Noncanonical Third-Order Advanced Differential Equations of Unstable Type: Oscillation and Property B via Canonical Transform

1. Introduction

This paper is concerned with the oscillation and asymptotic behavior of solutions of third-order unstable type equation of the form

$${\left(a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }=q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right),t\ge t_0>0.\left(E\right)$$

Here after we assume that:

(H₁)

$\alpha$ is a quotient of odd positive integers;

(H₂)

$a_2,a_1,q\in C([t_0,\infty ),(0,\infty )),$$\sigma \in C^1([t_0,\infty ),\mathbb{R})$ such that ${\sigma }^{\prime }\left(t\right)>0$ and $\sigma \left(t\right)\ge t$ for all $t\ge t_0$;

(H₃)

the equation $\left(E\right)$ is in noncanonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_2\left(t\right)}}dt<\infty \mathrm{and}{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_1\left(t\right)}}dt<\infty .$$

By a solution of equation $\left(E\right),$ it is meant a function $y:[T_y,\infty )\to \mathbb{R},$$T_x\ge t_0$ such that $y\left(t\right),$$a_1\left(t\right)y^{\prime }\left(t\right)$ and $a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }$ are continuously differentiable and satisfy $\left(E\right)$ on $[T_y,\infty ).$ We consider only those solutions $y\left(t\right)$ of $\left(E\right)$ that satisfy $sup\left\{\right|y\left(t\right)|:t\ge T\}>0$ for all $T\ge T_y.$ Such a solution is said to be oscillatory if it has a sequence of zeros tending to infinity, and nonoscillatory otherwise.

Advanced type differential equations are used to describe the phenomenon in which the evolution of the system depends on both present and future time. The possibility of introducing an advance into the equation to take into account future influence that may actually affect the present, makes such equations a useful tool in various economic problems, mathematics of networks, optimization, as well as, mathematical modeling of engineering problems such as, concerning electrical power systems, materials and energy. So, the advanced differential equations have appeared as a response to the urgent need for formulating models that includes a description of a future time, which is taken into consideration in many of life’s problems, see [17] and the references cited therein.

In view of the above mentioned practical applications of various types of advanced differential equations as well as a number of mathematical problems involved in [6]. The area of the qualitative theory of such equations attracted a large amount of research interest in the last three decades. The oscillatory properties of the following differential equation

$${\left(a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)=0,\left(E_1\right)$$

or its particular cases or its generalizations has been extensively investigated in the literature. See, for example, the monographs [11,16,19], the research papers [1,4,7,10,14,15,17,21] and the references cited therein. However from the review of literature, there are a few results available in the literature concerning the oscillatory and asymptotic behavior of solutions of the advanced type equation $\left(E\right)$ and its particular cases or its generalizations, see, for example, the monographs [13,19], the research papers [3,5,8,9,10,14,15,17] and the references cited therein.

In [1], the authors studied the oscillatory and property B for the equation $\left(E\right)$ when it is in canonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_2\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_1\left(t\right)}}dt=\infty$$

and in [12], the authors discussed the oscillatory properties of equation $\left(E\right)$ when it is in semi-canonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_2\left(t\right)}}dt<\infty \mathrm{and}{a}_1\left(t\right)\equiv 1.$$

Therefore in this paper, we consider the equation $\left(E\right)$ in noncanonical form and then we investigate the oscillatory and property B for the equation $\left(E\right)$. This is obtained by transforming the equation $\left(E\right)$ into canonical type equation and then using Koplatadze et al. [14] method to get Property B and oscillation of all solutions of $\left(E\right).$ Hence, the results obtained in this paper fill the gap in the oscillation theory of unstable type functional differential equations. Examples are provided to illustrate the importance and novelty of the main results.

2. Preliminary Results

First, we transform the equation $\left(E\right)$ into a canonical form. In view of $\left(H_3\right),$ we can use the following notation:

$$\begin{array}{ccc}\hfill {\Omega }_j\left(t\right)& =& {\int }_t^{\infty }{\displaystyle \frac{ds}{a_j\left(s\right)}},j=1,2,\Omega \left(t\right)={\int }_t^{\infty }{\displaystyle \frac{{\Omega }_2\left(s\right)}{a_1\left(s\right)}}ds,\hfill \\ \hfill {\Omega }_{*}\left(t\right)& =& {\int }_t^{\infty }{\displaystyle \frac{{\Omega }_1\left(s\right)}{a_2\left(s\right)}}ds,b_1\left(t\right)={\displaystyle \frac{a_1\left(t\right){\Omega }^2\left(t\right)}{{\Omega }_{*}\left(t\right)}},b_2\left(t\right)={\displaystyle \frac{a_2\left(t\right){\Omega }_{*}^2\left(t\right)}{\Omega \left(t\right)}}.\hfill \end{array}$$

Next, we use Theorem 2.1 of [10] instead of the result due to Trench [22] to transform the equation $\left(E\right)$ in the equivalent canonical form as

$${\left(b_2\left(t\right){\left(b_1\left(t\right){\left({\displaystyle \frac{y\left(t\right)}{\Omega \left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }={\Omega }_{*}\left(t\right)q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right).$$

(1)

Now, letting $z\left(t\right)={\displaystyle \frac{y\left(t\right)}{\Omega \left(t\right)}}$ in (1) and using the notation

$$Q\left(t\right)={\Omega }_{*}\left(t\right){\Omega }^{\alpha }\left(\sigma \left(t\right)\right)q\left(t\right),$$

the following results in [10] are immediate.

