -2)}\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

From Lemma 2, we find

$$y\left(\upsilon \right)\ge \frac{{\theta }_1}{\left(\kappa -2\right)!}{\upsilon }^{\kappa -2}{y}^{(\kappa -2)}\left(\upsilon \right).$$

(22)

Thus, we have

$${\varphi }^{\prime }\left(\upsilon \right)=\frac{-kq\left(\upsilon \right)y^{\beta }\left(\eta \left(\upsilon \right)\right)}{{\left(y^{(\kappa -2)}\left(\eta \left(\upsilon \right)\right)\right)}^{\beta }}\frac{{\left(y^{(\kappa -2)}\left(\eta \left(\upsilon \right)\right)\right)}^{\beta }}{{\left(y^{(\kappa -2)}\left(\upsilon \right)\right)}^{\beta }}-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

From (22), we obtain

$${\varphi }^{\prime }\left(\upsilon \right)\le -kq\left(\upsilon \right){\left(\frac{{\theta }_1{\eta }^{\kappa -2}\left(\upsilon \right)}{\left(\kappa -2\right)!}\right)}^{\beta }{\left(\frac{\zeta \left(\eta \left(\upsilon \right)\right)}{\zeta \left(\upsilon \right)}\right)}^{\beta }-\frac{\beta {\varphi }^{\beta /\beta +1}\left(\upsilon \right)}{a^{1/\beta }\left(\upsilon \right)}.$$

(23)

From [37], we can see that Equation (11) is non-oscillatory, which is a contradiction.

Theorem 1 is proved. □

Remark 1.

It is well known (see [15]) that if

$${\int }_{{\upsilon }_0}^{\infty }\frac{1}{a\left(\upsilon \right)}\mathrm{d}\upsilon <\infty ,and\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s>\frac{1}{4},$$

then Equations (9)–(11) with $\beta =1$ are oscillatory.

Based on the above results and Theorem 1, we can easily obtain the following Hille and Nehari type oscillation criteria for (1) with $\beta =1.$

Theorem 2.

Let $\beta =k=1$ and assume that (2) holds. If for $\theta ,{\theta }_1\in (0,1)$

$$\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^{\kappa -2}}{\left(\kappa -2\right)!a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^{\kappa -2}}{\left(\kappa -2\right)!a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s>\frac{1}{4},$$

(24)

with

$${\int }_{{\upsilon }_0}^{\infty }\frac{\theta {\upsilon }^{\kappa -2}}{\left(\kappa -2\right)!a\left(\upsilon \right)}\mathrm{d}\upsilon <\infty ,$$

and if

$$\underset{\upsilon \to \infty }{liminf}\upsilon {\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{\left(\kappa -4\right)!}{\int }_v^{\upsilon }{\left(\varsigma -\upsilon \right)}^{\kappa -4}{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\upsilon }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma \mathrm{d}v>\frac{1}{4},$$

(25)

$$\underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{a\left(s\right)}\mathrm{d}s\right)}^2\frac{{\theta }_1\zeta \left(\eta \left(s\right)\right){\eta }^{\kappa -2}\left(s\right)q\left(s\right)}{\zeta \left(s\right)\left(\kappa -2\right)!}\mathrm{d}s>\frac{1}{4},$$

(26)

then every solution of (1) is either oscillatory or satisfies ${\displaystyle \underset{\upsilon \to \infty }{lim}}y\left(\upsilon \right)=0$.

In the next theorem, we employ the integral averaging technique to establish a Philos-type oscillation criteria for (1):

Theorem 3.

Let (2) holds. If there exist positive functions $\tau ,\vartheta \in C^1\left(\left[{\upsilon }_0,\infty \right),\mathbb{R}\right)$ such that

$$\underset{\upsilon \to \infty }{limsup}\frac{1}{H_1\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_1\left(\upsilon ,s\right)k\tau \left(s\right)q\left(s\right)-\pi \left(s\right)\right)\mathrm{d}s=\infty ,$$

(27)

$${\displaystyle \underset{\upsilon \to \infty }{limsup}}\frac{1}{H_2\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_2\left(\upsilon ,s\right)\frac{\vartheta \left(s\right)}{\left(\kappa -4\right)!}\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\right)\mathrm{d}s=\infty ,$$

