Property (A) of Third-Order Differential Equations as a Consequence of Comparison Theorems

Abstract

The aim of this paper is to provide a general method for extending the criteria known for a simple canonical equation $x\prime \prime \prime \left(t\right)+q\left(t\right)x\left(\sigma \right(t\left)\right)=0$ to a noncanonical equation of the form ${\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+p\left(t\right)y\left(\tau \right(t\left)\right)=0$.

1. Introduction

We consider the functional differential equation with deviating argument

$$\left(r_2\right(t\left){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)\prime +p\left(t\right)y\left(\tau \right(t\left)\right)=0.$$

(1)

We always assume the following:

($H_1$)

$p\left(t\right)\in C\left([t_0,\infty )\right)$, $p\left(t\right)>0,$

($H_2$)

$r_i\left(t\right)\in C^1\left([t_0,\infty )\right)$, $r_i\left(t\right)>0,$ for $i=1,2,$

($H_3$)

$\tau \left(t\right)\in C^1\left([t_0,\infty )\right),$${\tau }^{\prime }\left(t\right)>0$ and ${\displaystyle \underset{t\to \infty }{lim}\tau (t)=\infty }$.

Our attention will be restricted to solutions $y\left(t\right)$ of (1) which satisfy the equation for all sufficiently large t and $sup\{\left|y\left(t\right)\right|:t\ge T\}>0$ for all $T\ge t_0.$ It is assumed that (1) does possess such a solution A solution is said to be oscillatory if it has a sequence of zeros clustering at $t=\infty$; otherwise, it is called nonoscillatory.

We assume that (1) is in strong noncanonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{r_i\left(s\right)}}\mathrm{d}s<\infty \mathrm{for}i=1,2$$

(2)

and therefore the functions

$${\rho }_i\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{1}{r_i\left(s\right)}}\mathrm{d}s,i=1,2$$

are well defined. To simplify our notation we introduce the following notations

$$L_{1}y\left(t\right)=r_1\left(t\right)y^{\prime }\left(t\right),L_{2}y\left(t\right)=r_2\left(t\right){\left(L_{1}y\left(t\right)\right)}^{\prime },L_{3}y\left(t\right)={\left(L_{2}y\left(t\right)\right)}^{\prime }.$$

There are many papers regarding the oscillation properties of the third-order differential equations. We mention some of them in the references [1,2,3,4,5,6,7,8,9,10].

First, let us clarify what property (A) means for a noncanonical equation. For this reason, we explore the solution space of (1). If we denote it by $\mathcal{N}$, then, in view of the well known lemma of Kiguradze [4], the set of all nonoscillatory solutions of (1), we have

$$\mathcal{N}={\mathcal{N}}_A\cup {\mathcal{N}}_B\cup {\mathcal{N}}_C\cup {\mathcal{N}}_D,$$

(3)

where positive solution $y\left(t\right)$ satisfies $L_{3}y\left(t\right)<0$ and

$$\begin{array}{cc}\hfill y\left(t\right)\in {\mathcal{N}}_A& \iff L_{1}y\left(t\right)<0,L_{2}y\left(t\right)>0,\hfill \\ \hfill y\left(t\right)\in {\mathcal{N}}_B& \iff L_{1}y\left(t\right)<0,L_{2}y\left(t\right)<0,\hfill \\ \hfill y\left(t\right)\in {\mathcal{N}}_C& \iff L_{1}y\left(t\right)>0,L_{2}y\left(t\right)>0,\hfill \\ \hfill y\left(t\right)\in {\mathcal{N}}_D& \iff L_{1}y\left(t\right)>0,L_{2}y\left(t\right)<0.\hfill \end{array}$$

By the way, when we consider the equation in canonical form (i.e., both integrals in (2) are divergent), only two classes of nonoscillatory solutions are admissible. We will further show that, for our equation too, only two classes, ${\mathcal{N}}_A$ and ${\mathcal{N}}_B$, are relevant, while classes ${\mathcal{N}}_C$ and ${\mathcal{N}}_D$ can be eliminated by a simple integral condition.

