1 Introduction

Advanced differential equations can be found application in numerous real-world problems where the evolution rate depends not only on the present, but also on the future. Therefore, an advanced argument could be introduced into the equation to describe the influence of potential future actions. For instance, representative fields where such phenomena are thought to occur are population dynamics, economical problems or mechanical control engineering; see [1, 2].

As a matter of fact, the investigation of half-linear equations has become an important area of research due to the fact that such equations arise in a variety of real world problems such as in the study of p-Laplace equations, non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in a porous medium; see [3, 4].

In reference [5,6,7,8], the oscillation of half-linear differential equations is studied. The articles on oscillation of differential equations with noncanonical operators can be found in literature [9], and qualitative properties of third-order differential equations can be seen in reference [10,11,12]. However, there are a few articles on oscillation of third-order advanced half-linear differential equation with noncanonical operators.

This paper deals with oscillation of the third-order neutral advanced half-linear differential equation of the form

$$\begin{aligned} \begin{aligned} \left\{ a(t)\left[ b(t)y'(t)\right] '\right\} '+q(t)f(x(\tau (t)))=0,\ \ \ t\geqslant t_0, \end{aligned} \end{aligned}$$
(1.1)

where \(y(t):=x(t)+p(t)x(\sigma (t))\). Throughout the paper, we will assume that

\((H_1)\) The functions \(a, b\in C([t_0,\infty ],(0,\infty ))\) and satisfy

$$\begin{aligned} \pi _1(t_0,\infty ):=\int _{t_0}^\infty \frac{\mathrm{d}s}{a(s)}<\infty \ \ \text {and}\ \ \pi _2(t_0,\infty ):=\int _{t_0}^\infty \frac{\mathrm{d}s}{b(s)}=\infty . \end{aligned}$$

\((H_2)\) \(\tau ,\ \sigma \in C^1([t_0,\infty ],{\mathbb {R}})\), \(\tau (t)<t\), \(\sigma (t)>t\), \(\tau '(t)>0\), \(\sigma '(t)\geqslant \sigma _0>0\), \(\sigma \circ \tau =\tau \circ \sigma \) and \(\lim \limits _{t\rightarrow \infty }\tau (t)=\lim \limits _{t\rightarrow \infty }\sigma (t)=\infty \).

\((H_3)\) \(p\in C([t_0,\infty ],[0,\infty ))\), \(q\in C([t_0,\infty ],(0,\infty ))\), \(p(t)\leqslant p_0<1\) and q does not vanish identically.

For the sake of brevity, we define the operators

$$\begin{aligned} L_0y=y,\ L_1y=by',\ L_2y=a(by')',\ L_3y=(a(by')')'. \end{aligned}$$

As is customary, a solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate.

Following classical results of Kiguradze and Kondratev (see, e.g., [13]), we say that (1.1) is almost oscillatory if any solution x of (1.1) is either oscillatory or satisfies \(\lim \limits _{t\rightarrow \infty }x(t) = 0\).

To start with, let us state a characterization of possible nonoscillatory solutions of (1.1). The following result is a modification of the well-known Kiguradze lemma [3, Lemma 1.1] based on (\(H_1\)).

Lemma 1.1

Let x be an eventually positive solution of (1.1). Then, there exists \(t_1\in [t_0,\infty )\) such that y is one of the following cases:

$$\begin{aligned}&case\ (1): y>0,\ \ L_1y<0,\ \ L_2y>0,\ \ L_3y<0, \\&case\ (2): y>0,\ \ L_1y>0,\ \ L_2y>0,\ \ L_3y<0, \\&case\ (3): y>0,\ \ L_1y>0,\ \ L_2y<0,\ \ L_3y<0, \end{aligned}$$

for \(t\geqslant t_1\). Solutions x whose corresponding function y satisfies case (1) are called Kneser solutions.

It is known that case (1) is not an empty set in both canonical and noncanonical cases. So, the elimination of Kneser solutions is an important method to study oscillation of the equation.

The aims of this work are to present new criteria for almost oscillation and oscillation of the third-order delay differential equation (1.1). In the existing papers on third-order differential equations under condition \((H_1)\), there are sufficient conditions of three first-order equations which are oscillatory to guarantee their oscillation; see [14, 15] etc.

In [15], R. P. Agarwal et al. considered third-order nonlinear delay differential equation of the form

$$\begin{aligned} \begin{aligned} \left\{ a(t)\left[ b(t)y'(t)\right] '\right\} '+q(t)y^\gamma (\tau (t))=0,\ \ \ t\geqslant t_0. \end{aligned} \end{aligned}$$
(1.2)

Their main result is as follows.