Theorem 1.

The noncanonical nonlinear differential equation $\left(E\right)$ possesses a solution $x\left(t\right)$ if and only if the canonical equation

$${\left(b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }=Q\left(t\right)z^{\alpha }\left(\sigma \left(t\right)\right)\left(E_c\right)$$

has the solution $z\left(t\right)={\displaystyle \frac{y\left(t\right)}{\Omega \left(t\right)}}.$

Corollary 1.

The noncanonical nonlinear differential equation $\left(E\right)$ has an eventually positive solution if and only the equation $\left(E_c\right)$ has an eventually positive solution.

Corollary 1 essentially simplifies the investigation of $\left(E\right).$ Since, for the equation $\left(E_c\right),$ we deal with only two classes of eventually positive solutions instead of four, namely either

$$N_1:z\left(t\right)>0,b_1\left(t\right)z^{\prime }\left(t\right)>0,b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }<0,{\left(b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0,$$

or

$$N_3:z\left(t\right)>0,b_1\left(t\right)z^{\prime }\left(t\right)>0,b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }>0,{\left(b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0,$$

for sufficiently large $t.$

Following [7,14,15,18], we say that $\left(E_c\right)$ has property B if every nonoscillatory solution of $\left(E\right)$ satisfies the class $N_3,$ that is, the class $N_1$ is empty.

The importance of property B consists in the following fact. For the particular case of $\left(E_c\right),$ namely, for the differential equation

$$y^{\prime \prime \prime }\left(t\right)=q\left(t\right)y\left(t\right)$$

there always exists a nonoscillatory solution satisfying class $N_3.$ Therefore, the interest of the researchers has been aimed in finding criteria for all nonoscillatory solutions of such equations satisfy just class $N_3.$ Hence various kinds of sufficient conditions for property B appeared. If we consider the equation $\left(E\right)$ directly, then the nonoscillatory solutions of $\left(E\right)$ satisfies one of the following four classes:

(I)

$y\left(t\right)>0,$$a_1\left(t\right)y^{\prime }\left(t\right)>0,$$a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }<0,$${\left(a_2\left(t\right){\left(a_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0;$

(II)

$y\left(t\right)>0,$$a_1\left(t\right)y^{\prime }\left(t\right)>0,$$a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }>0,$${\left(a_2\left(t\right){\left(a_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0;$

(III)

$y\left(t\right)>0,$$a_1\left(t\right)y^{\prime }\left(t\right)<0,$$a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }>0,$${\left(a_2\left(t\right){\left(a_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0;$

(IV)

$y\left(t\right)>0,$$a_1\left(t\right)y^{\prime }\left(t\right)<0,$$a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }<0,$${\left(a_2\left(t\right){\left(a_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0,$

for all $t\ge t_1.$ Hence to get property B of $\left(E\right)$ one has to eliminate the three cases (I), (III) and (IV). However, in this paper, by transforming the equation $\left(E\right)$ into canonical form the number of cases of nonoscillatory solutions reduced to two without assuming any extra conditions. Therefore, the results obtained in this paper are new and complement to the existing ones. Examples are provided to illustrate the significance and novelty of the main results.

3. Main Results

First, we derive some important monotonic properties and estimates of nonoscillatory solutions, that will be used in our main results.

For convenient let us denote

$$B_j\left(t\right)={\int }_{t_{*}}^t{\displaystyle \frac{1}{b_j\left(s\right)}}ds,j=1,2,Q_1\left(t\right)={\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)ds$$

where $t_{*}$ is large enough.

Lemma 1.

Let $z\left(t\right)\in N_3$ be a positive solution of $\left(E_c\right).$ Then $z\left(t\right)$ is decreasing and

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim}}z\left(t\right)=\infty \mathrm{and}{\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\mathrm{is}\mathrm{eventually}\mathrm{increasing}.\end{array}$$

(2)

Proof. 