Following Kusano and Naito [7], in order to establish property (A) for the noncanonical Equation (1), we introduce the concept of principal system (1). By a principal system for (1), we mean a set of three solutions $Y_1\left(t\right),Y_2\left(t\right),Y_3\left(t\right)$ of the equation $L_{3}y\left(t\right)=0$ which are eventually positive and satisfy the relation

$$\underset{t\to \infty }{lim}{\displaystyle \frac{Y_i\left(t\right)}{Y_j\left(t\right)}}=0\mathrm{for}1\le i<j\le 3.$$

A basic property of principal systems is that, if both $\left\{Y_1\left(t\right),Y_2\left(t\right),Y_3\left(t\right)\right\}$ and $\{{\tilde{Y}}_1(t),{\tilde{Y}}_2(t)$, $Y_3\left(t\right)\}$ are principal systems for (1), then for each $i=1,2,3,$$Y_i\left(t\right)$ and ${\tilde{Y}}_i(t)$ we have the same order of growth (or decay) as $t\to \infty ,$ that is, the limits

$$\underset{t\to \infty }{lim}{\displaystyle \frac{{\tilde{Y}}_i(t)}{Y_i\left(t\right)}}>0\mathrm{for}i=1,2,3$$

exist and are finite.

Definition 1.

Let $\left\{Y_1\left(t\right),Y_2\left(t\right),Y_3\left(t\right)\right\}$ be a principal system for (1). Equation (1) is said to have property (A) if every nonoscillatory solution $y\left(t\right)$ of (1) satisfies

$$\underset{t\to \infty }{lim}{\displaystyle \frac{y\left(t\right)}{Y_1\left(t\right)}}=0.$$

(4)

A simple calculation shows that $\left\{{\rho }_{12}\left(t\right),{\rho }_1\left(t\right),1\right\}$ is one of the principal systems for (1), where

$${\rho }_{12}(t)={\int }_t^{\infty }{\displaystyle \frac{1}{r_1\left(s\right)}}{\rho }_2\left(s\right)\mathrm{d}s$$

and so (4) can take the form

$$\underset{t\to \infty }{lim}{\displaystyle \frac{y\left(t\right)}{{\rho }_{12}\left(t\right)}}=0.$$

(5)

From the definition of property (A) and the structure of the solution space (3), it follows that, in order to establish property (A), it is necessary to eliminate the classes ${\mathcal{N}}_B,{\mathcal{N}}_C$, and ${\mathcal{N}}_D$ (since solutions from these classes cannot satisfy (5)). Moreover, it must be shown that every solution $y\left(t\right)\in {\mathcal{N}}_A$ satisfies (5).

To achieve our goals, property (A) of (1), we will use the technique of comparison theorems. The main idea of transforming a noncanonical equation into a canonical one is that the investigation of a canonical equation is simpler only from the point of view of the number of classes of possible nonoscillatory solutions. Moreover, the criteria for a property (A) are known for the canonical equations and can be applied in the final result after transformations to the studied noncanonical equation. In the first step we compare our noncanonical Equation (1) with the simpler noncanonical inequality of the following form:

$$\left(r\right(t\left)\right(r\left(t\right)z^{\prime }\left(t\right))\prime \prime )+p\left(t\right)z\left(\tau \right(t\left)\right)\le 0,$$

(6)

where the natural choice is $r\left(t\right)=\sqrt{r_1\left(t\right)r_2\left(t\right)}$, i.e., the geometric mean of $r_1\left(t\right)$ and $r_2\left(t\right)$. The next very important step is the transformation of noncanonical inequality (6) into canonical equation

$$x^{\prime \prime \prime }(t)+q(t)x\left(\sigma \right(t\left)\right)=0$$

(7)

in the sense that the property (A) of simpler Equation (7) will give us the desired property of (6) and, finally, property (A) of (1). This will allow us to immediately and effectively rewrite the criteria known for property (A) of (7) to the more general Equation (1). The idea of obtaining a comparison principle that connects a noncanonical equation with a canonical one is new in oscillation theory. Only the comparison theorems in which both equations are either canonical or noncanonical are listed in the articles.

On the other hand, the most famous comparison theorem for noncanonical equations formulated by Kusano and Naito (see Theorem 3 in [7]) claims the following:

Theorem 1.

If

$$p\left(t\right)\ge q\left(t\right)\ge 0,$$

(8)

then Equation (1) has property (A) if equation

$$\left(r_2\left(t\right){\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)\prime +q(t)y\left(\tau \right(t\left)\right)=0$$

has property (A).

This theorem requires the same deviating argument and the same operator, $L_{3}y\left(t\right).$ In the process of developing our article, we will expand on this theorem.