(28)

and,

$$\underset{\upsilon \to \infty }{limsup}\frac{1}{H_3\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\eta }^{\kappa -2}\left(s\right)}{\left(\kappa -2\right)!}\right)}^{\beta }{\zeta }^{\beta }\left(\eta \left(s\right)\right)-\tilde{\pi }\left(s\right)\right)\mathrm{d}s=\infty ,$$

where

$$\pi \left(s\right)=\frac{h_1^{\beta +1}\left(\upsilon ,s\right)H_1^{\beta }\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{{\left(\left(\kappa -2\right)!\right)}^{\beta }\tau \left(s\right)a\left(s\right)}{{\left(\theta s^{\kappa -2}\right)}^{\beta }}$$

and

$$\tilde{\pi }\left(s\right)=\frac{{\beta }^{\beta +1}{H}_3\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{1}{a^{1/\beta }\left(s\right)\zeta \left(s\right)}.$$

Then every solution of (1) is either oscillatory or satisfies ${\displaystyle \underset{\upsilon \to \infty }{lim}}y\left(\upsilon \right)=0.$

Proof. 

Assume to the contrary that y is a positive solution of (1). Then, we can suppose that $y\left(\upsilon \right)$and $y\left(\eta \left(\upsilon \right)\right)$ are positive for all $\upsilon \ge {\upsilon }_1$ sufficiently large. From Lemma 4, we have three possible cases $\left({\mathbf{S}}_1\right)$, $\left({\mathbf{S}}_2\right)$ and $\left({\mathbf{S}}_3\right)$. Assume that $\left({\mathbf{S}}_1\right)$ holds. From Theorem 1, we get that (15) holds. Multiplying (15) by $H_1\left(\upsilon ,s\right)$and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$ we find that

$$\begin{array}{lll}{\int }_{{\upsilon }_1}^{\upsilon }{H}_1\left(\upsilon ,s\right)k\tau \left(s\right)q\left(s\right)\mathrm{d}s& \le & \phi \left({\upsilon }_1\right)H_1\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }\left(\frac{\partial }{\partial s}{H}_1\left(\upsilon ,s\right)+\frac{{\tau }^{\prime }\left(s\right)}{\tau \left(s\right)}{H}_1\left(\upsilon ,s\right)\right)\phi \left(s\right)\mathrm{d}s\\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta \theta s^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\mathrm{d}s.\end{array}$$

From (3), we get

$$\begin{array}{lll}{\int }_{{\upsilon }_1}^{\upsilon }{H}_1\left(\upsilon ,s\right)k\tau \left(s\right)q\left(s\right)\mathrm{d}s& \le & \phi \left({\upsilon }_1\right)H_1\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }{h}_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)\phi \left(s\right)\mathrm{d}s\\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta \theta s^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\mathrm{d}s.\end{array}$$

(29)

Using Lemma 3 with $V=\beta \theta s^{\kappa -2}/\left(\left(\kappa -2\right)!{\left(\tau \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}\right)H_1\left(\upsilon ,s\right),U=h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)$

And $y=\phi \left(s\right)$, we get

$$\begin{array}{ll}& h_1\left(\upsilon ,s\right)H_1^{\beta /\left(\beta +1\right)}\left(\upsilon ,s\right)\phi \left(s\right)-\frac{\beta \theta s^{\kappa -2}}{\left(\kappa -2\right)!{\left(\tau \left(s\right)a\left(s\right)\right)}^{\frac{1}{\beta }}}{H}_1\left(\upsilon ,s\right){\phi }^{\frac{\beta +1}{\beta }}\left(s\right)\\ \le & \frac{h_1^{\beta +1}\left(\upsilon ,s\right)H_1^{\beta }\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{{\left(\left(\kappa -2\right)!\right)}^{\beta }\tau \left(s\right)a\left(s\right)}{{\left(\theta s^{\kappa -2}\right)}^{\beta }},\end{array}$$

which, with (29) gives

$$\frac{1}{H_1\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_1\left(\upsilon ,s\right)k\tau \left(s\right)q\left(s\right)-\pi \left(s\right)\right)\mathrm{d}s\le \phi \left({\upsilon }_1\right),$$

which contradicts (27). Assume that $\left({\mathbf{S}}_2\right)$ holds. From Theorem 1, we get that (18) holds. Multiplying (18) by $H_2\left(\upsilon ,s\right)$and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$, we obtain

$$\begin{array}{lll}{\int }_{{\upsilon }_1}^{\upsilon }{H}_2\left(\upsilon ,s\right)\frac{\vartheta \left(s\right)}{\left(\kappa -4\right)!}\varpi \left(s\right)\mathrm{d}s& \le & \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)\\ & & +{\int }_{{\upsilon }_1}^{\upsilon }\left(\frac{\partial }{\partial s}{H}_2\left(\upsilon ,s\right)+\frac{{\vartheta }^{\prime }\left(s\right)}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right)\right)\psi \left(s\right)\mathrm{d}s\\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right){\psi }^2\left(s\right)\mathrm{d}s.\end{array}$$