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right)$ belonging to $N_3$, let us say for $t\ge t_{*}.$ Then $z\left(t\right)$ and $b_1\left(t\right)z^{\prime }\left(t\right)$ are increasing for all $t\ge t_1\ge t_{*}.$ Now

$$\begin{array}{c}\hfill z\left(t\right)=z\left(t_1\right)+{\int }_{t_1}^t{\displaystyle \frac{b_1\left(s\right)z^{\prime }\left(s\right)}{b_1\left(s\right)}}ds\ge b_1\left(t_1\right)z^{\prime }\left(t_1\right){\int }_{t_1}^t{\displaystyle \frac{1}{b_1\left(s\right)}}ds,\end{array}$$

(3)

which implies that $z\left(t\right)\to \infty$ as $t\to \infty .$ Since $b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }$ is also increasing, one can easily prove that $b_1\left(t\right)z^{\prime }\left(t\right)\to \infty$ as $t\to \infty .$ Again from (3), we have

$$\begin{array}{c}\hfill z\left(t\right)\le z\left(t_1\right)-b_1\left(t\right)z^{\prime }\left(t\right){\int }_{t_{*}}^t{\displaystyle \frac{1}{b_1\left(s\right)}}ds+b_1\left(t\right)z^{\prime }\left(t\right)B_1\left(t\right).\end{array}$$

Using $b_1\left(t\right)z^{\prime }\left(t\right)\to \infty$ as $t\to \infty$ in the last inequality, we obtain

$$z\left(t\right)\le B_1\left(t\right)b_1\left(t\right)z^{\prime }\left(t\right)$$

and this yields

$${\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)}^{\prime }={\displaystyle \frac{B_1\left(t\right)b_1\left(t\right)z^{\prime }\left(t\right)-z\left(t\right)}{b_1\left(t\right)B_1^2\left(t\right)}}\ge 0.$$

Hence ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is increasing and this ends the proof. □

Lemma 2.

Let $z\left(t\right)\in N_1$ be a positive solution of $\left(E_c\right).$ Then

$$\begin{array}{c}\hfill {\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\mathrm{is}\mathit{eventually}\mathit{decreasing}.\end{array}$$

(4)

Proof. 

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right)$ such that $z\left(t\right)\in N_1$ for all $t\ge t_{*}.$ Since $b_1\left(t\right)z^{\prime }\left(t\right)$ is decreasing, we see that

$$\begin{array}{c}\hfill z\left(t\right)=z\left(t_{*}\right)+{\int }_{t_{*}}^t{\displaystyle \frac{b_1\left(s\right)z^{\prime }\left(s\right)}{b_1\left(s\right)}}ds\ge b_1\left(t\right)z^{*}\left(t\right)B_1\left(t\right).\end{array}$$

This implies that

$${\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)}^{\prime }={\displaystyle \frac{B_1\left(t\right)b_1\left(t\right)z^{\prime }\left(t\right)-z\left(t\right)}{b_1\left(t\right)B_1^2\left(t\right)}}\le 0.$$

Hence ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is decreasing and this ends the proof. □

Lemma 3.

Let $z\left(t\right)\in N_1$ be a positive solution of $\left(E_c\right).$ Then

$$\begin{array}{c}\hfill z\left(\sigma \left(t\right)\right)\ge B(\sigma \left(t\right),t)b_1\left(\sigma \left(t\right)\right)z^{\prime }\left(\sigma \left(t\right)\right),\end{array}$$

(5)

where

$$B(\sigma \left(t\right),t)={\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}ds.$$

Proof. 

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right)$ such that $z\left(t\right)\in N_1$ for all $t\ge t_{*}.$ Using $b_1\left(t\right)z^{\prime }\left(t\right)$ being decreasing, we have

$$\begin{array}{ccc}\hfill z\left(\sigma \right(t\left)\right)& =& z\left(t\right)+{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{b\left(s\right)z^{\prime }\left(s\right)}{b\left(s\right)}}ds\hfill \\ & \ge & B(\sigma \left(t\right),t)b\left(\sigma \left(t\right)\right)z^{\prime }\left(\sigma \left(t\right)\right).\hfill \end{array}$$

The proof of the lemma is complete. □

Next, we provide some criteria for the class $N_1$ of $\left(E_c\right)$ to be empty, which in turn implies the property B of $\left(E\right).$

Theorem 2.

Let $y\left(t\right)$ be a positive solution of $\left(E\right)$. If $\alpha =1$ and

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; inf}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}{\int }_s^{\infty }{\displaystyle \frac{1}{b_2\left(s_1\right)}}{\int }_{s_1}^{\infty }Q\left(s_2\right)d{s}_{2}d{s}_{1}ds>{\displaystyle \frac{1}{e}},\end{array}$$

(6)

then equation $\left(E\right)$ has property B.

Proof. 