2. Main Results

We will state the integral condition for emptying the classes ${\mathcal{N}}_C$ and ${\mathcal{N}}_D$.

Theorem 2.

Assume that

$${\int }_{t_0}^{\infty }{\rho }_2\left(s\right)p\left(s\right)\mathrm{d}s=\infty .$$

(9)

Then ${\mathcal{N}}_C=\varnothing$ and ${\mathcal{N}}_D=\varnothing .$

Proof. 

Assume that $y\left(t\right)$ is a positive solution of (1) which belongs to either ${\mathcal{N}}_C$ or ${\mathcal{N}}_D$. Then $y\left(t\right)\uparrow$. It follows that there exists a positive constant k such that $y\left(\tau \right(t\left)\right)\ge k.$ If $y\left(t\right)\in {\mathcal{N}}_C$, then an integration of (1) from $t_0$ to ∞ yields

$$L_{2}y\left(t_0\right)\ge {\int }_{t_0}^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\ge k{\int }_{t_0}^{\infty }p\left(s\right)\mathrm{d}s.$$

But this is inconsistent with (9).

On the other hand, if $y\left(t\right)\in {\mathcal{N}}_D$, then an integration of (1) from $t_0$ to t provides

$$-L_{2}y\left(t\right)\ge {\int }_{t_0}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\ge k{\int }_{t_0}^{t}p\left(s\right)\mathrm{d}s.$$

Integrating the last inequality from $t_0$ to ∞, we are led to

$$L_{1}y\left(t_0\right)\ge k{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{r_2\left(s_1\right)}}{\int }_{t_0}^{s_1}p\left(s\right)\mathrm{d}s=k{\int }_{t_0}^{t}p\left(s\right){\rho }_2\left(s\right)\mathrm{d}s$$

which contradicts (9). The proof is complete. □

The next task is to show that every solution from the class ${\mathcal{N}}_A$ satisfies (5).

Theorem 3.

Assume that

$${\int }_{t_0}^{\infty }{\rho }_{12}\left(\tau \left(s\right)\right)p\left(s\right)\mathrm{d}s=\infty .$$

(10)

Then every solution $y\left(t\right)\in {\mathcal{N}}_A$ satisfies (5).

Proof. 

Assume that $y\left(t\right)\in {\mathcal{N}}_A$ is a positive solution of (1). It is easy to see that (10) implies

$$y\left(t\right)\downarrow 0,L_{2}y\left(t\right)\downarrow 0.$$

Now, it follows from L’Hospital’s rule that

$$\underset{t\to \infty }{lim}{\displaystyle \frac{y\left(t\right)}{{\rho }_{12}\left(t\right)}}=\underset{t\to \infty }{lim}{L}_{2}y\left(t\right).$$

Therefore, $\underset{t\to \infty }{lim}{\displaystyle \frac{y\left(t\right)}{{\rho }_{12}\left(t\right)}}$ exists and is non-negative. If we admit that (5) does not hold, then there exists $k>0$ such that $y\left(t\right)\ge k{\rho }_{12}\left(t\right)$. An integration of (1) from $t_0$ to ∞ yields

$$L_{2}y\left(t_0\right)\ge {\int }_{t_0}^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\ge k{\int }_{t_0}^{\infty }{\rho }_{12}\left(\tau \left(s\right)\right)p\left(s\right)\mathrm{d}s.$$

This is a contradiction and the proof is finished. □

Remark 1.

If $\tau \left(t\right)\le t$, then condition (9) follows from (10).

We are prepared to provide the first comparison result. We introduce auxiliary functions

$$r(t)=\sqrt{r_1\left(t\right)r_2\left(t\right)}\mathrm{and}\rho (t)={\int }_t^{\infty }{\displaystyle \frac{1}{r\left(s\right)}}\mathrm{d}s.$$

(11)

For future references, we denote

$$P(t)={\int }_{t_0}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s,$$

where $y\left(t\right)$ is a positive solution of (1).

Theorem 4.

Let (9), (10), and (11) hold. Assume that

$${\displaystyle \frac{r_1\left(t\right)}{r_2\left(t\right)}}\downarrow .$$

(12)

Then Equation (1) has property (A) if differential inequality (6) has property (A).

Proof. 