Thus, from (4), we obtain

$$\begin{array}{lll}{\int }_{{\upsilon }_1}^{\upsilon }{H}_2\left(\upsilon ,s\right)\frac{\vartheta \left(s\right)}{\left(\kappa -4\right)!}\varpi \left(s\right)\mathrm{d}s& \le & \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }{h}_2\left(\upsilon ,s\right)\sqrt{H_2\left(\upsilon ,s\right)}\psi \left(s\right)\mathrm{d}s\\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{1}{\vartheta \left(s\right)}{H}_2\left(\upsilon ,s\right){\psi }^2\left(s\right)\mathrm{d}s\\ & \le & \psi \left({\upsilon }_1\right)H_2\left(\upsilon ,{\upsilon }_1\right)+{\int }_{{\upsilon }_1}^{\upsilon }\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\mathrm{d}s\end{array}$$

and so

$$\frac{1}{H_2\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_2\left(\upsilon ,s\right)\frac{\vartheta \left(s\right)}{\left(\kappa -4\right)!}\varpi \left(s\right)-\frac{\vartheta \left(s\right)h_2^2\left(\upsilon ,s\right)}{4}\right)\mathrm{d}s\le \psi \left({\upsilon }_1\right),$$

which contradicts (28). Assume that $\left({\mathbf{S}}_3\right)$ holds. Using (19) and (21), we see that

$$-\varphi \left(\upsilon \right){\zeta }^{\beta }\left(\upsilon \right)\le 1$$

(30)

due to (30). Multiplying this inequality by ${\zeta }^{\beta }\left(\upsilon \right)$ and integrating the resulting inequality from ${\upsilon }_1$ to $\upsilon$, we get

$$\begin{array}{ll}& {\zeta }^{\beta }\left(\upsilon \right)\varphi \left(\upsilon \right)-{\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)+\beta {\int }_{{\upsilon }_1}^{\upsilon }{a}^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)ds\\ \le & -{\int }_{{\upsilon }_1}^{\upsilon }kq\left(s\right){\left(\frac{{\theta }_1{\eta }^{\kappa -2}\left(s\right)}{\left(\kappa -2\right)!}\right)}^{\beta }{\zeta }^{\beta }\left(\eta \left(s\right)\right)ds-\beta {\int }_{{\upsilon }_1}^{\upsilon }\frac{{\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)ds.\end{array}$$

(31)

Multiplying (31) by $H_3\left(\upsilon ,s\right)$, we find that

$$\begin{array}{lll}{\int }_{{\upsilon }_1}^{\upsilon }{H}_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\eta }^{\kappa -2}\left(s\right)}{\left(\kappa -2\right)!}\right)}^{\beta }{\zeta }^{\beta }\left(\eta \left(s\right)\right)\mathrm{d}s& \le & {\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)H_3\left(\upsilon ,{\upsilon }_1\right)-{\zeta }^{\beta }\left(\upsilon \right)\varphi \left(\upsilon \right)H_3\left(\upsilon ,{\upsilon }_1\right)\\ & & +{\int }_{{\upsilon }_1}^{\upsilon }\beta a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)H_3\left(\upsilon ,s\right)\mathrm{d}s\\ & & -{\int }_{{\upsilon }_1}^{\upsilon }\frac{\beta {\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)\mathrm{d}s.\end{array}$$

Using Lemma 3 with $V={\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)/a^{1/\beta }\left(s\right),U=a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)H_3\left(\upsilon ,s\right)$and $y=\varphi \left(s\right)$, we get

$$\begin{array}{ll}& \beta a^{-1/\beta }\left(s\right){\zeta }^{\beta -1}\left(s\right)\varphi \left(s\right)H_3\left(\upsilon ,s\right)-\frac{\beta {\varphi }^{\beta /\beta +1}\left(s\right)}{a^{1/\beta }\left(s\right)}{\zeta }^{\beta }\left(s\right)H_3\left(\upsilon ,s\right)\\ \le & \frac{{\beta }^{\beta +1}{H}_3\left(\upsilon ,s\right)}{{\left(\beta +1\right)}^{\beta +1}}\frac{1}{a^{1/\beta }\left(s\right)\zeta \left(s\right)}\end{array}$$

and easily, we find that

$$\frac{1}{H_3\left(\upsilon ,{\upsilon }_1\right)}{\int }_{{\upsilon }_1}^{\upsilon }\left(H_3\left(\upsilon ,s\right)kq\left(s\right){\left(\frac{{\theta }_1{\eta }^{\kappa -2}\left(s\right)}{\left(\kappa -2\right)!}\right)}^{\beta }{\zeta }^{\beta }\left(\eta \left(s\right)\right)-\tilde{\pi }\left(s\right)\right)\mathrm{d}s\le {\zeta }^{\beta }\left({\upsilon }_1\right)\varphi \left({\upsilon }_1\right)+1,$$

which contradicts (27). This completes the proof. □

Example 1.