Theorem 1.1

Assume that there exist numbers \(\alpha \leqslant \gamma \), \(\beta \geqslant \gamma \), and two functions \(\xi ,\sigma \in C([t_0,\infty ),{\mathbb {R}})\) such that \(\alpha \), \(\beta \) are the ratios of odd positive integers, \(\xi (t)>t\), \(\xi (t)\) is nondecreasing, \(\tau (\xi (\xi (t)))<t\), \(\sigma (t)\) is nondecreasing, and \(\sigma (t)>t\). If for all sufficiently large \(t_1\geqslant t_0\) and for \(t_2>t_1\), the first-order delay differential equation

$$\begin{aligned} \omega '(t)+c_1^{\gamma -\alpha }q(t)\left( \int _{t_2}^{\tau (t)}\frac{\int _{t_1}^s\frac{\mathrm{d}u}{a(u)}}{b(s)}\mathrm{d}s\right) ^\alpha \omega ^\alpha (\tau (t))=0,\ \ t>t_2 \end{aligned}$$

is oscillatory for all constants \(c_1>0\), the first-order delay differential equation

$$\begin{aligned} \nu '(t)+\left( \frac{1}{b(t)}\int _t^{\xi (t)}\frac{1}{a(s_2)}\int _{s_2}^\xi (s_2q(s_1))\mathrm{d}s_1\mathrm{d}s_2\right) v^\gamma (\tau (\xi (\xi (t))))=0,\ \ t>t_2 \end{aligned}$$

is oscillatory, and the first-order advanced differential equation

$$\begin{aligned} z'(t)-c_2^{\gamma -\beta }q(t)\left( \int _{\sigma (t)}^\infty \frac{\mathrm{d}s}{a(s)}\right) ^\beta \left( \int _{t_1}^{\tau (t)} \frac{\mathrm{d}s}{b(s)}\right) ^\gamma z^\beta (\sigma (t))=0,\ \ t>t_2 \end{aligned}$$

is oscillatory for all constants \(c_2>0\), then (1.2) is oscillatory.

Compared to existing results of [15], oscillation of (1.1) is attained by easier conditions. Our ideas are partly based on [16, 17]. We consider oscillation of third-order differential equation by establishing sufficient conditions for nonexistence of Kneser solutions rather than entirely using transformation techniques such as the aforementioned one due to Trench [18].

The organization is as follows. Firstly, we present a simplified theorem for almost oscillation of (1.1). Secondly, we establish sufficient conditions for nonexistence of Kneser solutions. Thirdly, we establish oscillation criteria for (1.1). Finally, we illustrate the importance of the results obtained.

2 Main Results

First, we study equation (1.1) under the following conditions: \((H_{4a})\) \(f\in C([t_0,\infty ),{\mathbb {R}})\), and \(xf(x)>0\); there exists a positive constant k such that

$$\begin{aligned} \frac{f(x)}{x^\gamma }\geqslant k,\ \ x\ne 0, \end{aligned}$$

where \(\gamma \) is the ratio of positive odd integers.

Next, we shall establish a new criterion for almost oscillation of (1.1).

As usual, all occurring functional inequalities are assumed to hold eventually; that is, they are satisfied for all t large enough.

Theorem 2.1

If there exists a positive real-valued differentiable function m(t) such that, for any real number d

$$\begin{aligned} \begin{aligned} \frac{d}{b(t)}+\frac{m'(t)}{m(t)}<0,\ \ \inf _{t\geqslant t_0}\left\{ 1-p(\tau (t))\frac{m(\tau (t))}{m(\sigma (\tau (t)))}\right\} >0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty \frac{1}{a(u)}\int _{t_0}^u q(s)\mathrm{d}s\mathrm{d}u=\infty , \end{aligned} \end{aligned}$$
(2.1)

then (1.1) is almost oscillatory.

Proof

First of all, it is important to note that if both \((H_1)\) and (2.1) hold, then

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty q(s)\mathrm{d}s=\infty . \end{aligned} \end{aligned}$$
(2.2)

Now suppose for the sake of contradiction that x is a nonoscillatory solution of (1.1) on \([t_0,\infty )\). Without loss of generality, we may take \(t_1 \geqslant t_0\) such that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\). By Lemma 1.1, three possible cases may occur for \(t\geqslant t_1\). We will consider each of these cases separately.

