Oscillation of Third-Order Thomas–Fermi-Type Nonlinear Differential Equations with an Advanced Argument

Abstract

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of Thomas–Fermi-type third-order nonlinear differential equations with advanced argument of the form ${\left(a_2\left(t\right){\left(a_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)=0,$ under the assumptions that ${\displaystyle {\int }_{t_0}^{\infty }\frac{1}{a_2\left(t\right)}dt<\infty }$ and ${\displaystyle {\int }_{t_0}^{\infty }\frac{1}{a_1\left(t\right)}dt=\infty }$. The results are achieved by transforming the equation into a canonical-type equation and then applying integral averaging techniques and the comparison method to obtain oscillation criteria for the transformed equation. This in turn will imply the oscillation of the original equation. Several examples are provided to illustrate the significance of the main results.

1. Introduction

Advanced-type differential equations are used to describe phenomena in which the evolution of the system depends on both present and future time. The possibility of introducing an advanced argument into an equation to take into account future influences that may actually affect the present makes such equations a useful tool in various economic problems, in the mathematics of networks, and in optimization, as well as in modeling engineering problems such as in electrical power systems, materials, and energy (see [1]).

Oscillation theory represents one of the basic research fields in both physics and mathematics that deals with the behavior of systems that exhibit periodic motions such as in population dynamics, rhythmic beating of the human heart, and electrical circuits. Also, the study of oscillations is important not only in such natural phenomena as the movement of tides and planetary motion, but also in technological systems such as mechanical engineering, control systems, and signal processing (see [2,3]). It is also a well-established area of research on its own.

Delay differential equations of various orders, that is, equations containing both present and past history, have long been the focus of scientists and researchers, and numerous oscillation results have emerged as a consequence; see for example, the monograph [2] and the references therein. However, the ability of advanced-type differential equations to describe and analyze complex phenomena in the real world makes their importance increasingly significant. They are important in medicine, where advanced-type differential equations help to understand drug interactions within the body, and in the study of disease dynamics. They are also useful in weather forecasting, studying ocean movement, and predicting earthquakes.

In view of the variety of applications described above, it is easy to see why the qualitative theory of such equations has gained increased momentum in recent years, including the study of the oscillatory and asymptotic behavior of solutions of third-order functional differential (see the monograph [3] and the references cited therein). In the references [4,5,6,7,8,9], the authors studied the oscillatory behavior of third order delay differential equations, while in [10,11,12,13] advanced-type equations are considered. Moreover in [14,15,16,17], the authors examined similar problems for differential equations with deviating arguments, and see [18,19,20,21] for differential equations with mixed arguments.

In this paper, we wish to develop conditions for the oscillation of the advanced Thomas–Fermi-type functional differential equation

$$L_{3}y\left(t\right)-q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)=0,t\ge t_0,$$

(1)

where

$$L_{0}y=y,L_{i}y=a_i{\left(L_{i-1}y\right)}^{\prime },i=1,2,L_{3}y={\left(L_{2}y\right)}^{\prime }.$$

Without further mention, we assume:

Hypothesis 1.

$a_1\left(t\right)\in C^2([t_0,\infty ),(0,\infty ))$, $a_2\left(t\right)\in C^1([t_0,\infty ),(0,\infty ))$, and $q\left(t\right)\in C([t_0,\infty ),$$(0,\infty ))$;

Hypothesis 2.

$\sigma \left(t\right)\in C^1([t_0,\infty ),R)$, ${\sigma }^{\prime }\left(t\right)>0$, and $\sigma \left(t\right)\ge t$ for $t\ge t_0$;

Hypothesis 3.

α is the ratio of odd positive integers.

By a solution y of (1), we mean a function $y\in C([T_y,\infty ),\mathbf{R})$, $T_y\ge t_0$, where $L_{1}y\left(t\right)$ and $L_{2}y\left(t\right)$ are continuously differentiable and satisfy (1). We restrict our attention to solutions $y\left(t\right)$ of (1) that are continuable; that is, they satisfy $sup\left\{\right|y\left(t\right)|:t\ge T\}>0$ for any $T\ge T_y$. A solution is oscillatory if it possesses a sequence of zeros that tends to infinity, and is nonoscillatory otherwise.