In view of Theorems 2 and 3, it is sufficient to show that ${\mathcal{N}}_B=\varnothing .$ Assume, on the contrary, that (1) possesses a solution $y\left(t\right)\in {\mathcal{N}}_B$. An integration of (1) from $t_0$ to t yields

$$-L_{2}y\left(t\right)\ge P\left(t\right).$$

Integrating the last inequality again from $t_0$ to t, we are led to

$$-L_{1}y\left(t\right)\ge {\int }_{t_0}^t{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2$$

and finally integrating the last inequality from t to ∞, we get

$$y\left(t\right)\ge {\int }_t^{\infty }{\displaystyle \frac{1}{r_1\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1.$$

(13)

We claim that

$${\int }_t^{\infty }{\displaystyle \frac{1}{r_1\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1\ge {\int }_t^{\infty }{\displaystyle \frac{1}{r\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1,t\ge t_0.$$

We can prove this inequality by using standard methods of calculus. We introduce the function

$$F(t)={\int }_t^{\infty }{\displaystyle \frac{1}{r_1\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1-{\int }_t^{\infty }{\displaystyle \frac{1}{r\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1.$$

Since $F(\infty )=0$, it is enough to verify that $F^{\prime }\left(t\right)\le 0$ for $t\ge t_0$. Because

$$F^{\prime }(t)=-{\displaystyle \frac{1}{r_1\left(t\right)}}{\int }_{t_0}^t{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2+{\displaystyle \frac{1}{r\left(t\right)}}{\int }_{t_0}^t{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2$$

just verify that

$$G(t)=-{\int }_{t_0}^t{\displaystyle \frac{1}{r_2\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2+{\displaystyle \frac{r_1\left(t\right)}{r\left(t\right)}}{\int }_{t_0}^t{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\le 0,t\ge t_0.$$

As $G\left(t_0\right)=0$ we shall show that $G^{\prime }\left(t\right)\le 0$. Simple calculations reveal that

$$\begin{array}{cc}\hfill G^{\prime }\left(t\right)& =-{\displaystyle \frac{1}{r_2\left(t\right)}}P\left(t\right)+{\left({\displaystyle \frac{r_1\left(t\right)}{r\left(t\right)}}\right)}^{\prime }{\int }_{t_0}^t{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2+{\displaystyle \frac{r_1\left(t\right)}{r^2\left(t\right)}}P\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{P\left(t\right)}{r_2\left(t\right)}}\left(-1+{\displaystyle \frac{r_1\left(t\right)r_2\left(t\right)}{r^2\left(t\right)}}\right)=0.\hfill \end{array}$$

We just verified that indeed $F\left(t\right)>0$ which together with (13) yields

$$y(t)\ge {\int }_t^{\infty }{\displaystyle \frac{1}{r\left(s_1\right)}}{\int }_{t_0}^{s_1}{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2\mathrm{d}{s}_1.$$

(14)

Let us denote the right-hand side of (14) by $z\left(t\right)$. Then

$$\begin{array}{cc}\hfill y\left(t\right)& \ge z\left(t\right)>0,r\left(t\right)z^{\prime }\left(t\right)=-{\int }_{t_0}^t{\displaystyle \frac{1}{r\left(s_2\right)}}P\left(s_2\right)\mathrm{d}{s}_2<0,\hfill \\ \hfill r\left(t\right)& {\left(r\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }=-P\left(t\right)<0,(r\left(t\right){\left(r\left(t\right)z^{\prime }\left(t\right))\prime \right)}^{\prime }=-p\left(t\right)y\left(\tau \right(t\left)\right)<0,\hfill \end{array}$$

(15)

which means that $z\left(t\right)\in {\mathcal{N}}_B$ and $z\left(t\right)$ is a solution of the differential inequality

$${\left(r\left(t\right){\left(r\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+p\left(t\right)z\left(\tau \right(t\left)\right)\le 0.$$

Moreover, (15) implies that

$$\underset{t\to \infty }{lim}{\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}=1/2\underset{t\to \infty }{lim}{\left(r\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }<0.$$

It is easy to see that $\left\{{\rho }^2\left(t\right),\rho \left(t\right),1\right\}$ is a principal system for (6) and therefore property (A) of (6) means that every solution $z\left(t\right)$ satisfies

$$\underset{t\to \infty }{lim}{\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}=0.$$

This is a contradiction. The proof is finished. □

Now we are prepared to connect Equation (1) and

$$x^{\prime \prime \prime }\left(t\right)+q\left(t\right)x\left(\sigma \right(t\left)\right)=0.$$

(16)

For function $\rho \left(t\right)$ defined by (11) we denote

$$R(t)={\displaystyle \frac{1}{\rho \left(t\right)}}$$

(17)

and $R^{-1}\left(t\right)$ is the inverse function to $R\left(t\right)$.