Since $y\left(t\right)$ is a positive solution of $\left(E\right),$ then by Corollary 1, the corresponding function $z\left(t\right)={\displaystyle \frac{y\left(t\right)}{\Omega \left(t\right)}}$ is a positive solution of $\left(E_c\right)$ belonging to $N_1$ or $N_3$ for all $t\ge t_1.$ To prove the result, we have to show that the class $N_1$ is empty. Assume the contrary that $z\left(t\right)\in N_1$ for all $t\ge t_1.$ Integrating $\left(E_c\right)$ from t to $\infty ,$ one gets

$$-b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\ge {\int }_t^{\infty }Q\left(s\right)z\left(\sigma \left(s\right)\right)ds,$$

or

$$-{\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\ge {\displaystyle \frac{z\left(\sigma \right(t\left)\right)}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)ds.$$

Another integration from t to ∞ yields

$$z^{\prime }\left(t\right)\ge {\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{z\left(\sigma \right(s\left)\right)}{b_2\left(s\right)}}{\int }_s^{\infty }Q\left(u\right)duds.$$

It follows from the last inequality that $z\left(t\right)$ is a positive solution of the differential inequality

$$z^{\prime }\left(t\right)-\left({\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_s^{\infty }Q\left(u\right)duds\right)z\left(\sigma \left(t\right)\right)\ge 0.$$

(7)

But, by Theorem 2.4.1 of [16], condition (6) ensures that (7) has no positive solutions. This is a contradiction, and we conclude that $\left(E\right)$ has property B. The proof of the theorem is complete. □

Remark 1.

It follows from the proof of Theorem 2 that if at least one of the following conditions is satisfied:

$$\begin{array}{c}\hfill {\int }_{t_0}^{\infty }Q\left(t\right)dt=\infty ,\\ \hfill {\int }_{t_0}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)dsdt=\infty ,\end{array}$$

(8)

then any nonoscillatory solution $z\left(t\right)$ of $\left(E_c\right)$ cannot satisfy the class $N_1.$ Therefore, we may assume that the corresponding integrals in (8) are convergent.

Theorem 3.

Let $y\left(t\right)$ be an eventually positive solution of $\left(E\right)$. If

$$\begin{array}{c}\hfill {\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_t^{\infty }{Q}_1\left(s\right)dsdt=\infty \end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \underset{t\to \infty }{lim\; sup}}}\left({\displaystyle \frac{1}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_{t_{*}}^t{B}_1^{\alpha }\left(\sigma \left(s\right)\right)B_1\left(s\right)Q_1\left(s\right)ds+{\int }_t^{\sigma \left(t\right)}{B}_1\left(s\right)Q_1\left(s\right)ds\right.\\ \hfill \left.+B_1\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)ds\right)\left\{\begin{array}{c}>1if\alpha =1\hfill \\ >0if\alpha >1,\hfill \end{array}\right.\end{array}$$

(10)

then equation $\left(E\right)$ has property B.

Proof. 

Let $y\left(t\right)$ be an eventually positive solution of $\left(E\right).$ Then proceeding as in the proof of Theorem 2, we see that the function $z\left(t\right)\in N_1$ or $z\left(t\right)\in N_3.$ To prove the theorem, we have to show that the class $N_1$ is empty. Assume the contrary that $z\left(t\right)\in N_1$ for all $t\ge t_1.$ Integrating $\left(E_c\right)$ from t to ∞ twice and using the monotonicity of $z\left(\sigma \right(t\left)\right),$ we get

$$b_1\left(t\right)z^{\prime }\left(t\right)\ge {\int }_{t_{*}}^{\infty }{Q}_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds.$$

Again integrating the last inequality form $t_{*}$ to t and changing the order of integration, we obtain

$$z\left(t\right)\ge {\int }_{t_{*}}^t{Q}_1\left(s\right)B_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds+B_1\left(t\right){\int }_t^{\infty }{Q}_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds.$$

Therefore,

$$\begin{array}{ccc}\hfill z\left(\sigma \left(t\right)\right)\ge {\int }_{t_{*}}^t{Q}_1\left(s\right)B_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds& +& {\int }_t^{\sigma \left(t\right)}{Q}_1\left(s\right)B_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds\hfill \\ & +& B_1\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds.\hfill \end{array}$$

Using the fact that $z\left(t\right)$ is increasing and ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is decreasing, we get

$$\begin{array}{ccc}\hfill z\left(\sigma \left(t\right)\right)\ge {\displaystyle \frac{z^{\alpha }\left(\sigma \left(t\right)\right)}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_{t_{*}}^t{B}_1^{\alpha }\left(\sigma \left(s\right)\right)Q_1\left(s\right)B_1\left(s\right)ds& +& z^{\alpha }\left(\sigma \left(t\right)\right){\int }_t^{\sigma \left(t\right)}{Q}_1\left(s\right)B_1\left(s\right)ds\hfill \\ & +& z^{\alpha }\left(\sigma \left(t\right)\right)B_1\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)ds.\hfill \end{array}$$

That is,

$$\begin{array}{c}\hfill z^{1-\alpha }\left(\sigma \left(t\right)\right)\ge {\displaystyle \frac{1}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_{t_{*}}^t{B}_1^{\alpha }\left(\sigma \left(s\right)\right)Q_1\left(s\right)B_1\left(s\right)ds+{\int }_t^{\sigma \left(t\right)}{Q}_1\left(s\right)B_1\left(s\right)ds\\ \hfill +B_1\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)ds.\end{array}$$

(11)

When $\alpha =1$ in (11), we get a contradiction with (10), and from (9) we see that $z\left(t\right)\to \infty$ as $t\to \infty .$ So taking lim sup as $t\to \infty$ on both sides of (11), we are led to a contradiction again with (10) for $\alpha >1.$ The proof of the theorem is complete. □

Theorem 4.