From a review of the literature, we see that most papers on qualitative theory of functional differential equations are devoted to Emden–Fowler -type equations, viz.,

$$L_{3}y\left(t\right)+q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)=0,t\ge t_0,$$

under one or more of the conditions

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

(2)

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

(3)

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

(4)

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

(5)

For example, in [5], the authors considered the Emden–Fowler-type delay differential equation

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

In [4], the authors studied this equation with $a_1\left(t\right)\equiv 1$, and in [6,8,9] with $\alpha =1$. As for Thomas–Fermi-type equations, in [15,16,17], the equation

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

is analyzed, and in [5],

$$L_{3}y=q\left(t\right)y\left(\delta \left(t\right)\right)$$

where $\delta \left(t\right)\ge t$ or $\delta \left(t\right)\le t$ is studied. In [12], the authors considered the advanced-type differential equation

$${(a_2(t){({(a_1(t)y^{\prime }(t))}^{\prime })}^{\alpha })}^{\prime }=p\left(t\right)f(y\left(\sigma \left(t\right)\right),$$

in [11] the equation

$$L_{3}y\left(t\right)=f(t,y\left(\sigma \left(t\right)\right),$$

and in [13]

$$y^{\left(n\right)}\left(t\right)=p\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)$$

all with $\sigma \left(t\right)\ge t$.

Mixed-type equations such as

$${(a(t){(y^{″}(t))}^{\alpha })}^{\prime }=q(t)f(y(\tau (t))+p(t)h(y(\sigma (t))))$$

(6)

(see [7]) and

$${(a(t){(y^{″}(t))}^{\alpha })}^{″}=q(t)f(y(\tau (t))+p(t)h(y(\sigma (t))))$$

(see [20]) under the conditions

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a^{1/\alpha }\left(t\right)}}dt=\infty or{\int }_{t_0}^{\infty }{\displaystyle \frac{1}{a^{1/\alpha }\left(t\right)}}dt<\infty .$$

have also been considered.

In [22] (also see [9]), the author introduced the terminology of “canonical, noncanonical, and semicanonical” equations (depending on the behavior of ${\int }_{t_0}^t{\displaystyle \frac{1}{a_i\left(s\right)}}ds$ as $t\to \infty$ for $i=1,2$) for third-order Emden–Fowler-type equations. There does not seem to be any reason not to introduce such terminology for the Thomas–Fermi Equation (1) above. Therefore, we say that (1) is in canonical form (C) if (2) holds, in noncanonical form (NC) if (5) holds, and in semicanonical form (${\mathrm{SC}}_1$) if (3) holds or semicanonical form (${\mathrm{SC}}_2$) if (4) holds.

Even though Thomas–Fermi-type nonlinear differential equations appear in a number of applications (see, for example [2,6]), oscillation theory for such equations has received far less attention than have Emden–Fowler-type equations. The first detailed investigation in this direction appears that in the monograph of Chanturia and Kiguradze [15]. Their results are especially concerned with what is known as Property B for solutions of the linear equation

$$y^{\prime \prime \prime }\left(t\right)-q\left(t\right)y\left(t\right)=0$$

(7)

By Property B of (7), we mean that every nonoscillatory solution y is strongly increasing; that is, $|y^{\left(i\right)}\left(t\right)|\to +\infty$ as $t\to \infty$ for $i=1,2$. Oscillation theory for equations related to (1) has been investigated in [4,12,18,20]. For example, in [18], the authors discussed the oscillation and asymptotic behavior of (1) if (2) holds, and in [10,14,23], the authors established criteria for Property B to hold as well as finding criteria for the oscillation of the mixed type Equation (6) with $\tau \left(t\right)<t$ and $\sigma \left(t\right)>t$, and condition (2) or (4) holding. Since this equation is of semi-canonical type, it has three classes of nonoscillatory solutions, and to obtain oscillation criteria, they eliminated these three types of nonoscillatory solutions. Their methods needed the presence of both delay and advanced parts.

The point here is to obtain oscillation criteria for Equation (1); we achieve this by first transforming the semi-canonical Equation (1) into canonical form. This approach significantly simplifies the study of Equation (1) in the sense that the number of classes of nonoscillatory solutions are reduced from three to two without assuming any additional conditions. Then, we employ integral averaging and comparisons to obtain oscillation criteria for (1). Some examples are given to demonstrate the scope of our main results.

2. Main Results

For simplicity, we define the following quantities:

$$A_2\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{1}{a_2\left(s\right)}}ds\mathrm{and}{A}_{12}\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{A_2\left(s\right)}{a_1\left(s\right)}}ds.$$

Theorem 1.

The operator $L_{3}y$ can be written in the canonical form

$$L_{3}y=\left\{\begin{array}{cc}{\displaystyle \frac{1}{A_2\left(t\right)}}{\left(a_2{A}_2^2{\left({\displaystyle \frac{a_1}{A_2}}{y}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right),\hfill & if{A}_{12}\left(t_0\right)=\infty ,\hfill \\ {\displaystyle \frac{1}{A_2\left(t\right)}}{\left({\displaystyle \frac{a_2{A}_2^2}{A_{12}}}{\left({\displaystyle \frac{a_1{A}_{12}^2}{A_2}}{\left({\displaystyle \frac{y}{A_{12}}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right),\hfill & if{A}_{12}\left(t_0\right)<\infty .\hfill \end{array}\right.$$

Proof. 

