Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria

Abstract

The purpose of this paper is to add some new asymptotic and oscillatory results for third-order neutral delay differential equations with noncanonical operators. Without assuming any extra conditions, by using the canonical transform technique, the studied equation is changed to a canonical type equation, and this reduces the number of classes of nonoscillatory solutions into two instead of four. Then, we obtain Myshkis type sufficient conditions for the nonexistence of Kneser type solutions for the studied equation. Finally, employing these newly obtained criteria, we provide conditions for the oscillation of all solutions of the studied equation. Examples are presented to illustrate the importance and the significance of the main results.

1. Introduction

This paper deals with the oscillatory and asymptotic properties of the third-order neutral delay differential equation

$${\left({\mu }_2{\left({\mu }_1{\varphi }^{\prime }\right)}^{\prime }\right)}^{\prime }\left(t\right)+f\left(t\right)\theta \left(\delta \left(t\right)\right)=0,t\ge t_0>0,$$

(E)

where $\varphi \left(t\right)=\theta \left(t\right)+g\left(t\right)\theta \left(\tau \right(t\left)\right)$. We assume the following hypotheses:

$\left(H_1\right)$

$\delta ,\tau \in C^1([t_0,\infty ),\mathbb{R}),\delta \left(t\right)<t,{\tau }^{\prime }\left(t\right)\ge {\tau }_0>0$ and ${lim}_{t\to \infty }\delta \left(t\right)={lim}_{t\to \infty }\tau \left(t\right)=\infty ;$

$\left(H_2\right)$

$g,f\in C([t_0,\infty ),{\mathbb{R}}^{+}),0\le g\left(t\right)\le g_0<\infty$ and f does not vanish identically;

$\left(H_3\right)$

${\mu }_i\in C^{(3-i)}([t_0,\infty ),(0,\infty ))$, $i=1,2,$ and $\left(E\right)$ is in noncanonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{dt}{{\mu }_1\left(t\right)}}<\infty and{\int }_{t_0}^{\infty }{\displaystyle \frac{dt}{{\mu }_2\left(t\right)}}<\infty ;$$

$\left(H_4\right)$

$\tau \left(t\right)\le t$ such that $\tau$ and $\delta$ are commute.

For the sake of simplicity, we define the operators $L_0\varphi =\varphi ,L_1\varphi ={\mu }_1{\varphi }^{\prime }$, $L_2\varphi ={\mu }_2{\left({\mu }_1{\varphi }^{\prime }\right)}^{\prime },L_3\varphi ={\left({\mu }_2{\left({\mu }_1{\varphi }^{\prime }\right)}^{\prime }\right)}^{\prime }.$

The above stated hypotheses $\left(H_1\right)$–$\left(H_4\right)$ are essential and necessary for the existence of a solution of Equation $\left(E\right).$ By a solution of Equation $\left(E\right)$, we mean a nontrivial function $\theta \in C([T_{\theta },\infty ),\mathbb{R})$ with $T_{\theta }\ge t_0$, which has the property $L_i\varphi \in C^1([T_{\theta },\infty ),\mathbb{R}),i=1,2$ and satisfies $\left(E\right)$ on $[T_{\theta },\infty )$. We only consider those solutions of $\left(E\right)$ which exist on some half-line $[T_{\theta },\infty )$ and satisfy the condition $sup\left\{\right|\theta \left(t\right)|:T\le t<\infty \}$ for any $T\ge T_{\theta }$.

As usual, a solution $\theta$ of $\left(E\right)$ is said to be oscillatory if it has a sequence of zeros tending to infinity, and nonoscillatory otherwise. If all solutions are oscillatory, then the equation itself is called oscillatory. Further, we say that $\left(E\right)$ has property A if any solution $\theta$ of $\left(E\right)$ is either oscillatory or satisfies $\theta \left(t\right)\to 0$ as $t\to \infty$.

To begin with, let us state the structure of the possible nonoscillatory solutions of $\left(E\right)$, see, Lemma 1 of [1] or Lemma 1.3 of [2].