Let $y\left(t\right)$ be an eventually positive solution of $\left(E\right)$ and let $0<\alpha <1.$ If

$$\begin{array}{c}\hfill {\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)B_1^{\alpha }\left(\sigma \left(s\right)\right)dsdt=\infty \end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \underset{t\to \infty }{lim\; sup}}}\left({\displaystyle \frac{1}{B_1\left(\sigma \left(t\right)\right)}}{\int }_{t_{*}}^t{B}_1^{\alpha }\left(\sigma \left(s\right)\right)B_1\left(s\right)Q_1\left(s\right)ds+B_1^{\alpha -1}\left(\sigma \left(t\right)\right){\int }_t^{\sigma \left(t\right)}{B}_1\left(s\right)Q_1\left(s\right)ds\right.\\ \hfill \left.+B_1^{\alpha }\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)ds\right)>0,\end{array}$$

(13)

then the equation $\left(E\right)$ has property B.

Proof. 

Proceeding as in the proof of Theorem 2, we assume that $z\left(t\right)\in N_1$ for all $t\ge t_{*}.$ From (4), we have ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is decreasing and in view of (12), we shall prove that

$$\begin{array}{c}\hfill {\displaystyle {\displaystyle \underset{t\to \infty }{lim}}}{\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}=0.\end{array}$$

(14)

If not, let us assume that ${lim}_{t\to \infty }{\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}=l>0.$ Then $z\left(t\right)\ge l{B}_1\left(t\right)$ and using this in $\left(E_c\right)$ we get

$${\left(b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\ge l^{\alpha }Q\left(t\right)B_1^{\alpha }\left(\sigma \left(t\right)\right).$$

Integrating the last inequality twice yields

$$b_1\left(t_{*}\right)z^{\prime }\left(t_{*}\right)\ge l^{\alpha }{\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)B_1^{\alpha }\left(\sigma \left(s\right)\right)dsdt,$$

which contradicts (12) and so we conclude that (14) holds. Now, setting $x\left(t\right)={\displaystyle \frac{z\left(\sigma \right(t\left)\right)}{B_1\left(\sigma \left(t\right)\right)}}$ then (7) yields

$$\begin{array}{c}\hfill x^{1-\alpha }\left(t\right)\ge \left({\displaystyle \frac{1}{B_1\left(\sigma \left(t\right)\right)}}{\int }_{t_{*}}^t{B}_1^{\alpha }\left(\sigma \left(s\right)\right)B_1\left(s\right)Q_1\left(s\right)ds+B_1^{1-\alpha }\left(\sigma \left(t\right)\right){\int }_t^{\sigma \left(t\right)}{B}_1\left(s\right)Q_1\left(s\right)ds\right.\\ \hfill \left.+B_1^{\alpha }\left(\sigma \left(t\right)\right){\int }_{\sigma \left(t\right)}^{\infty }{Q}_1\left(s\right)ds\right).\end{array}$$

(15)

Taking $limsup$ as $t\to \infty$ on both sides of (15) and using (14), we get a contradiction with (2). The proof of the theorem is complete. □

In the following, we eliminate class $N_3$ to get criteria for the oscillation of all solutions of $\left(E\right).$

Theorem 5.

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right).$ Assume that there exists a function $\tau \left(t\right)\in C^1\left([t_0,\infty )\right)$ such that

$${\tau }^{\prime }\left(t\right)\ge 0,\tau \left(t\right)<t,\delta \left(t\right)=\sigma \left(\tau \left(\tau \left(t\right)\right)\right)>t.$$

(16)

If the first-order advanced differential equation

$$w^{\prime }\left(t\right)-\left({\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)duds\right)w^{\alpha }\left(\delta \left(t\right)\right)=0$$

(17)

is oscillatory, then class $N_3$ cannot hold.

Proof. 