The proof of the theorem can be found from Theorem 2 in [9]; hence, the details are omitted. □

Based on Theorem 1, we can rewrite Equation (1) as the equivalent canonical equation

$$D_{3}z\left(t\right)=Q\left(t\right)z^{\alpha }\left(\sigma \left(t\right)\right)$$

(8)

with

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{b_1\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}dt=\infty ,$$

where $D_{0}z=z$, $D_{i}z=b_i{\left(D_{i-1}z\right)}^{\prime }$, $i=1,2$, $D_{3}z={\left(D_{2}z\right)}^{\prime }$, and

$$b_1\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{a_1\left(t\right)}{A_2\left(t\right)}},\hfill & if{A}_{12}\left(t_0\right)=\infty ,\hfill \\ {\displaystyle \frac{a_1\left(t\right)A_{12}^2\left(t\right)}{A_2\left(t\right)}},\hfill & if{A}_{12}\left(t_0\right)<\infty \hfill \end{array}\right.$$

$$b_2\left(t\right)=\left\{\begin{array}{cc}{a}_2\left(t\right)A_2^2\left(t\right),\hfill & if{A}_{12}\left(t_0\right)=\infty ,\hfill \\ {\displaystyle \frac{a_2\left(t\right)A_2^2\left(t\right)}{A_{12}\left(t\right)}},\hfill & if{A}_{12}\left(t_0\right)<\infty ,\hfill \end{array}\right.$$

$$Q\left(t\right)=\left\{\begin{array}{cc}{A}_2\left(t\right)q\left(t\right),\hfill & if{A}_{12}\left(t_0\right)=\infty ,\hfill \\ A_2\left(t\right)q\left(t\right)A_{12}^{\alpha }\left(\sigma \left(t\right)\right),\hfill & if{A}_{12}\left(t_0\right)<\infty ,\hfill \end{array}\right.$$

and

$$z\left(t\right)=\left\{\begin{array}{cc}y\left(t\right),\hfill & if{A}_{12}\left(t_0\right)=\infty ,\hfill \\ {\displaystyle \frac{y\left(t\right)}{A_{12}\left(t\right)}},\hfill & if{A}_{12}\left(t_0\right)<\infty .\hfill \end{array}\right.$$

From the above discussion, we immediately obtain the following results.

Corollary 1.

The semi-canonical nonlinear differential Equation (1) possesses a solution $y\left(t\right)$ if and only if $z\left(t\right)$ is a solution of the canonical Equation (8).

Corollary 2.

The semi-canonical nonlinear differential Equation (1) has an eventually positive solution if and only if the canonical Equation (8) has an eventually positive solution.

Since the Equation (8) is of the canonical type, it is possible to use Kiguradze’s lemma [15] to obtain the following structure of its possible positive nonoscillatory solutions.

Lemma 1.

Assume that z is an eventually positive solution of (8). Then,

$$z\in S_1=\{z:z\left(t\right)>0,D_{1}z\left(t\right)>0,D_{2}z\left(t\right)<0,D_{3}z\left(t\right)>0\}$$

or

$$z\in S_3=\{z:z\left(t\right)>0,D_{1}z\left(t\right)>0,D_{2}z\left(t\right)>0,D_{3}z\left(t\right)>0\}$$

eventually.

Following [5,15,23], we say that Equation (8) has Property B if every nonoscillatory solution of (8) belongs to the class $S_3$.

Next, we derive some useful monotonic properties of nonoscillatory solutions that will be employed in proving our main results. For simplicity and convenience, we let

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

and

$$Q_1\left(t\right)={\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)ds,$$

where $t_{*}$ is appropriately large enough.

Lemma 2.

Let $z\left(t\right)\in S_3$ be a positive solution of (8). Then, $z\left(t\right)$ is increasing and

$$\underset{t\to \infty }{lim}z\left(t\right)=\infty \mathit{and}{\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\mathit{is}\mathit{eventually}\mathit{increasing}.$$

(9)

Proof. 