Lemma 1. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold, and θ is a nonoscillatory solution of $\left(E\right)$. Then, there are four possible cases for ϕ:

$$\begin{array}{ccc}\hfill S_0& =& \left\{\varphi \left(t\right):\varphi \left(t\right)L_1\varphi \left(t\right)<0,\varphi \left(t\right)L_2\varphi \left(t\right)>0,t\ge T\ge t_0\right\};\hfill \\ \hfill S_1& =& \left\{\varphi \left(t\right):\varphi \left(t\right)L_1\varphi \left(t\right)<0,\varphi \left(t\right)L_2\varphi \left(t\right)<0,t\ge T\ge t_0\right\};\hfill \\ \hfill S_2& =& \left\{\varphi \left(t\right):\varphi \left(t\right)L_1\varphi \left(t\right)>0,\varphi \left(t\right)L_2\varphi \left(t\right)>0,t\ge T\ge t_0\right\};\hfill \\ \hfill S_3& =& \left\{\varphi \left(t\right):\varphi \left(t\right)L_1\varphi \left(t\right)>0,\varphi \left(t\right)L_2\varphi \left(t\right)<0,t\ge T\ge t_0\right\}.\hfill \end{array}$$

So, if we want to establish conditions for the oscillation of Equation $\left(E\right)$, one has to eliminate the above mentioned four classes, and for the nonexistence of Kneser type solutions, we have to eliminate the two classes $S_0$ and $S_1.$ However, if we transform Equation $\left(E\right)$ into canonical form, then, the number of classes of nonoscillatory solutions are reduced to only two, and this greatly simplifies the examination of $\left(E\right)$.

In view of $\left(H_3\right)$, we can use the following notation:

$${\Omega }_j\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{ds}{{\mu }_j\left(s\right)}},j=1,2,\Omega \left(t\right)={\int }_t^{\infty }{\displaystyle \frac{{\Omega }_2\left(s\right)}{{\mu }_1\left(s\right)}}ds,$$

$${\Omega }_{*}\left(t\right)={\int }_t^{\infty }{\displaystyle \frac{{\Omega }_1\left(s\right)}{{\mu }_2\left(s\right)}}ds,{\beta }_1\left(t\right)={\displaystyle \frac{{\mu }_1\left(t\right){\Omega }^2\left(t\right)}{{\Omega }_{*}\left(t\right)}},{\beta }_2\left(t\right)={\displaystyle \frac{{\mu }_2\left(t\right){\Omega }_{*}^2\left(t\right)}{\Omega \left(t\right)}}.$$

Instead of using the result of Trench [3], we employ ([4], Theorem 2.1) to transform Equation (E) in the equivalent canonical form as

$${\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\left({\displaystyle \frac{\varphi \left(t\right)}{\Omega \left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+{\Omega }_{*}\left(t\right)f\left(t\right)\theta \left(\delta \left(t\right)\right)=0.$$

(1)

Now by letting $\alpha \left(t\right)={\displaystyle \frac{\varphi \left(t\right)}{\Omega \left(t\right)}}$ in (1) and using the notation $q\left(t\right)={\Omega }_{*}\left(t\right)f\left(t\right)$, the following results are immediate.

Lemma 2. 

The noncanonical neutral delay differential Equation (E) has a solution $\theta \left(t\right)$ if and only if the canonical equation

$${\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\left(\alpha \left(t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+q\left(t\right)\theta \left(\delta \left(t\right)\right)=0$$

(E_c)

also has the same solution.

Lemma 3. 

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

Note that from the above lemma, we see that the canonical transform used here preserves the properties of the solutions of Equation $\left(E\right)$ and that of canonical Equation $\left(E_c\right).$ Further, Lemma 3 essentially simplifies the investigation of $\left(E\right)$. Because due to Equation $\left(E_c\right),$ we deal with only two classes of an eventually positive (nonoscillatory) solution, namely, either

$$N_0:\alpha \left(t\right)>0,{\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)<0,{\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\le 0$$

or

$$N_2:\alpha \left(t\right)>0,{\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)>0,{\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }\le 0$$

for sufficiently large t.

The solutions $\theta$ whose corresponding function $\alpha \in N_0$ are called Kneser type solutions. It is clear that $\left(E\right)$ has property A if and only if any nonoscillatory solution $\theta$ is of Kneser type and ${lim}_{t\to \infty }\theta \left(t\right)=0$.

Recent years have seen an increasing interest paid to studying the oscillation property of third-order neutral type functional differential Equation $\left(E\right)$. It is known from the literature that depending on various ranges of g, there are many results reported for property A of $\left(E\right)$ or its particular cases when it is in canonical form, see, for example, [1,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and the references cited therein. In [10], the authors studied the oscillation properties of the noncanonical Equation $\left(E\right)$ for the case $g\left(t\right)>1$, and in [14], the authors discussed the case $0\le g\left(t\right)<1$ for the semi-canonical case. On the other hand, the authors in [1] established oscillation criteria for the noncanonical Equation $\left(E\right)$ when $g\left(t\right)=0.$ Further, the results in [1,10,14] are established without using a canonical transform method.