Assume that case (1) holds. Then, from \(L_1y<0\), we see that y is decreasing, that is, there exists a finite constant \(l\geqslant 0\) such that \(\lim \limits _{t\rightarrow \infty }y(t)=l\). We claim that \(l=0\). Assume on the contrary that \(l>0\). Then, there exists \(t_2\geqslant t_1\) such that \(y(\tau (t))\geqslant l\) for \(t\geqslant t_2\). Because of \((H_2)\)-\((H_3)\) and \(y(t)=x(t)+p(t)x(\sigma (t))\), we obtain

$$\begin{aligned}&x(t)=y(t)-p(t)x(\sigma (t))\geqslant y(t)-p(t)y(\sigma (t))\nonumber \\&\geqslant y(t)-p(t)y(t)\geqslant y(t)(1-p_0). \end{aligned}$$
(2.3)

Then,

$$\begin{aligned} \begin{aligned} x(\tau (t))\geqslant l\left( 1-p_0\right) . \end{aligned} \end{aligned}$$
(2.4)

From (1.1) and \((H_{4a})\), we have

$$\begin{aligned} \begin{aligned} -L_3y(t)\geqslant kq(t)x^\gamma (\tau (t)). \end{aligned} \end{aligned}$$
(2.5)

We can obtain the following inequality from (2.4) and (2.5)

$$\begin{aligned} \begin{aligned} -L_3y(t)\geqslant kl^\gamma \left( 1-p_0\right) ^\gamma q(t). \end{aligned} \end{aligned}$$
(2.6)

Integrating (2.6) from \(t_2\) to t, we see that

$$\begin{aligned} L_2y(t)\leqslant L_2y(t_2)-kl^\gamma \left( 1-p_0\right) ^\gamma \int _{t_0}^t q(s)\mathrm{d}s\rightarrow -\infty ,\ \ t\rightarrow \infty , \end{aligned}$$

where we used (2.2). This contradicts to the positivity of \(L_2y\) and thus, \(\lim \limits _{t\rightarrow \infty }y(t)=0\).

Assume that case (2) holds. Let us define a function

$$\begin{aligned} w(t)=\frac{L_2y(t)}{y^\gamma (\tau (t))},\ \ t\geqslant t_1. \end{aligned}$$

Clearly, w is positive for \(t\geqslant t_1\). By (1.1), we have

$$\begin{aligned} w'(t)=\frac{L_3y(t)}{y^\gamma (\tau (t))}-\frac{L_2y(t)\gamma y'(\tau (t))\tau '(t)}{y^{\gamma +1}(\tau (t))}\leqslant \frac{L_3y(t)}{y^\gamma (\tau (t))} \leqslant \frac{L_3y(t)}{x^\gamma (\tau (t))}\leqslant -kq(t). \end{aligned}$$

Integrating the above inequality from \(t_1\) to t and taking (2.2) into account, we obtain

$$\begin{aligned} w(t)\leqslant w(t_1)-k\int _{t_1}^tq(s)\mathrm{d}s\rightarrow -\infty ,\ \ t\rightarrow \infty , \end{aligned}$$

a contradiction.

Assume that case (3) holds. Then, from \(L_1y>0\), \(L_2y<0\), i.e., \(y'(t)>0\), \((b(t)y'(t))'<0\). We can easily see that

$$\begin{aligned} \left( \frac{b(t)y'(t)}{y(t)}\right) '<0. \end{aligned}$$

Let \(d=\frac{b(t_1)y'(t_1)}{y(t_1)}\). We have

$$\begin{aligned} \frac{b(t)y'(t)}{y(t)}<d,\ \ t>t_1, \end{aligned}$$

namely,

$$\begin{aligned} \frac{y'(t)}{y(t)}<\frac{d}{b(t)}. \end{aligned}$$

Then, we have

$$\begin{aligned} \begin{aligned} (y(t)m(t))'&=y'(t)m(t)+y(t)m'(t)=m(t)y(t)\left( \frac{y'(t)}{y(t)}+\frac{m'(t)}{m(t)}\right) \\&\leqslant m(t)y(t)\left( \frac{d}{b(t)}+\frac{m'(t)}{m(t)}\right) <0, \end{aligned} \end{aligned}$$

and thus, y(t)m(t) is nonincreasing. Because of \((H_2)\)–\((H_3)\) and \(y(t)=x(t)+p(t)x(\sigma (t))\), we have

$$\begin{aligned} x(t)= & {} y(t)-p(t)x(\sigma (t))\geqslant y(t)-p(t)y(\sigma (t))\geqslant y(t)-p(t)\frac{m(t)}{m(\sigma (t))}y(t)\\= & {} y(t)\left( 1-p(t)\frac{m(t)}{m(\sigma (t))}\right) . \end{aligned}$$