Choose $t\ge t_{*}$ such that $z\left(t\right)$ is a positive solution of (8) belonging to $S_3$ for all $t\ge t_{*}$, for some $t_{*}$ large enough. From the definition of the class $S_3$, it is easy to see that $z\left(t\right)$ and $D_{1}z\left(t\right)$ are increasing for all $t\ge t_1\ge t_{*}$. Now,

$$z\left(t\right)=z\left(t_1\right)+{\int }_{t_1}^t{\displaystyle \frac{D_{1}z\left(s\right)}{b_1\left(s\right)}}ds\ge D_{1}z\left(t_1\right){\int }_{t_1}^t{\displaystyle \frac{1}{b_1\left(s\right)}}ds,$$

(10)

which implies that $z\left(t\right)$ tends to ∞ as $t\to \infty$. Since $D_{2}z\left(t\right)$ is also increasing, it is clear that $D_{1}z\left(t\right)$ also tends to ∞ as $t\to \infty$. Again from (2.2), we have

$$z\left(t\right)\le z\left(t_1\right)-D_{1}z\left(t\right){\int }_{t_{*}}^{t_1}{\displaystyle \frac{1}{b_1\left(s\right)}}ds+D_{1}z\left(t\right){\int }_{t_{*}}^t{\displaystyle \frac{1}{b_1\left(s\right)}}ds.$$

Using the fact that $D_{1}z\left(t\right)\to \infty$ as $t\to \infty$ in the last inequality, we see that

$$z\left(t\right)\le B_1\left(t\right)D_{1}z\left(t\right).$$

This implies that

$${\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)}^{\prime }={\displaystyle \frac{B_1\left(t\right)D_{1}z\left(t\right)-z\left(t\right)}{b_1\left(t\right)B_1^2\left(t\right)}}\ge 0.$$

Thus, $\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)$ is eventually increasing, and this completes the proof. □

Lemma 3.

Let $z\left(t\right)\in S_1$ be a positive solution of (8). Then, ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is eventually decreasing.

Proof. 

Let $z\left(t\right)$ be an eventually positive solution of (8) that belongs to $S_1$ for all $t\ge t_{*}$. Then, $D_{1}z\left(t\right)$ is decreasing, and so we have

$$z\left(t\right)\ge {\int }_{t_8}^t{\displaystyle \frac{D_{1}z\left(s\right)}{b_1\left(s\right)}}ds\ge D_{1}z\left(t\right){\int }_{t_{*}}^t{\displaystyle \frac{1}{b_1\left(s\right)}}ds.$$

This implies that

$${\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)}^{\prime }={\displaystyle \frac{B_1\left(t\right)D_{1}z\left(t\right)-z\left(t\right)}{b_1\left(t\right)B_1^2\left(t\right)}}\le 0.$$

Hence, $\left({\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\right)$ is eventually decreasing, and this proves the lemma. □

Remark 1.

Lemma 2 is new, and Lemma 3 is an extension of [18] (Lemma 2.3) for the semi-canonical Equation (1) once it has been transformed to canonical form.

Now, we present criteria for the class $S_1$ to be empty.

Theorem 2.

Assume that

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

(11)

If

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

(12)

where $M=1$ if $\alpha =1$ and $M=0$ if $\alpha >1$, then the class $S_1$ is empty.

Proof. 

Assume, to the contrary, that $z\left(t\right)$ is a positive solution of (8) belonging to the class $S_1$ for all $t\ge t_{*}$. Integrating (8) from t to ∞ twice yields

$$\begin{array}{cc}\hfill D_{1}z\left(t\right)& \ge {\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_s^{\infty }Q\left(u\right)z^{\alpha }\left(\sigma \left(u\right)\right)duds\hfill \\ \hfill & \ge {\int }_{t_{*}}^{\infty }{\displaystyle \frac{z^{\alpha }\left(\sigma \left(s\right)\right)}{b_2\left(s\right)}}{\int }_s^{\infty }Q\left(u\right)duds={\int }_{t_{*}}^{\infty }{Q}_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds,\hfill \end{array}$$

where we have used the monotonicity of $z\left(\sigma \right(t\left)\right)$. Integrating the last inequality from $t_{*}$ to t and then interchanging the order of the integrations, we obtain

$$\begin{array}{cc}\hfill z\left(t\right)& \ge {\int }_{t_{*}}^t{\displaystyle \frac{1}{b_1\left(s\right)}}{\int }_s^{\infty }{Q}_1\left(u\right)z^{\alpha }\left(\sigma \left(u\right)\right)duds\hfill \\ \hfill & ={\int }_{t_{*}}^t{Q}_1\left(s\right)B_1\left(s\right)z^{\alpha }(\sigma \left(s\right)ds+B_1\left(t\right){\int }_t^{\infty }{Q}_1\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)ds.\hfill \end{array}$$

Therefore,

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

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 have

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

That is,

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

(13)

If $\alpha =1$ in (13), we obtain a contradiction to (12). From (11), we see that $z\left(t\right)\to \infty$ as $t\to \infty$, so taking the lim sup as $t\to \infty$ on both sides of (13), we are again led to a contradiction to (12) for $\alpha >1$. This finishes the proof. □

Theorem 3.

Let $0<\alpha <1$ and assume that

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

(14)

If

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

(15)

 then the class $S_1$ is empty.