In view of the above observations, we see that not much is known regarding the oscillation of all solutions of the noncanonical Equation $\left(E\right)$. Therefore, in the present paper, our purpose is to fill this gap by obtaining new sufficient conditions by eliminating Kneser type solutions of $\left(E\right)$ under the assumptions (H₁)–(H₄) and using the canonical transform method. Finally, by using this newly obtained ones, we present criteria for the oscillation of all solutions of $\left(E\right)$. Examples are provided to illustrate the significance and the novelty of the main results.

2. Main Results

For convenience, we adopt the following notation:

$$B_i\left(t\right)={\int }_{t_{*}}^t{\displaystyle \frac{ds}{{\beta }_i\left(s\right)}},i=1,2,B_3\left(t\right)={\int }_{t_{*}}^t{\displaystyle \frac{B_2\left(s\right)}{{\beta }_1\left(s\right)}}ds,$$

$$G\left(t\right)=min\left\{q\right(t),q(\tau \left(t\right)\left)\right\},E\left(t\right)=G\left(t\right)\Omega \left(\delta \right(t\left)\right)$$

for $t\ge t_{*}\ge t_0$. Also, we define

$$D_0\alpha =\alpha ,D_1\alpha ={\beta }_1{\alpha }^{\prime },D_2\alpha ={\beta }_2{\left({\beta }_1{\alpha }^{\prime }\right)}^{\prime },D_3\alpha ={\left({\beta }_2{\left({\beta }_1{\alpha }^{\prime }\right)}^{\prime }\right)}^{\prime }.$$

Lemma 4. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. If θ is a positive solution of $\left(E\right)$, then, the corresponding function α satisfies the inequality

$$D_3\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_3\alpha \left(\tau \left(t\right)\right)+E\left(t\right)\alpha \left(\delta \left(t\right)\right)\le 0$$

(2)

for all $t\ge t_1\ge t_0$.

Proof. 

Let $\theta$ be an eventually positive solution of $\left(E\right)$. Then, there is a $t_1\ge t_0$ such that $\theta \left(t\right)>0,\theta \left(\tau \right(t\left)\right)>0$ and $\theta \left(\delta \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. Then, the corresponding function $\alpha \left(t\right)$ is positive and satisfies $\left(E_c\right)$. So from $\left(E_c\right),\left(H_1\right)$ and $\left(H_4\right)$, we see that

$$\begin{array}{ccc}\hfill 0& =& {\displaystyle \frac{g_0}{{\tau }^{\prime }\left(t\right)}}{\left(D_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+g_{0}q\left(\tau \left(t\right)\right)\theta \left(\delta \left(\tau \left(t\right)\right)\right)\hfill \\ & \ge & {\displaystyle \frac{g_0}{{\tau }_0}}{\left(D_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+g_{0}q\left(\tau \left(t\right)\right)\theta \left(\delta \left(\tau \left(t\right)\right)\right)\hfill \\ & =& {\displaystyle \frac{g_0}{{\tau }_0}}{\left(D_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+g_{0}q\left(\tau \left(t\right)\right)\theta \left(\tau \left(\delta \left(t\right)\right)\right).\hfill \end{array}$$

Combining $\left(E_c\right)$ along with the last inequality, we obtain

$$\begin{array}{ccc}\hfill 0& \ge & D_3\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{\left(D_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+q\left(t\right)\theta \left(\delta \left(t\right)\right)+g_{0}q\left(\tau \left(t\right)\right)\theta \left(\tau \left(\delta \left(t\right)\right)\right)\hfill \\ & \ge & D_3\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{\left(D_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+G\left(t\right)(\theta \left(\delta \left(t\right)\right)+g_0\theta \left(\tau \left(\delta \left(t\right)\right)\right)).\hfill \end{array}$$

(3)

Using $\left(H_2\right)$ in the definition of $\varphi \left(t\right)$, we obtain

$$\begin{array}{ccc}\hfill \Omega \left(\delta \right(t\left)\right)\alpha \left(\delta \right(t\left)\right)=\varphi \left(\delta \right(t\left)\right)& =& \theta \left(\delta \right(t\left)\right)+g\left(\delta \right(t\left)\right)\theta \left(\tau \right(\delta \left(t\right)\left)\right)\hfill \\ & \le & \theta \left(\delta \left(t\right)\right)+g_0\theta \left(\tau \left(\delta \left(t\right)\right)\right).\hfill \end{array}$$