Integrating (2.5) from \(t_1\) to t yields

$$\begin{aligned} \begin{aligned} -L_2y(t)&\geqslant -L_2y(t_1)+k\int _{t_1}^tq(s)x^\gamma (\tau (s))\mathrm{d}s\\&\geqslant k\int _{t_1}^t y^\gamma (\tau (s))\left( 1-p(\tau (s))\frac{m(\tau (s))}{m(\sigma (\tau (s)))}\right) ^\gamma q(s)\mathrm{d}s\\&\geqslant ky^\gamma (\tau (t_1))\inf _{t\geqslant t_0}\left\{ 1-p(\tau (t))\frac{m(\tau (t))}{m(\sigma (\tau (t)))}\right\} ^\gamma \int _{t_1}^t q(s)\mathrm{d}s, \end{aligned} \end{aligned}$$

i.e.,

$$\begin{aligned} -(L_1y)'(t)\geqslant \frac{K}{a(t)}\int _{t_1}^tq(s)\mathrm{d}s, \end{aligned}$$

where \(K:=ky^\gamma (\tau (t_1))\inf \limits _{t\geqslant t_0}\left\{ 1-p(\tau (t))\frac{m(\tau (t))}{m(\sigma (\tau (t)))}\right\} ^\gamma \). Integrating above inequality from \(t_1\) to t and using (2.1), we have

$$\begin{aligned} L_1y(t)\leqslant L_1y(t_1)-K\int _{t_1}^t\frac{1}{a(u)}\int _{t_1}^uq(s)\mathrm{d}s\mathrm{d}u\rightarrow -\infty , \end{aligned}$$

a contradiction. The proof is complete. \(\square \)

Remark 2.1

We can see that any nonoscillatory solution about equation (1.1) satisfies case (1) which are called Kneser solutions from the proof of Theorem 2.1.

Theorem 2.2

If there exists a function \(\xi (t)\in C([t_0 ,\infty ),(0,\infty ))\) satisfying \(\tau (\sigma (t))<\xi (t)\), such that the first-order delay differential equation

$$\begin{aligned} \begin{aligned} \omega '(t)+\frac{2k\sigma _0^\gamma (1-p_0)^\gamma }{(1+\sigma _0)^\gamma }Q(t)\pi ^\gamma (\tau (\sigma (t)),\xi (t)) \omega ^\gamma (\xi (t))=0 \end{aligned} \end{aligned}$$
(2.7)

is oscillatory, where \(Q(t):=\min \{q(t),q(\sigma (t))\}\) and \(\pi (u,v):=\int _u^v\frac{\pi _1(s,v)}{b(s)}\mathrm{d}s\), then (1.1) has no Kneser solution.

Proof

Assume to the contrary that x is a Kneser solution of (1.1). Without loss of generality, we may take \(x(t)>0,\ x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\geqslant t_0\). Because of Lemma 1.1, we obtain

$$\begin{aligned} y> 0,\ \ L_1y<0,\ \ L_2y>0,\ \ L_3y<0,\ \ \ [t_1,\infty ). \end{aligned}$$

From (2.5), we see that

$$\begin{aligned} \begin{aligned} 0\geqslant \frac{1}{\sigma '(t)}(L_2y(\sigma (t)))'+k q(\sigma (t))x^\gamma (\tau (\sigma (t))). \end{aligned} \end{aligned}$$

Combining (1.1) along with the above inequality and using \((H_{2})\), we obtain

$$\begin{aligned} \begin{aligned} 0&\geqslant L_3y(t)+\frac{1}{\sigma '(t)}(L_2y(\sigma (t)))'+kq(t)x^\gamma (\tau (t))+ kq(\sigma (t))x^\gamma (\tau (\sigma (t)))\\&\geqslant L_3y(t)+\frac{1}{\sigma _0}(L_2y(\sigma (t)))'+kQ(t)(x^\gamma (\tau (t))+ x^\gamma (\tau (\sigma (t)))), \end{aligned} \end{aligned}$$
(2.8)

where \(Q(t)=\min \{q(t),q(\sigma (t))\}\).