Proof. 

Assume that $z\left(t\right)$ is a positive solution of (8) belonging to $S_1$ for all $t\ge t_{*}$. By Lemma 3, the function ${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}$ is decreasing. To show

$$\underset{t\to \infty }{lim}{\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}=0,$$

(16)

assume that ${\displaystyle \underset{t\to \infty }{lim}\frac{z\left(t\right)}{B_1\left(t\right)}=m>0}$. Then,

$${\displaystyle \frac{z\left(t\right)}{B_1\left(t\right)}}\ge m$$

for $t\ge t_{*}$, and so

$$z^{\alpha }\left(\sigma \left(t\right)\right)\ge m^{\alpha }{B}_1^{\alpha }\left(\sigma \left(t\right)\right).$$

Integrating (8) twice gives

$$\begin{array}{cc}{D}_{1}z\left(t_{*}\right)\ge {\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)z^{\alpha }\left(\sigma \left(s\right)\right)dsdt\hfill & \\ & \hfill \ge m^{\alpha }{\int }_{t_{*}}^{\infty }{\displaystyle \frac{1}{b_2\left(t\right)}}{\int }_t^{\infty }Q\left(s\right)B_1^{\alpha }\left(\sigma \left(s\right)\right)dsdt.\end{array}$$

This contradicts assumption (14), and so we conclude that (16) holds. Now if we set

$$x\left(t\right)={\displaystyle \frac{z\left(\sigma \right(t\left)\right)}{B_1\left(\sigma \left(t\right)\right)}},$$

then (13) implies

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

(17)

Taking the lim sup as $t\to \infty$ and using (16) gives a contradiction to (15). This proves the theorem. □

Remark 2.

From Theorems 2 and 3, we conclude that Equation (8) has Property B. To obtain an oscillation result by making use of theses two theorems, we would need to eliminate this possibility, that is, we would need to show that $S_3$ is empty.

Theorem 4.

If $\sigma \left(t\right)>t$ and

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

(18)

where $M=1$ if $\alpha =1$ and $M=0$ if $\alpha >1,$ then $S_3=\varnothing$.

Proof. 

To show that $S_3$ is empty, assume that $z\left(t\right)$ is a positive solution of (8) that belongs to $S_3$ for all $t\ge t_{*}$. Integrating (8) from s to t and applying Lemma 2 gives

$${\left(D_{1}z\left(s\right)\right)}^{\prime }\ge {\displaystyle \frac{1}{b_2\left(s\right)}}{\int }_t^{s}Q\left(u\right)z^{\alpha }\left(\sigma \left(u\right)\right)du\ge {\displaystyle \frac{z^{\alpha }\left(\sigma \left(t\right)\right)}{b_2\left(s\right)B_1^{\alpha }\left(\sigma \left(t\right)\right)}}{\int }_t^{s}Q\left(u\right)B_1^{\alpha }\left(\sigma \left(u\right)\right)du.$$

Integrating this, we obtain

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

Another integration yields

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

Setting $s=\sigma \left(t\right)$ gives

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

(19)

For $\alpha =1$ in (19), we obtain a contradiction to (18). If $\alpha >1$, then since Lemma 2 implies $z\left(t\right)\to \infty$ as $t\to \infty$, we again have a contradiction to (18). The proof of the theorem is now complete. □

Next, we present another criterion to eliminate the class $S_3$.

Theorem 5.

Assume that there exist nondecreasing functions $\tau \left(t\right)$, $\theta \left(t\right)\in C^{\prime }([t_0,\infty ),R)$ such that $\sigma \left(t\right)>\tau \left(t\right)>\theta \left(t\right)>t$ for all $t\ge t_0$. If the first-order advanced differential equation

$${\omega }^{\prime }\left(t\right)-Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}ds\right)}^{\alpha }{\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(s\right)}}ds\right)}^{\alpha }{\omega }^{\alpha }\left(\theta \left(t\right)\right)=0$$

(20)

is oscillatory, then the class $S_3$ is empty.

Proof. 

Assume, to the contrary, that $z\left(t\right)$ is a positive solution of (8) belonging to $S_3$ for all $t\ge t_{*}$. For $t\ge s\ge t_{*}$,

$$z\left(t\right)-z\left(s\right)={\int }_s^t{\displaystyle \frac{b_1\left(u\right)z^{\prime }\left(u\right)}{b_1\left(u\right)}}du\ge b_1\left(s\right)z^{\prime }\left(s\right){\int }_s^t{\displaystyle \frac{1}{b_1\left(u\right)}}du,$$

(21)

so by letting $s=\tau \left(t\right)$ and $t=\sigma \left(t\right)$, we obtain

$$z\left(\sigma \left(t\right)\right)\ge \left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(u\right)}}du\right)b_1\left(\tau \left(t\right)\right)z^{\prime }\left(\tau \left(t\right)\right).$$