By virtue of the latter inequality, (3) becomes

$$D_3\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_3\alpha \left(\tau \left(t\right)\right)+E\left(t\right)\alpha \left(\delta \left(t\right)\right)\le 0$$

or

$${\left(D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+E\left(t\right)\alpha \left(\delta \left(t\right)\right)\le 0,$$

(4)

which proves (2). The proof of the lemma is complete. □

Define

$$A(t,t_1)={\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_s^t{\displaystyle \frac{d{s}_1}{{\beta }_2\left(s_1\right)}}ds.$$

Theorem 1. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. If there exists a function $\zeta \left(t\right)\in C([t_0,\infty ),(0,\infty ))$ satisfying $\delta \left(t\right)<\zeta \left(t\right)<\tau \left(t\right)$ such that

$$\underset{t\to \infty }{lim}inf{\int }_{{\tau }^{-1}\left(\zeta \left(t\right)\right)}^{t}E\left(s\right)A(\zeta \left(s\right),\delta \left(s\right))ds>{\displaystyle \frac{{\tau }_0+g_0}{{\tau }_{0}e}},$$

(5)

then, $N_0$ is empty.

Proof. 

Assume to the contrary that $\theta \in N_0$. Without loss of generality, we may assume that $\theta \left(t\right)>0,\theta \left(\tau \right(t\left)\right)>0$ and $\theta \left(\delta \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. Then, the corresponding function $\alpha \left(t\right)\in N_0$, that is,

$$\alpha \left(t\right)>0,D_1\alpha \left(t\right)<0,D_2\alpha \left(t\right)>0,D_3\alpha \left(t\right)\le 0fort\ge t_1.$$

Now, proceeding as in Lemma 4, we see that (4) holds. On the other hand, from the monotonicity of $D_2\alpha$, we have

$$-D_1\alpha \left(u\right)\ge D_1\alpha \left(v\right)-D_1\alpha \left(u\right)={\int }_u^v{\displaystyle \frac{D_2\alpha \left(s\right)}{{\beta }_2\left(s\right)}}ds\ge D_2\alpha \left(v\right){\int }_u^v{\displaystyle \frac{ds}{{\beta }_2\left(s\right)}}$$

for $v\ge u\ge t_1$. Integrating the above inequality from u to $v\ge u$, we obtain

$$\alpha \left(u\right)\ge D_2\alpha \left(v\right){\int }_u^v{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}{\int }_s^v{\displaystyle \frac{1}{{\beta }_2\left(x\right)}}dxds=D_2\alpha \left(v\right)A(v,u).$$

(6)

Setting $u=\delta \left(t\right)$ and $v=\zeta \left(t\right),t\ge s\ge t_1$ in (6), we obtain

$$\alpha \left(\delta \left(t\right)\right)\ge D_2\alpha \left(\zeta \left(t\right)\right)A(\zeta \left(t\right),\delta \left(t\right))).$$

(7)

Using (7) in (4) yields that

$${\left(D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+E\left(t\right)A(\zeta \left(t\right),\delta \left(t\right))D_2\alpha \left(\zeta \left(t\right)\right)\le 0.$$

(8)

Now, let

$$\omega \left(t\right)=D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)>0.$$

Using the fact that $\tau \left(t\right)\le t$ and $D_2\alpha$ is nonincreasing, we see that

$$\omega \left(t\right)\le \left(1+{\displaystyle \frac{g_0}{{\tau }_0}}\right)D_2\alpha \left(\tau \left(t\right)\right)$$

or equivalently,

$$D_2\alpha \left(\zeta \left(t\right)\right)\ge {\displaystyle \frac{{\tau }_0}{{\tau }_0+g_0}}\omega \left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right).$$

(9)

Combining (9) with (8), we see that $\omega$ is a positive solution of the first-order delay differential inequality

$${\omega }^{\prime }\left(t\right)+{\displaystyle \frac{{\tau }_0}{{\tau }_0+g_0}}E\left(t\right)A(\zeta \left(t\right),\delta \left(t\right))\omega \left({\tau }^{-1}\left(\zeta \left(t\right)\right)\right)\le 0.$$