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right),$ satisfying class $N_3.$ Integrating $\left(E_c\right)$ from $\tau \left(t\right)$ to t, we have

$$b_2\left(t\right){\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-b_2\left(\tau \left(t\right)\right){\left(b_1\left(\tau \left(t\right)\right)z^{\prime }\left(\tau \left(t\right)\right)\right)}^{\prime }={\int }_{\tau \left(t\right)}^{t}Q\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds$$

or

$${\left(b_1\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\ge {\displaystyle \frac{z^{\alpha }\left(\sigma \left(\tau \left(t\right)\right)\right)}{b_2\left(t\right)}}{\int }_{\tau \left(t\right)}^{t}Q\left(s\right)ds.$$

An integration from $\tau \left(t\right)$ to t yields

$$\begin{array}{ccc}\hfill b_1\left(t\right)z^{\prime }\left(t\right)& \ge & {\int }_{\tau \left(t\right)}^t{\displaystyle \frac{z^{\alpha }\left(\sigma \left(\tau \left(s\right)\right)\right)}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)duds\hfill \\ & \ge & z^{\alpha }\left(\delta \left(t\right)\right){\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)duds.\hfill \end{array}$$

Consequently, $z\left(t\right)$ is a positive solution of the advanced differential inequality

$$\begin{array}{c}\hfill z^{\prime }\left(t\right)-\left({\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)duds\right)z^{\alpha }\left(\delta \left(t\right)\right)\ge 0.\end{array}$$

(18)

Hence, by Lemma 2.3 of [16], the corresponding differential equation (17) has also a positive solution, which is a contradiction. The proof of the theorem is complete. □

For $\alpha =1,$ using Theorem 2.4.1 of [11], we obtain the following corollary.

Corollary 2.

Let $\alpha =1$ and there exists a function $\tau \left(t\right)\in C^1\left([t_0,\infty )\right)$ such that (16) holds. If

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; inf}}{\int }_t^{\delta \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}{\int }_{\tau \left(s\right)}^s{\displaystyle \frac{1}{b_2\left(s_1\right)}}{\int }_{\tau \left(s_1\right)}^{s_1}Q\left(s_2\right)d{s}_{2}d{s}_{1}ds>{\displaystyle \frac{1}{e}}\end{array}$$

(19)

then the class $N_3$ cannot hold.

Next, we provide an explicit condition for the class $N_3$ is empty when $\alpha >1.$

Corollary 3.

Let $\alpha >1$ and there exists a function $\tau \left(t\right)\in C^1\left([t_0,\infty )\right)$ such that (16) holds. If

$$\begin{array}{c}\hfill {\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)dudsdt=\infty ,\end{array}$$

(20)

then the class $N_3$ cannot hold.

Proof. 

Proceeding as in Theorem 5, we arrive at (18). Since $z\left(t\right)$ is increasing and $\delta \left(t\right)>t,$ we have from (18)

$${\displaystyle \frac{z^{\prime }\left(t\right)}{z\left(t\right)}}\ge {\displaystyle \frac{1}{b_1\left(t\right)}}\left({\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)duds\right).$$

Integrating the last inequality from $t_1$ to $\infty ,$ we get

$${\displaystyle \frac{1}{(\alpha -1)z^{\alpha -1}\left(t_1\right)}}\ge {\int }_{t_1}^{\infty }{\displaystyle \frac{1}{b_1\left(t\right)}}{\int }_{\tau \left(t\right)}^t{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_{\tau \left(s\right)}^{s}Q\left(u\right)dudsdt,$$

which contradicts (20). The proof of the corollary is complete.

Next, we present another condition for the elimination of class $N_3.$ □

Theorem 6.

Let $z\left(t\right)$ be an eventually positive solution of $\left(E_c\right).$ If

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; sup}}{\displaystyle \frac{1}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}{\int }_t^s{\displaystyle \frac{1}{b_2\left(s_1\right)}}{\int }_t^{s_1}Q\left(s_2\right)B_1^{\alpha }\left(s_2\right)d{s}_{2}d{s}_{1}ds>M\end{array}$$

(21)

where $M=1$ if $\alpha =1$ and $M=0$ if $\alpha >1,$ then the class $N_3$ cannot hold.

Proof. 

Let $z\left(t\right)$ be a positive of $\left(E_c\right)$ belonging to $N_3$ for all $t\ge t_{*}.$ Integrating $\left(E_c\right)$ from s to t and using (2), we have

$$\begin{array}{ccc}\hfill {\left(b_1\left(s\right)z^{\prime }\left(s\right)\right)}^{\prime }& \ge & {\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_t^{s}Q\left(u\right)z^{\alpha }\left(\sigma \left(u\right)\right)du\hfill \\ & \ge & {\displaystyle \frac{z^{\alpha }\left(\sigma \left(t\right)\right)}{B_1^{\alpha }\left(\sigma \left(t\right)\right)b_2\left(s\right)}}{\int }_t^{s}Q\left(u\right)B_1^{\alpha }\left(\sigma \left(u\right)\right)du.\hfill \end{array}$$

Integrating the last inequality in $s,$ we obtain

$$\begin{array}{ccc}\hfill z^{\prime }\left(s\right)& \ge & {\displaystyle \frac{z^{\alpha }\left(\sigma \left(t\right)\right)}{B_1^{\alpha }\left(\sigma \left(t\right)\right)b_1\left(s\right)}}{\int }_t^s{\displaystyle \frac{1}{b_2\left(s_1\right)}}{\int }_t^{s_1}Q\left(s_2\right)B_1^{\alpha }\left(s_2\right)d{s}_{2}d{s}_1.\hfill \end{array}$$