We consider the equation

$${\left({\upsilon }^5{y}^{‴}\left(\upsilon \right)\right)}^{\prime }+\upsilon q_{0}y\left(3\upsilon \right)=0,\upsilon \ge 1,$$

(32)

where $q_0>0$ is a constant. Note that $\beta =1,\kappa =4,a\left(\upsilon \right)={\upsilon }^5,q\left(\upsilon \right)=\upsilon q_0$ and$\eta \left(\upsilon \right)=3\upsilon$. If we set $k=1,$ then condition (24) becomes

$$\begin{array}{ll}& \underset{\upsilon \to \infty }{liminf}{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^{\kappa -2}}{\left(\kappa -2\right)!a\left(s\right)}\mathrm{d}s\right)}^{-1}{\int }_{\upsilon }^{\infty }{\left({\int }_{{\upsilon }_0}^{\upsilon }\frac{\theta s^{\kappa -2}}{\left(\kappa -2\right)!a\left(s\right)}\mathrm{d}s\right)}^{2}q\left(s\right)\mathrm{d}s\\ =& \underset{\upsilon \to \infty }{liminf}\left(4{\upsilon }^2\right){\int }_{\upsilon }^{\infty }\frac{q_0}{16s^3}ds=\underset{\upsilon \to \infty }{liminf}\left(4{\upsilon }^2\right)\left(\frac{q_0}{32{\upsilon }^2}\right)\\ =& \frac{q_0}{8}>\frac{1}{4},\end{array}$$

while condition (25) becomes

$$\begin{array}{lll}\underset{\upsilon \to \infty }{liminf}\upsilon {\int }_{{\upsilon }_0}^{\upsilon }\frac{1}{\left(\kappa -4\right)!}{\int }_v^{\upsilon }{\left(\varsigma -\upsilon \right)}^{\kappa -4}{\left(\frac{1}{a\left(\varsigma \right)}{\int }_{\varsigma }^{\upsilon }q\left(s\right)\mathrm{d}s\right)}^{1/\beta }\mathrm{d}\varsigma \mathrm{d}v& =& \underset{\upsilon \to \infty }{liminf}\upsilon \left(\frac{q_0}{4\upsilon }\right)\\ & =& \frac{q_0}{4}>\frac{1}{4},\end{array}$$

and hence condition (26) is satisfied. Therefore, from Theorem 2, all solutions of Equation (32) are oscillatory if $q_0>2$.

Remark 2.

One can easily see that the results obtained in [18,19] cannot be applied to conditions in Theorem 2, so our results are new.

Remark 3.

We can generalize our results by studying the equation in the form

$${\left(a\left(\upsilon \right){\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }+\sum _{i=1}^j{q}_i\left(\upsilon \right)y^{\beta }\left({\eta }_i\left(\upsilon \right)\right)=0,\mathrm{where}\upsilon \ge {\upsilon }_0,j\ge 1.$$

For this we leave the results to researchers interested.

3. Conclusions

In this article we studied we provided three new Theorems on the oscillatory and asymptotic behavior of a class of even-order advanced differential equations with a non-canonical operator in the form of (1).

For researchers interested in this field, and as part of our future research, there is a nice open problem which is finding new results in the following cases:

$$\begin{array}{ll}\left({\mathbf{S}}_1\right)& y\left(\upsilon \right)>0,y^{\prime }\left(\upsilon \right)>0,y^{\left(\kappa -2\right)}\left(\upsilon \right)>0,y^{\left(\kappa -1\right)}\left(\upsilon \right)\le 0,{\left(a\left(\upsilon \right){\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }\le 0,\\ \left({\mathbf{S}}_2\right)& y\left(\upsilon \right)>0,y^{\left(r\right)}\left(\upsilon \right)<0,y^{(r+1)}\left(\upsilon \right)>0,\forall r\in \{1,3,\dots ,\kappa -3\},\\ & \mathrm{and} y^{(\kappa -1)}\left(\upsilon \right)<0,{\left(a\left(\upsilon \right){\left(y^{\left(\kappa -1\right)}\left(\upsilon \right)\right)}^{\beta }\right)}^{\prime }\le 0.\end{array}$$

For all this there is some research in progress.