(10)

In view of a well-known result (Theorem 2 of [18] and Theorem 1 of [23]) and (5), we see that $\omega \left(t\right)$ is not a positive solution of (10). This contradiction completes the proof of the theorem. □

Theorem 2. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. If there exists a function $\eta \left(t\right)\in C([t_0,\infty ),(0,\infty ))$ satisfying $\eta \left(t\right)<t$ and $\delta \left(t\right)<\tau \left(\eta \right(t\left)\right)$ such that

$$\underset{t\to \infty }{lim}sup{\int }_{\eta \left(t\right)}^{t}E\left(s\right)A(\tau \left(\eta \left(t\right)\right),\delta \left(s\right))ds>{\displaystyle \frac{g_0+{\tau }_0}{{\tau }_0}},$$

(11)

then, $N_0$ is empty.

Proof. 

Proceeding as in the proof of Theorem 1, we have (4) and (6). Setting $u=\delta \left(s\right)$ and $v=\tau \left(\eta \left(t\right)\right),t\ge s\ge t_1$ in (6), we find

$$\alpha \left(\delta \left(s\right)\right)\ge D_2\alpha \left(\tau \left(\eta \left(t\right)\right)\right)A(\tau \left(\eta \left(t\right)\right),\delta \left(s\right)).$$

(12)

On the other hand, integrating (4) from $\eta \left(t\right)$ to t and using (12), we have

$$\begin{array}{ccc}\hfill D_2\alpha \left(\eta \left(t\right)\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(\eta \left(t\right)\right)\right)& \ge & D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)+{\int }_{\eta \left(t\right)}^{t}E\left(s\right)\alpha \left(\delta \left(s\right)\right)ds\hfill \\ & \ge & {\int }_{\eta \left(t\right)}^{t}E\left(s\right)A(\tau \left(\eta \left(t\right)\right),\delta \left(s\right))D_2\alpha \left(\tau \left(\eta \left(t\right)\right)\right)dt.\hfill \end{array}$$

Since $\tau \left(\eta \right(t\left)\right)<\tau \left(t\right)$ and $D_2\alpha$ is nonincreasing, we have

$$D_2\alpha \left(\tau \left(\eta \left(t\right)\right)\right)\left(1+{\displaystyle \frac{g_0}{{\tau }_0}}\right)\ge D_2\alpha \left(\tau \left(\eta \left(t\right)\right)\right){\int }_{\eta \left(t\right)}^{t}E\left(s\right)A(\tau \left(\eta \left(t\right)\right),\delta \left(s\right))ds,$$

that is,

$${\displaystyle \frac{{\tau }_0+g_0}{{\tau }_0}}\ge {\int }_{\eta \left(t\right)}^{t}A(\tau \left(\eta \left(t\right)\right),\delta \left(s\right))E\left(s\right)ds.$$

Taking the lim sup on both sides of the last inequality, we obtain a contradiction to (11). The proof of the theorem is complete. □

Setting $\eta \left(t\right)=\tau \left(t\right)$ in Theorem 2, we obtain the following result.

Corollary 1. 

Let (H₁)–(H₄) hold and $\delta \left(t\right)<\tau \left(\tau \right(t\left)\right)$. If

$$\underset{t\to \infty }{lim}supA(\tau \left(\tau \left(t\right)\right),\delta \left(t\right)){\int }_{\tau \left(t\right)}^{t}E\left(s\right)ds>{\displaystyle \frac{{\tau }_0+g_0}{{\tau }_0}},$$

(13)

then, $N_0$ is empty.

Note that the criteria (6) and (12) are usually called as Myshkis type criteria because results of that kind were first achieved by Myshkis, see, for example [23].

So far, we have provided new criteria for the nonexistence of Kneser type solutions of Equation $\left(E\right)$, under the assumptions $\left(H_1\right)$–$\left(H_4\right)$. In order to obtain the oscillation of all solutions of $\left(E\right),$ we begin with a theorem to eliminate the class $N_2.$

Theorem 3. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. If

$${\int }_{t_{*}}^{\infty }E\left(t\right)B_1\left(t\right)dt=\infty ,$$

(14)

then, $N_2$ is empty.

Proof. 