Using this in (8) yields

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

Let $x\left(t\right)=b_1\left(t\right)z^{\prime }\left(t\right)$, $t\ge t_1\ge t_{*}$; we see that

$${\left(b_2\left(t\right)x^{\prime }\left(t\right)\right)}^{\prime }\ge Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(u\right)}}du\right)}^{\alpha }{x}^{\alpha }\left(\tau \left(t\right)\right).$$

(22)

Also, similar to (21), we have

$$x\left(t\right)\ge b_2\left(s\right)z^{\prime }\left(s\right)\left({\int }_s^t{\displaystyle \frac{1}{b_2\left(u\right)}}du\right)$$

for $t\ge s\ge t_1$, so letting $s=\theta \left(t\right)$ and $t=\tau \left(t\right)$,

$$x\left(\tau \left(t\right)\right)\ge b_2\left(\theta \left(t\right)\right)z^{\prime }\left(\theta \left(t\right)\right)\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(u\right)}}du\right).$$

(23)

If we let $\omega \left(t\right)=b_2\left(t\right)x^{\prime }\left(t\right)$ and use (23) in (22), we have

$${\omega }^{\prime }\left(t\right)-Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(u\right)}}du\right)}^{\alpha }{\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(u\right)}}du\right)}^{\alpha }{\omega }^{\alpha }\left(\theta \left(t\right)\right)\ge 0.$$

(24)

Thus, $\omega \left(t\right)$ is a positive solution of the above inequality, and by ([24], Corollary 2.4.2), we see that the corresponding Equation (20) also has a positive solution. This contradiction ends the proof. □

For $\alpha =1$, using Theorem 1 (ii) in [7], we obtain the following corollary.

Corollary 3.

Let $\alpha =1$ and assume that there exists nondecreasing functions $\tau \left(t\right)$, $\theta \left(t\right)\in C^{\prime }([t_0,\infty ),R)$ such that $\sigma \left(t\right)>\tau \left(t\right)>\theta \left(t\right)>t$ for all $t\ge t_{*}$. If

$$\underset{t\to \infty }{lim\; inf}\left[{\int }_t^{\theta \left(t\right)}Q\left(s\right)\left({\int }_{\tau \left(s\right)}^{\sigma \left(s\right)}{\displaystyle \frac{1}{b_1\left(u\right)}}du\right)\left({\int }_{\theta \left(s\right)}^{\tau \left(s\right)}{\displaystyle \frac{1}{b_2\left(u\right)}}du\right)ds\right]>{\displaystyle \frac{1}{e}},$$

(25)

then the class $S_3$ is empty.

Corollary 4.

Let $\alpha >1$ and assume that there exist nondecreasing functions $\tau \left(t\right)$, $\theta \left(t\right)\in C^{\prime }([t_0,\infty ),R)$ such that $\sigma \left(t\right)>\tau \left(t\right)>\theta \left(t\right)>t$ for all $t\ge t_{*}$. If

$${\int }_{t_{*}}^{\infty }Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(u\right)}}du\right)}^{\alpha }{\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(u\right)}}du\right)}^{\alpha }dt=\infty ,$$

(26)

then the class $S_3$ is empty.

Proof. 

Proceeding as in the proof of Theorem 5, we arrive at (24). Since $\omega \left(t\right)$ is increasing and $\theta \left(t\right)>t$, from (24) we obtain

$${\displaystyle \frac{{\omega }^{\prime }\left(t\right)}{{\omega }^{\alpha }\left(t\right)}}>Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}ds\right)}^{\alpha }{\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(s\right)}}ds\right)}^{\alpha }.$$

An integration gives

$${\displaystyle \frac{1}{(\alpha -1){\omega }^{\alpha -1}\left(t_1\right)}}\ge {\int }_{t_1}^{\infty }Q\left(t\right){\left({\int }_{\tau \left(t\right)}^{\sigma \left(t\right)}{\displaystyle \frac{1}{b_1\left(s\right)}}ds\right)}^{\alpha }{\left({\int }_{\theta \left(t\right)}^{\tau \left(t\right)}{\displaystyle \frac{1}{b_2\left(s\right)}}ds\right)}^{\alpha }dt,$$

which contradicts (26) and completes the proof. □

Combining the criteria obtained for both of the classes $S_1$ and $S_3$ to be empty, we obtain the following criteria for the oscillation of Equation (1).

Theorem 6.

Let $\alpha =1$ or $\alpha >1$. If (11), (12), and (18) hold, then Equation (1) is oscillatory.

Proof. 