Again integrating, we get

$$\begin{array}{ccc}\hfill z\left(s\right)& \ge & {\displaystyle \frac{z^{\alpha }\left(\sigma \left(t\right)\right)}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_t^s{\displaystyle \frac{1}{b_1\left(s_1\right)}}{\int }_t^{s_1}{\displaystyle \frac{1}{b_2\left(s_2\right)}}{\int }_t^{s_2}Q\left(s_3\right)B_1^{\alpha }\left(s_3\right)d{s}_{3}d{s}_{2}d{s}_1.\hfill \end{array}$$

Setting $s=\sigma \left(t\right)$ in the last inequality yields

$$\begin{array}{c}\hfill z^{1-\alpha }\left(\sigma \left(t\right)\right)\ge {\displaystyle \frac{1}{B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}{\int }_t^s{\displaystyle \frac{1}{b_2\left(s_1\right)}}{\int }_t^{s_1}Q\left(s_2\right)B_1^{\alpha }\left(s_2\right)d{s}_{2}d{s}_{1}ds.\end{array}$$

(22)

Letting $\alpha =1$ in (22), we get a contradiction with (21) and for $\alpha >1$ in (22) and using the fact from (2) that $z\left(t\right)\to \infty$ as $t\to \infty ,$ we again obtain a contradiction with (21). The proof of the theorem is complete. □

Combining the criteria obtained for both classes $N_1$ and $N_3$ to be empty, we get a criterion for oscillation of $\left(E\right).$

Theorem 7.

Let $\alpha =1.$ If all conditions of Theorem 2 and Corollary 2 hold, then equation $\left(E\right)$ is oscillatory.

Proof. 

Assume the contrary that $y\left(t\right)$ is a positive solution of $\left(E\right)$ for all $t\ge t_{*}\ge t_0.$ Then by Corollary 1, we see that $z\left(t\right)={\displaystyle \frac{y\left(t\right)}{\Omega \left(t\right)}}$ is a positive solution of $\left(E_c\right)$ and belongs to either $N_1$ or $N_3$ for all $t\ge t_{*}.$ From Theorem 2, we obtain that the class $N_1$ is empty and from Corollary 2, the class $N_3$ is empty. Therefore, we conclude that equation $\left(E_c\right)$ is oscillatory which in turn implies $\left(E\right)$ is oscillatory. The proof of the theorem is complete. □

Similarly, we can prove the following theorems.

Theorem 8.

Let $\alpha =1$ or $\alpha >1$. If all conditions of Theorem 3 and Theorem 6 hold, then equation $\left(E\right)$ is oscillatory.

Theorem 9.

Let $\alpha >1$. If all conditions of Theorem 3 and Corollary 3 hold, then equation $\left(E\right)$ is oscillatory.

4. Examples

In this section, we present three examples to show the importance of our main results.

Example 1.

Consider the third-order noncanonical advanced differential equation

$${\left(t^2{\left(t^2{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-\mu ty\left(\lambda t\right)=0,t\ge 1,$$

(23)

where $\mu >0$ and $\lambda >1.$

By a simple calculation, we see that the transformed equation as

$$z^{\prime \prime \prime }\left(t\right)-{\displaystyle \frac{\mu }{{\lambda }^2{t}^3}}z\left(\lambda t\right)=0,t\ge 1,$$

which is in canonical form. Here $b_1\left(t\right)=b_2\left(t\right)=1$ and $Q\left(t\right)={\displaystyle \frac{\mu }{{\lambda }^2{t}^3}}$ and $\alpha =1.$ The condition (6) becomes

$${\displaystyle \frac{\mu }{2{\lambda }^2}}ln\lambda >{\displaystyle \frac{1}{e}},$$

that is, condition (6) holds if $\mu >{\displaystyle \frac{2{\lambda }^2}{eln\lambda }}.$

Choose $\tau \left(t\right)={\lambda }_{1}t$ with ${\lambda }_1>{\displaystyle \frac{1}{\sqrt{\lambda }}}$ and $\delta \left(t\right)=\lambda {\lambda }_1^{2}t.$ So that condition (16) holds. The condition (19) becomes

$${\displaystyle \frac{\mu }{2{\lambda }^2{\lambda }_1^3}}ln\lambda {\lambda }_1^2>{\displaystyle \frac{1}{e}},$$

that is, condition (19) holds if $\mu >{\displaystyle \frac{2{\lambda }^2{\lambda }_1}{eln\lambda {\lambda }_1^2}}.$

Therefore, by Theorem 7, the Equation (23) is oscillatory if

$$\mu >max\left\{{\displaystyle \frac{2{\lambda }^2}{eln\lambda }},{\displaystyle \frac{2{\lambda }^2{\lambda }_1}{eln\lambda {\lambda }_1^2}}\right\}.$$

Example 2.