Assume the contrary that $\theta \in N_2$. Without loss of generality, we may assume that $\theta \left(t\right)>0,\theta \left(\tau \right(t\left)\right)>0$ and $\theta \left(\delta \right(t\left)\right)>0$ for $t\ge t_1\ge t_0$. Then, the corresponding function $\alpha \left(t\right)\in N_2$, that is,

$$\alpha \left(t\right)>0,D_1\alpha \left(t\right)>0,D_2\alpha \left(t\right)>0,D_3\alpha \left(t\right)\le 0fort\ge t_1.$$

(15)

From Lemma 4, (4) holds, that is,

$${\left(D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+E\left(t\right)\alpha \left(\delta \left(t\right)\right)\le 0.$$

(16)

Since $D_1\alpha \left(t\right)={\beta }_1\left(t\right){\alpha }^{\prime }\left(t\right)$ is increasing, there exists a $t_2$ and a constant M such that

$${\alpha }^{\prime }\left(t\right)\ge {\displaystyle \frac{M}{{\beta }_1\left(t\right)}},t\ge t_2.$$

Integrating the last inequality from $t_2$ to t, we obtain

$$\alpha \left(t\right)\ge M{B}_1\left(t\right)$$

(17)

and using (17) in (16), we obtain

$${\left(D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+ME\left(t\right)B_1\left(\delta \left(t\right)\right)\le 0,t\ge t_2.$$

Integrating the last inequality from $t_2$ to t, we have

$$M{\int }_{t_2}^{t}E\left(s\right)B_1\left(\delta \left(s\right)\right)ds<D_2\alpha \left(t_2\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t_2\right)\right)<\infty$$

which contradicts (14) as $t\to \infty$. The proof of the theorem is complete. □

Theorem 4. 

Let $\left(H_1\right)$–$\left(H_4\right)$ hold. If

$$\underset{t\to \infty }{lim}inf{\int }_{{\tau }^{-1}\left(\delta \left(t\right)\right)}^{t}E\left(s\right)B_3\left(\delta \left(s\right)\right)ds>{\displaystyle \frac{{\tau }_0+g_0}{{\tau }_{0}e}},$$

(18)

then, $N_2$ is empty.

Proof. 

Proceeding as in the proof of Theorem 3, we have (15) and (16). From (15), we see that

$$D_1\alpha \left(t\right)=D_1\alpha \left(t_1\right)+{\int }_{t_1}^t{\displaystyle \frac{D_2\alpha \left(s\right)}{{\beta }_2\left(s\right)}}ds\ge B_2\left(t\right)D_2\alpha \left(t\right)$$

(19)

and so

$${\left({\displaystyle \frac{D_1\alpha \left(t\right)}{B_2\left(t\right)}}\right)}^{\prime }={\displaystyle \frac{B_2\left(t\right)D_2\alpha \left(t\right)-D_1\alpha \left(t\right)}{{\beta }_2\left(t\right)B_2^2\left(t\right)}}\le 0.$$

That is,

$${\displaystyle \frac{D_1\alpha \left(t\right)}{B_2\left(t\right)}}\mathrm{is} \mathrm{decreasing}.$$

Now,

$$\begin{array}{ccc}\hfill \alpha \left(t\right)& =& \alpha \left(t_1\right)+{\int }_{t_1}^t{\alpha }^{\prime }\left(s\right)ds\hfill \\ & \ge & {\int }_{t_1}^t{\displaystyle \frac{B_2\left(s\right)}{{\beta }_1\left(s\right)}}{\displaystyle \frac{D_1\alpha \left(s\right)}{B_2\left(s\right)}}ds\ge B_3\left(t\right){\displaystyle \frac{D_1\alpha \left(t\right)}{B_2\left(t\right)}}\hfill \end{array}$$

and so from (9),

$$\alpha \left(t\right)\ge B_3\left(t\right)D_2\alpha \left(t\right).$$

Using this in (4), we obtain

$${\left(D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\right)}^{\prime }+E\left(t\right)B_3\left(\delta \left(t\right)\right)D_2\alpha \left(\delta \left(t\right)\right)\le 0.$$

(20)

Let

$$\begin{array}{ccc}\hfill \omega \left(t\right)& =& D_2\alpha \left(t\right)+{\displaystyle \frac{g_0}{{\tau }_0}}{D}_2\alpha \left(\tau \left(t\right)\right)\hfill \\ & \le & \left(1+{\displaystyle \frac{g_0}{{\tau }_0}}\right)D_2\alpha \left(\tau \left(t\right)\right),\hfill \end{array}$$

that is,

$$\omega \left({\tau }^{-1}\left(t\right)\right)\le \left(1+{\displaystyle \frac{g_0}{{\tau }_0}}\right)D_2\alpha \left(t\right).$$