If $y\left(t\right)$ is a positive solution of (1) for all $t\ge t_{*}$, then by Corollary 1, we see that $z\left(t\right)$ is a positive solution of (8) and by Lemma 1, $z\left(t\right)\in S_1$ or $z\left(t\right)\in S_3$ for all $t\ge t_{*}$. In view of Theorem 2, we see that the class $S_1$ is empty and by Theorem 4, we have the class $S_3$ is empty. Therefore, we conclude that Equation (8) is oscillatory, which in turn implies (1) is oscillatory. This proves the theorem. □

Theorem 7.

Let $\alpha =1$ ($\alpha >1$). If all conditions of Theorem 2 and Corollary 3 (Corollary 4) hold, then Equation (1) is oscillatory.

Proof. 

The proof is similar to that of Theorem 6, and so the details are omitted. □

3. Examples

In this section, we present examples to illustrate importance of our results over previously known ones. The first four examples are devoted to illustrating our oscillation results. The last example is concerned with solutions possessing Property B.

Example 1.

Consider the third-order linear semi-canonical advanced differential equation

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

(27)

where $a>0$ and $\lambda >1$. Here, we have $a_1\left(t\right)=1/t$, $a_2\left(t\right)=t^2$, $q\left(t\right)=a/t^2$, $\sigma \left(t\right)=\lambda t$, $\lambda >1$, and $\alpha =1$. Simple computations show that $A_2\left(t\right)={\displaystyle \frac{1}{t}}$ and $A_{12}\left(1\right)={\int }_1^{\infty }dt=\infty$, so the condition $A_{12}\left(t_0\right)=\infty$ holds. Now $b_1\left(t\right)=1$, $b_2\left(t\right)=1$, $Q\left(t\right)=a/t^3$, and $z\left(t\right)=y\left(t\right)$, so the transformed equation is

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

which we see is in canonical form. Further calculations show that

$$B_1\left(t\right)\approx t,Q_1\left(t\right)={\displaystyle \frac{a}{2t^2}},$$

and condition (11) becomes

$${\int }_1^{\infty }{\int }_t^{\infty }{\displaystyle \frac{a}{2s^2}}dsdt=\infty ,$$

so it holds. Condition (12) for $\alpha =1$ takes the form

$$\underset{t\to \infty }{lim\; sup}\left({\displaystyle \frac{1}{\lambda t}}{\int }_1^t{\displaystyle \frac{\lambda a}{2}}ds+{\int }_t^{\lambda t}{\displaystyle \frac{a}{2s}}ds+\lambda t{\int }_{\lambda t}^{\infty }{\displaystyle \frac{a}{2s^2}}ds\right)=a+{\displaystyle \frac{a}{2}}ln\lambda >1,$$

and so it holds if $a>{\displaystyle \frac{2}{2+ln\lambda }}$. Condition (18) for $\alpha =1$ is

$$\underset{t\to \infty }{lim\; sup}\left({\displaystyle \frac{1}{\lambda t}}{\int }_t^{\lambda t}{\int }_t^s{\int }_t^{s_1}{\displaystyle \frac{a\lambda }{s_2^2}}d{s}_{2}d{s}_{1}ds\right)=a\left[{\displaystyle \frac{{\lambda }^2}{2}}-{\displaystyle \frac{1}{2}}-\lambda ln\lambda \right]>1,$$

which holds if $a>{\displaystyle \frac{2}{{\lambda }^2-1-2\lambda ln\lambda }}$. Therefore, by Theorem 6, Equation (1) is oscillatory if

$$a>max\left\{{\displaystyle \frac{2}{2+ln\lambda }},{\displaystyle \frac{2}{{\lambda }^2-1-2\lambda ln\lambda }}\right\}.$$

In particular, if we take $\lambda =2$, then for $a>8.7946386$, Equation (1) is oscillatory.

Next, by choosing $\tau \left(t\right)={\lambda }_{1}t$ and $\theta \left(t\right)={\lambda }_{2}t$ such that $\lambda >{\lambda }_1>{\lambda }_2>1$, we see that condition (25) holds if $a(\lambda -{\lambda }_1)({\lambda }_1-{\lambda }_2)ln{\lambda }_2>{\displaystyle \frac{1}{e}}.$ Therefore, by Theorem 7, Equation (1) is oscillatory if

$$a>max\left\{{\displaystyle \frac{2}{2+ln\lambda }},{\displaystyle \frac{1}{e(\lambda -{\lambda }_1)({\lambda }_1-{\lambda }_2)ln{\lambda }_2}}\right\}.$$

For $\lambda =2$, ${\lambda }_1=1.5$, and ${\lambda }_2=1.25$, we see that Equation (1) is oscillatory if $a>13.18898$. Hence, Theorem 6 gives better condition than Theorem 7.

Example 2.