Consider the nonlinear noncanonical third-order advanced differential equation

$${\left(t^2{\left(t^2{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-\mu t^5{y}^3\left(\lambda t\right)=0,t\ge 1,$$

(24)

where $\mu >0$ and $\lambda >1.$

By a simple calculation, the transformed equation as

$$z^{\prime \prime \prime }\left(t\right)-{\displaystyle \frac{\mu }{4{\lambda }^6{t}^3}}{z}^3\left(\lambda t\right)=0,t\ge 1,$$

which is in canonical form. Here $b_1\left(t\right)=b_2\left(t\right)=1$ and $Q\left(t\right)={\displaystyle \frac{\mu }{4{\lambda }^6{t}^3}}$ and $\alpha =3.$ The condition (9) becomes

$${\int }_1^{\infty }{\displaystyle \frac{\mu }{8{\lambda }^6}}{\displaystyle \frac{1}{t}}dt=\infty ,$$

that is, condition (9) holds. The condition (10) becomes

$${\displaystyle \frac{\mu }{{\lambda }^6}}\left({\displaystyle \frac{1}{6}}+{\displaystyle \frac{1}{8}}ln\lambda \right)>0,$$

that is, condition (10) holds if $\mu >0.$

Choose $\tau \left(t\right)={\lambda }_{1}t<t$ with ${\lambda }_1>{\displaystyle \frac{1}{\sqrt{\lambda }}}$ and $\delta \left(t\right)=\lambda {\lambda }_1^{2}t>t,$ so condition (16) holds. The condition (20) becomes

$${\int }_1^{\infty }{\displaystyle \frac{\mu }{8{\lambda }^6}}\left(1-{\displaystyle \frac{1}{{\lambda }_1}}\right)\left(1-{\displaystyle \frac{1}{{\lambda }_1^2}}\right){\displaystyle \frac{1}{t}}dt=\infty ,$$

that is, condition (20) holds if $\mu >0.$ Hence, the Equation (24) is oscillatory by Theorem 9 if $\mu >0.$

Example 3.

Consider the noncanonical third-order advanced differential equation

$${\left(e^t{\left(e^t{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-\mu e^{2t}{y}^{1/3}\left(\lambda t\right)=0,t\ge 1,$$

(25)

where $\mu >0$ and $\lambda >1$ are constants.

By a simple calculation, we find the transformed equation as

$${\left(e^{-t}{\left(e^{-t}{z}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-2^{2/3}\mu e^{-{\displaystyle \frac{2\lambda }{3}}t}{z}^{1/3}\left(\lambda t\right)=0,t\ge 1,$$

which is in canonical form. Here, $b_1\left(t\right)=b_2\left(t\right)=e^{-t},$$Q\left(t\right)=2^{2/3}\mu e^{{\displaystyle \frac{-2\lambda }{3}}t}$ and $\alpha ={\displaystyle \frac{1}{3}}.$ With a further calculation we see that $B_1\left(t\right)\approx e^t$ and $Q_1\left(t\right)={\displaystyle \frac{3\mu }{2^{1/3}\lambda }}{e}^{(1-{\displaystyle \frac{2\lambda }{3}})t}.$ The condition (12) becomes

$${\int }_1^{\infty }{2}^{2/3}\mu e^{(1-\lambda /3)t}dt=\infty$$

if $\lambda \le 3,$ that is condition (12) holds if $\lambda \in (1,3].$ The condition (13) becomes

$${\displaystyle \underset{t\to \infty }{lim}}{\displaystyle \frac{3\mu }{2^{1/3}\lambda }}\left({\displaystyle \frac{e^{2t}}{(2-\frac{\lambda }{3})}}+{\displaystyle \frac{e^{2\lambda t}-e^t}{2(1-\frac{\lambda }{3})}}+{\displaystyle \frac{e^t}{(\frac{2\lambda }{3}-1)}}\right)\to \infty ,$$

as $t\to \infty$ if $\lambda \in (3/2,3).$ That is, condition (13) holds if $\lambda \in (3/2,3).$ Hence, equation (25) has property B by Theorem 4 if $\lambda \in (3/2,3).$

5. Conclusions

In this paper, by using canonical transformation technique, we reduced the studied noncanonical equation $\left(E\right)$ into canonical type and this reduced the set of nonoscillatory solutions into two instead of four, without assuming any extra conditions. Therefore, the criteria obtained here are new and complement to the existing results. Further three examples are given to show the importance of the results obtained here since the criteria already reported cannot be applicable to the equations (23)-(25) since they are noncanonical type equations.Thus the oscillation results presented here are further contribution to the oscillation theory of advanced type differential equations. It is interesting to obtain similar results of this paper to equation $\left(E\right)$ when the following semi-canonical condition

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_2\left(t\right)}}dt=\infty \mathrm{and}{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a_1\left(t\right)}}dt<\infty$$

holds.