Theorem 5.

Let (9), (10), (11), (12), and (17) hold. Assume that

$$\begin{array}{cc}\hfill q\left(t\right)& =p\left(R^{-1}\left(t\right)\right)r\left(R^{-1}\left(t\right)\right){\rho }^4\left(R^{-1}\left(t\right)\right){\rho }^2\left(\tau \left(R^{-1}\left(t\right)\right)\right),\hfill \\ \hfill \sigma \left(t\right)& =\left(R\left(\tau \left(R^{-1}\left(t\right)\right)\right)\right).\hfill \end{array}$$

(18)

Then Equation (1) has property (A) if Equation (16) has property (A).

Proof. 

By Theorems 2 and 3, we shall show that ${\mathcal{N}}_B=\varnothing .$ Assume, on the contrary, that (1) has a positive solution $y\left(t\right)\in {\mathcal{N}}_B$. Using the same arguments as in Theorem 4, we arrive at the fact that (6) has a positive solution $z\left(t\right)\in {\mathcal{N}}_B$ such that

$$\underset{t\to \infty }{lim}{\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}>0.$$

(19)

Let us calculate derivatives of the following function, and using the definition of $\rho \left(t\right)$ in (11), we have

$${\left[{\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}\right]}^{\prime }={\displaystyle \frac{1}{r\left(r\right){\rho }^2\left(t\right)}}\left(r(t)z^{\prime }(t)+{\displaystyle \frac{2z\left(t\right)}{\rho \left(t\right)}}\right),$$

$${\left[r(t){\rho }^2(t){\left({\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}\right)}^{\prime }\right]}^{\prime }={\left(r(t)z^{\prime }(t)+{\displaystyle \frac{2z\left(t\right)}{\rho \left(t\right)}}\right)}^{\prime }={\left(r\right(t\left)z^{\prime }\right(t\left)\right)}^{\prime }+{\displaystyle \frac{2z^{\prime }\left(t\right)}{\rho \left(t\right)}}+{\displaystyle \frac{2z\left(t\right)}{r\left(t\right){\rho }^2\left(t\right)}},$$

$$\begin{array}{cc}\hfill {\left[r(t){\rho }^2(t){\left(r\left(t\right){\rho }^2\left(t\right){\left({\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right]}^{\prime }& ={\left(r\left(t\right){\left(r\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }{\rho }^2\left(t\right)+2z^{\prime }\left(t\right)r\left(t\right)\rho \left(t\right)+2z\left(t\right)\right)}^{\prime }\hfill \\ & ={\rho }^2\left(t\right){\left((r\left(t\right){\left(r\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }.\hfill \end{array}$$

Finally, we get that (6) can be written in the form

$${\displaystyle \frac{1}{{\rho }^2\left(t\right)}}{\left(r(t){\rho }^2(t){\left(r(t){\rho }^2(t){\left({\displaystyle \frac{z\left(t\right)}{{\rho }^2\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+p\left(t\right)z\left(\tau \right(t\left)\right)\le 0,$$

but it means that $\tilde{z}\left(t\right)=z\left(t\right)/{\rho }^2\left(t\right)$ is a solution of the differential inequality

$$\left(r\right(t\left){\rho }^2\right(t\left){\left(r\left(t\right){\rho }^2\left(t\right){\tilde{z}}^{\prime }\left(t\right)\right)}^{\prime }\right)\prime +{\rho }^2\left(t\right){\rho }^2\left(\tau \right(t\left)\right)p\left(t\right)\tilde{z}\left(\tau \right(t\left)\right)\le 0$$

(20)

and in view of (19), we have

$$\underset{t\to \infty }{lim}\tilde{z}\left(t\right)>0.$$

(21)