Consider the third-order semi-canonical advanced equation

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

(28)

where $a>0$ and $\lambda >1$. Here, $a_1\left(t\right)=1/t$, $a_2\left(t\right)=t^2$, $q\left(t\right)=a/t^2$, $\sigma \left(t\right)=\lambda t$, $\lambda >1$, and $\alpha =3$. Also, $A_2\left(t\right)={\displaystyle \frac{1}{t}}$ and $A_{12}\left(1\right)={\int }_1^{\infty }dt=\infty$, $b_1\left(t\right)=1$, $b_2\left(t\right)=1$, $Q\left(t\right)=a/t^3$, and $z\left(t\right)=y\left(t\right)$, so the transformed equation is

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

which is in canonical form. Additional computations give

$$B_1\left(t\right)\approx t,Q_1\left(t\right)={\displaystyle \frac{a}{2t^2}},$$

and (11) holds since

$${\int }_1^{\infty }{\int }_t^{\infty }{\displaystyle \frac{a}{2s^2}}dsdt=\infty .$$

Condition (12) with $\alpha >1$ reduces to

$$a\left({\displaystyle \frac{2}{3}}\lambda +{\displaystyle \frac{ln\lambda }{2}}\right)>0ifa>0,$$

so it holds, and condition (18) becomes

$$a{\displaystyle \frac{{(\lambda -1)}^3}{6}}>0,$$

which holds since $a>0$ and $\lambda >1$. Hence, by Theorem 6, Equation (28) is oscillatory for all $a>0$.

Example 3.

Consider the third-order nonlinear semi-canonical advanced differential equation

$${\left(t^2{\left({\displaystyle \frac{1}{t}}{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\displaystyle \frac{a}{t^{4/3}}}{y}^{1/3}\left(\lambda t\right)=0,t\ge 1,$$

(29)

where $a>0$ and $\lambda >1$. Here, $a_1\left(t\right)={\displaystyle \frac{1}{t}}$, $a_2\left(t\right)=t^2$, $q\left(t\right)={\displaystyle \frac{a}{t^{4/3}}}$, $\sigma \left(t\right)=\lambda t$, $\lambda >1$, $\alpha =1/3$, $A_2\left(t\right)={\displaystyle \frac{1}{t}}$ and $A_{12}\left(1\right)=\infty$. Also, $b_1\left(t\right)=b_2\left(t\right)=1$ and $Q\left(t\right)={\displaystyle \frac{a}{t^{7/3}}}$. The transformed equation in canonical form is

$$z^{\prime \prime \prime }\left(t\right)-{\displaystyle \frac{a}{t^{7/3}}}{z}^{1/3}\left(\lambda t\right)=0,t\ge 1,$$

We have

$$B_1\left(t\right)\approx t\mathit{and}{Q}_1\left(t\right)={\displaystyle \frac{3a}{4t^{4/3}}}.$$

Condition (14) becomes

$${\int }_1^{\infty }{\int }_t^{\infty }{\displaystyle \frac{a{\lambda }^{1/3}}{s^2}}dsdt=a{\lambda }^{1/3}{\int }_1^{\infty }{\displaystyle \frac{1}{t}}dt=\infty ,$$

that is, (14) holds since $a>0$. Condition (15) has the form

$${\displaystyle \frac{3a}{8}}\left(9-{\displaystyle \frac{1}{{\lambda }^{2/3}}}\right)>0$$

so it holds. Therefore, by Theorem 3, Equation (29) has Property B.

Remark 3.

Note, in particular, that the results in [10,14,18,23] cannot be applied to any of the above examples since they are all semi-canonical equations with an advanced argument. Hence, the results established here are new and make a significant contribution to oscillation theory for third-order differential equations in general and Thomas–Fermi-type equations in particular.

4. Conclusions

In this paper, we obtain some new sufficient conditions for the oscillation of the semi-canonical Equation (1). This is achieved by first transforming it into canonical form and then applying integral averaging techniques and a comparison to first-order advanced differential equations whose oscillatory behavior is known. We also found sufficient conditions for Equation (1) to have Property B in the case where $0<\alpha <1$. Hence, it would be of interest to find conditions for all solutions of (1) to be oscillatory if $0<\alpha <1$. We included examples to show the importance and novelty of our main results.

It would also be interesting to extend the results in this paper to equations of the form

$${(a_3(t){(a_2\left(t\right){(a_1\left(t\right)y\left(t\right))}^{\prime })}^{\prime })}^{\prime }=p\left(t\right)y^{\alpha }\left(\tau \left(t\right)\right)+q\left(t\right)y^{\alpha }\left(\sigma \left(t\right)\right)$$

where $\tau \left(t\right)\le t$ and $\sigma \left(t\right)\ge t$ and that are in semi-canonical or noncanonical form.