Using this in (20), we obtain

$${\omega }^{\prime }\left(t\right)+\left({\displaystyle \frac{{\tau }_0}{{\tau }_0+g_0}}\right)E\left(t\right)B_3\left(\delta \left(t\right)\right)\omega \left({\tau }^{-1}\left(\delta \left(t\right)\right)\right)\le 0.$$

The rest of the proof is similar to that of Theorem 1. The proof of the theorem is complete. □

Now, combining Theorem 3 or Theorem 4 with Theorem 1 or Theorem 2 or Corollary 1, we obtain the oscillation criteria for Equation $\left(E\right)$.

Theorem 5. 

Assume that condition (15) or (19) holds. If all assumptions of Theorem 1 or Theorem 2 or Corollary 1 are satisfied, then Equation $\left(E\right)$ is oscillatory.

Proof. 

From Theorem 3 or Theorem 4, we see that $N_2$ is empty, and the set $N_0$ is empty by Theorem 1 or Theorem 2 or Corollary 1. So, all solutions of $\left(E\right)$ are oscillatory. □

Remark 1. 

The noncanonical Equation $\left(E\right)$ is changed to a canonical type Equation $\left(E_c\right)$ without assuming any extra conditions, and so one can apply all known results regarding the oscillation of $\left(E_c\right)$ to obtain similar results for the noncanonical Equation $\left(E\right)$. Therefore, we hope this method is very useful to obtain the oscillation and asymptotic behavior of solutions of noncanonical equations from their canonical type.

3. Examples

In this section, we provide two examples to show the novelty and the importance of our main results.

Example 1. 

Consider the third-order neutral delay differential equation

$${\left(t^2{\left(t^2{\left(\theta \left(t\right)+g_0\theta \left({\lambda }_{1}t\right)\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+f_{0}t\theta \left({\lambda }_{2}t\right)=0,t\ge 1,$$

(21)

where $g_0>0,f_0>0,{\lambda }_1,{\lambda }_2\in (0,1)$ with ${\lambda }_1>{\lambda }_2$. Here, ${\mu }_1\left(t\right)={\mu }_2\left(t\right)=t^2,$$g\left(t\right)=g_0,\delta \left(t\right)={\lambda }_{2}t,\tau \left(t\right)={\lambda }_{1}t$ and $f\left(t\right)=f_{0}t.$ Clearly, the hypotheses $\left(H_1\right)$–$\left(H_4\right)$ are satisfied. By a simple calculation, we can transform Equation (21) into

$${\alpha }^{‴}\left(t\right)+{\displaystyle \frac{2f_0}{t}}\theta \left({\lambda }_{2}t\right)=0,t\ge 1,$$

which is in canonical form. With a further calculation, we see that $B_1\left(t\right)=B_2\left(t\right)\approx t,$$B_3\left(t\right)\approx {\displaystyle \frac{t^2}{2}},G\left(t\right)={\displaystyle \frac{2f_0}{t}},E\left(t\right)={\displaystyle \frac{f_0}{{\lambda }_2^2{t}^3}},A(t,t_1)={\displaystyle \frac{{(t-t_1)}^2}{2}}$. Choose $\zeta \left(t\right)={\lambda }_{3}t$, such that ${\lambda }_1>{\lambda }_3>{\lambda }_2$; then, ${\tau }^{-1}\left(\zeta \left(t\right)\right)={\displaystyle \frac{{\lambda }_3}{{\lambda }_1}}t<t$. Condition (5) becomes

$$\underset{t\to \infty }{lim}inf{\int }_{{\displaystyle \frac{{\lambda }_3}{{\lambda }_1}}t}^t{\displaystyle \frac{f_0}{{\lambda }_2^2{s}^3}}{s}^2{\displaystyle \frac{{({\lambda }_3-{\lambda }_2)}^2}{2}}dx={\displaystyle \frac{f_0}{2}}{\displaystyle \frac{{({\lambda }_3-{\lambda }_2)}^2}{{\lambda }_2^2}}\ell n\left({\displaystyle \frac{{\lambda }_1}{{\lambda }_3}}\right)>{\displaystyle \frac{{\lambda }_1+g_0}{{\lambda }_{1}e}},$$

that is, (5) holds if

$$f_0{\displaystyle \frac{{({\lambda }_3-{\lambda }_2)}^2}{{\lambda }_2^2}}\ell n\left({\displaystyle \frac{{\lambda }_1}{{\lambda }_3}}\right)>2\left({\displaystyle \frac{{\lambda }_1+g_0}{{\lambda }_{1}e}}\right).$$