On the other hand, setting $x\left(t\right)=\tilde{z}\left(R^{-1}\left(t\right)\right)$, one can verify that

$$x^{\prime }\left(t\right)=r\left(s\right){\rho }^2\left(s\right){\tilde{z}}^{\prime }\left(s\right){|}_{s=R^{-1}\left(t\right)},x^{\prime \prime }\left(t\right)=r\left(s\right){\rho }^2\left(s\right){\left(r\left(s\right){\rho }^2\left(s\right){\tilde{z}}^{\prime }\left(s\right)\right)}^{\prime }{|}_{s=R^{-1}\left(t\right)}$$

and finally

$$x^{\prime \prime \prime }\left(t\right)=r\left(s\right){\rho }^2\left(s\right){\left(r\left(s\right){\rho }^2\left(s\right){\left(r\left(s\right){\rho }^2\left(s\right){\tilde{z}}^{\prime }\left(s\right)\right)}^{\prime }\right)}^{\prime }{|}_{s=R^{-1}\left(t\right)}.$$

(22)

On the other hand, inequality (20) can be rewritten in the form

$$r{\rho }^2{\left(r,{\rho }^2,{\left(r{\rho }^2{z}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(s\right){|}_{s=R^{-1}\left(t\right)}+r\left(s\right){\rho }^4\left(s\right){\rho }^2\left(\tau \left(s\right)\right)p\left(s\right)\tilde{z}\left(\tau \left(s\right)\right){|}_{s=R^{-1}\left(t\right)}\le 0$$

which, in view of (18) and (22), yields that $x\left(t\right)$ is a positive solution of inequality

$$x^{\prime \prime \prime }\left(t\right)+q\left(t\right)x\left(\sigma \right(t\left)\right)\le 0.$$

But according to Theorem 2 in [7] the corresponding differential Equation (16) also has a solution $x\left(t\right)\in {\mathcal{N}}_B$. The mentioned theorem gives a result for a more general n-th order differential equation with quasiderivatives. Equation (16) is a special case of the equation studied by Kusano and Naito and all the conditions of Theorem 2 in [7] are satisfied.

Moreover, it follows from (21) that

$$\underset{t\to \infty }{lim}x(t)>0,$$

but that is contrary to the property (A) of (16). It is because of the fact that $\left\{1,t,t^2\right\}$ is a principal system for (16), and thus, property (A) of (16) means that every nonoscillatory solution satisfies

$$\underset{t\to \infty }{lim}x(t)=0.$$

The proof is complete. □

3. Criteria for Property (A)

We will use the obtained comparison theorem to transfer the criteria from the canonical to the noncanonical equation. Our approach works regardless of whether we consider delay, advanced, or equations without deviating argument. We will use the results form [5] as reference criteria.

Corollary 1.

Let (9), (10), (11), (12), (17), and (18) hold. Assume that $\tau \left(t\right)\le t$. If

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\left\{{\displaystyle \frac{1}{\rho \left(\tau \right(t\left)\right)}}{\int }_t^{\infty }{\rho }^2\left(x\right)\rho \left(\tau \left(x\right)\right)p\left(x\right)\mathrm{d}x+{\int }_{\tau \left(t\right)}^t{\rho }^2\left(x\right)p\left(x\right)\mathrm{d}x\right.\\ \left.\rho \left(\tau \left(t\right)\right){\int }_0^{\tau \left(t\right)}\rho \left(x\right)p\left(x\right)\mathrm{d}x\right\}>2,\hfill \end{array}$$

(23)

then Equation (1) has property (A).

Proof. 

By Theorem 2.1 in [5], condition

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\left\{\sigma \left(t\right){\int }_t^{\infty }\sigma \left(s\right)q\left(s\right)\mathrm{d}s+{\int }_{\sigma \left(t\right)}^t{\sigma }^2\left(s\right)q\left(s\right)\mathrm{d}s\right.\\ \left.{\displaystyle \frac{1}{\sigma \left(t\right)}}{\int }_0^{\sigma \left(t\right)}s{\sigma }^2\left(s\right)q\left(s\right)\mathrm{d}s\right\}>2\hfill \end{array}$$

(24)

guarantees property (A) of (16). Taking into account that, in view of (18) condition (24) is equivalent to (23), the assertion follows immediately from Theorem 5. □

Corollary 2.

Let (9), (10), (11), (12), (17), and (18) hold. Assume that $\tau \left(t\right)\ge t$. If

$$\begin{array}{c}\underset{t\to \infty }{lim\; sup}\left\{{\displaystyle \frac{1}{\rho \left(\tau \right(t\left)\right)}}{\int }_{\tau \left(t\right)}^{\infty }{\rho }^2\left(x\right)\rho \left(\tau \left(x\right)\right)p\left(x\right)\mathrm{d}x+{\int }_t^{\tau \left(t\right)}\rho \left(x\right)\rho \left(\tau \left(x\right)\right)p\left(x\right)\mathrm{d}x\right.\\ \left.\rho \left(\tau \left(t\right)\right){\int }_0^t\rho \left(x\right)p\left(x\right)\mathrm{d}x\right\}>2,\hfill \end{array}$$

(25)

then Equation (1) has property (A).