(22)

Thus, $N_0$ is empty if (22) holds. In particular, let ${\lambda }_1={\displaystyle \frac{1}{2}},{\lambda }_2={\displaystyle \frac{1}{4}},{\lambda }_3={\displaystyle \frac{1}{3}}$ and $g_0={\displaystyle \frac{1}{2}}$; we see that $N_0$ is empty if $f_0>32.6629$, that is, no Kneser type solution for (21) if $f_0>32.6629$.

Next, the condition (18) becomes

$$\underset{t\to \infty }{lim}inf{\int }_{{\displaystyle \frac{{\lambda }_2}{{\lambda }_1}}t}^t{\displaystyle \frac{f_0}{2s}}ds={\displaystyle \frac{f_0}{2}}\ell n\left({\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}\right)>{\displaystyle \frac{{\lambda }_1+g_0}{{\lambda }_{1}e}},$$

that is, $N_2$ is empty if $f_0\ell n\left({\displaystyle \frac{{\lambda }_1}{{\lambda }_2}}\right)>2\left({\displaystyle \frac{{\lambda }_1+g_0}{{\lambda }_{1}e}}\right)$. For ${\lambda }_1={\displaystyle \frac{1}{2}},{\lambda }_2={\displaystyle \frac{1}{4}},{\lambda }_3={\displaystyle \frac{1}{3}}$ and $g_0={\displaystyle \frac{1}{2}}$, we see that

$$N_{2}is empty,if{f}_0>2.12295,$$

(23)

that is, no non-Kneser type solution for (21) if (23) holds. Hence, Equation $\left(E\right)$ is oscillatory if $f_0>32.6629$.

Example 2. 

Consider again Equation (21) with $g_0=1,\tau \left(t\right)={\displaystyle \frac{t}{2}}$ and $\delta ={\displaystyle \frac{t}{4}}.$ We see the transformed equation is

$${\alpha }^{‴}\left(t\right)+{\displaystyle \frac{2f_0}{t}}\theta \left({\displaystyle \frac{t}{4}}\right)=0,t\ge 1.$$

Moreover, $B_1\left(t\right)=B_2\left(t\right)\approx t,B_3\left(t\right)\approx {\displaystyle \frac{t^2}{2}},G\left(t\right)={\displaystyle \frac{2f_0}{t}},E\left(t\right)={\displaystyle \frac{16f_0}{t^3}}$ and $A(t,t_1)={\displaystyle \frac{{(t-t_1)}^2}{2}}$. Choosing $\eta ={\displaystyle \frac{3}{4}}t$, we see that $\delta \left(t\right)<\tau \left(\eta \right(t\left)\right)$ the condition (11) becomes

$$\underset{t\to \infty }{lim}{\displaystyle \frac{{\left(\frac{3}{8}t-\frac{t}{4}\right)}^2}{2}}{\int }_{{\displaystyle \frac{3}{4}}t}^t{\displaystyle \frac{16f_0}{s^3}}ds={\displaystyle \frac{7f_0}{144}}>3.$$

Therefore, we see that $N_0$ is empty for $f_0>61.73143$. Therefore, Kneser type solutions do not exist when $f_0>61.73143$. Also for $g_0=1$, the set $N_2$ is empty if $f_0>3.18443$. Hence, Equation (21) with $g_0=1$ is oscillatory if $f_0>61.73143$.

Remark 2. 

Note that this conclusion cannot be obtained by applying [14] or [10] or [1] since $g_0=1$.

4. Conclusions

In this paper, we have obtained conditions for the nonexistence of Kneser type solutions for the studied Equation $\left(E\right)$. This is achieved via transforming the noncanonical Equation $\left(E\right)$ into a canonical type equation. This reduces the number of Kneser type solutions to one instead of two. Further, we obtain conditions for making the class $N_2$ empty. Using these results, we achieve the oscillation of all solutions of $\left(E\right).$ Hence, the results of this paper are new and a significant contribution to the oscillation theory of third-order neutral differential equations. It is interesting to extend the results of this paper to nonlinear noncanonical third-order neutral differential equations with or without the hypothesis $\left(H_4\right).$