Proof. 

Employing Theorem 2.3 in [5] together with Theorem 5 and proceeding in the same way as in the proof of Corollary 1, we can easily verify this criterion. □

For a final example of the use of our comparison theorem, we will use the results from [4].

Corollary 3.

Let (9), (10), (11), (12), (17), and (18) hold. Assume that $\tau \left(t\right)\equiv t$. If

$$\underset{t\to \infty }{lim\; inf}{\displaystyle \frac{1}{\rho \left(t\right)}}{\int }_t^{\infty }{\rho }^3\left(x\right)p\left(x\right)\mathrm{d}x>{\displaystyle \frac{2}{3\sqrt{3}}},$$

(26)

then Equation (1) has property (A).

Proof. 

By Theorem 2.12 in [4], condition

$$\underset{t\to \infty }{lim\; inf}t{\int }_t^{\infty }sq\left(s\right)\mathrm{d}s>{\displaystyle \frac{2}{3\sqrt{3}}}$$

(27)

guarantees property (A) of (16) for $\sigma \left(t\right)\equiv t$. The assertion immediately follows from Theorem 5. □

As we could see from the proofs of Corollaries 1–3, the rewriting of criteria from a canonical equation to a noncanonical one is, thanks to Theorem 5, immediate.

4. Examples

We support our criteria by the following illustrative example, where we shall discuss all three possibilities, namely, $\tau \left(t\right)\le t,$$\tau \left(t\right)\ge t$, and $\tau \left(t\right)\equiv t.$

Example 1.

Consider the Euler delay differential equation

$${\left(t^{\beta }{\left(t^{\alpha }{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+p_0{t}^{\alpha +\beta -3}y\left(\lambda t\right)=0,p_0>0,\lambda >0,\beta \ge \alpha >1.$$

(28)

Now

$${\rho }_1(t)={\displaystyle \frac{t^{1-\alpha }}{\alpha -1}},{\rho }_2(t)={\displaystyle \frac{t^{1-\beta }}{\beta -1}},{\rho }_{12}(t)={\displaystyle \frac{t^{2-\alpha -\beta }}{(\beta -1)(\alpha +\beta -2)}}.$$

It follows from (11) that

$$r(t)=t^{{\displaystyle \frac{\alpha +\beta }{2}}},\rho (t)={\displaystyle \frac{2t^{\frac{2-\alpha -\beta }{2}}}{\alpha +\beta -2}}.$$

It is easy to see that (9), (10), and (12) hold true.

•

Case $0<\lambda <1.$

In our example, condition (23) takes the form

$${\displaystyle \frac{2p_0}{{(\alpha +\beta -2)}^3}}\left(4+(\alpha +\beta -2)ln{\displaystyle \frac{1}{\lambda }}\right)>1$$

(29)

which by Corollary 1 guarantees property (A) of (28).

•

Case $\lambda >1.$

Now, condition (25) simplifies to

$${\displaystyle \frac{2p_0}{{(\alpha +\beta -2)}^3}}{\lambda }^{{\displaystyle \frac{2-\alpha -\beta }{2}}}\left(4+(\alpha +\beta -2)ln\lambda \right)>1$$

which is, by Corollary 2, sufficient for property (A) of (28).

•

Case $\lambda =1.$

It is easy to verify that condition (26) reduces to

$${\displaystyle \frac{p_0}{{(\alpha +\beta -2)}^3}}>{\displaystyle \frac{1}{12\sqrt{3}}}$$

which, by Corollary 3, yields property (A) of (28).

5. Concluding Remarks

This article introduces a new type of comparison theorems that allow us to investigate a noncanonical differential equation with quasiderivatives using a suitable simple canonical equation. In the work, all cases are investigated, namely, the delayed and advanced argument, as well as the equation without a deviating argument. The progress achieved has been demonstrated on the Euler equation. The presented comparative approach to the investigation of noncanonical equations opens up new applications for the future. Natural extensions include higher-order equations, nonlinear equations, and neutral differential equations.