By virtue of (2.3), (2.8) becomes

$$\begin{aligned} L_3y(t)+\frac{1}{\sigma _0}(L_2y(\sigma (t)))'+2(1-p_0)^\gamma k Q(t)y^\gamma (\tau (\sigma (t)))\leqslant 0, \end{aligned}$$

that is

$$\begin{aligned} \begin{aligned} (L_2y(t)+\frac{1}{\sigma _0} L_2y(\sigma (t)))'+2(1-p_0)^\gamma kQ(t)y^\gamma (\tau (\sigma (t)))\leqslant 0. \end{aligned} \end{aligned}$$
(2.9)

On the other hand, it follows from the monotonicity of \(L_2y\) that

$$\begin{aligned} -L_1y(s)\geqslant L_1y(v)-L_1y(s)=\int _s^v\frac{L_2y(i)}{a(i)}\mathrm{d}i\geqslant L_2y(v)\pi _1(s,v), \end{aligned}$$

for \(v\geqslant s\geqslant t_1\). Integrating the above inequality from \(u\geqslant t_1\) to v for s, we obtain

$$\begin{aligned} \begin{aligned} y(u)\geqslant L_2y(v)\int _u^v\frac{\pi _1(s,v)}{b(s)}\mathrm{d}s:= L_2y(v)\pi (u,v). \end{aligned} \end{aligned}$$
(2.10)

Hence, we have

$$\begin{aligned} \begin{aligned} y(\tau (\sigma (t)))\geqslant L_2y(\xi (t))\pi (\tau (\sigma (t)),\xi (t)),\ \ \ \xi (t)>\tau (\sigma (t)). \end{aligned} \end{aligned}$$
(2.11)

Because of (2.9) and (2.11), we have

$$\begin{aligned} \begin{aligned} (L_2y(t)+\frac{1}{\sigma _0} L_2y(\sigma (t)))'+2(1-p_0)^\gamma kQ(t)L_2^\gamma y(\xi (t))\pi ^\gamma (\tau (\sigma (t)),\xi (t))\leqslant 0. \end{aligned}\nonumber \\ \end{aligned}$$
(2.12)

Now, set

$$\begin{aligned} \omega (t):=L_2y(t)+\frac{1}{\sigma _0}L_2y(\sigma (t))>0. \end{aligned}$$

From \((H_3)\) and the fact that \(L_2y\) is decreasing, we have

$$\begin{aligned} \omega (t)\leqslant L_2y(t)(1+\frac{1}{\sigma _0}), \end{aligned}$$

or equivalently,

$$\begin{aligned} \begin{aligned} L_2y(\xi (t))\geqslant \omega (\xi (t))\frac{\sigma _0}{1+\sigma _0}. \end{aligned} \end{aligned}$$
(2.13)

Using (2.13) in (2.12), we see that \(\omega (t)\) is a positive solution of the first-order delay differential inequality

$$\begin{aligned} \omega '(t)+ \frac{2k\sigma _0^\gamma (1-p_0)^\gamma }{(1+\sigma _0)^\gamma }Q(t)\pi ^\gamma (\tau (\sigma (t)),\xi (t)) \omega ^\gamma (\xi (t)) \leqslant 0. \end{aligned}$$

In view of Corollary 1 in [19], the associated delay differential equation (2.7) also has a positive solution, which is a contradiction. Thus, the class of Kneser solutions is empty and the proof is complete. \(\square \)

Remark 2.2

Theorem 2.2 is easy to apply when \(\tau (\sigma (t))<\xi (t)<t\). However, when \(\xi (t)>\tau (\sigma (t))>t\) or \(\xi (t)>t>\tau (\sigma (t))\), the results of oscillation of the first-order advanced superlinear equation are few and need to be further studied.

Therefore, we can conclude oscillation of equation by combining the above two theorems.

Corollary 2.1

If all assumptions of Theorems 2.1 and 2.2 are satisfied, then (1.1) is oscillatory.

Now, we study (1.1) under the following condition that \(\gamma =1\) in \((H_{4a})\). \((H_{4b})\) \(f\in C([t_0,\infty ),{\mathbb {R}})\), and \(xf(x)>0\); there exists a positive constant \(k_1\) such that

$$\begin{aligned} \frac{f(x)}{x}\geqslant k_1,\ \ x\ne 0. \end{aligned}$$

Theorem 2.3

If there exists a function \(\eta (t)\in C([t_0,\infty ),(0,\infty ))\) satisfying \(t>\eta (t)>\tau (t)\), such that

$$\begin{aligned} \begin{aligned} \liminf _{t\rightarrow \infty }\pi (\tau (t),\eta (t))\int _{\eta (t)}^tQ(s)\mathrm{d}s>\frac{\sigma _0+p_0}{k_1\sigma _0}, \end{aligned} \end{aligned}$$
(2.14)

then (1.1) has no Kneser solution.

Proof

Assume to the contrary that x is a Kneser solution of (1.1). Without loss of generality, we may take \(x(t)>0,\ x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\geqslant t_0\). From Lemma 1.1, we have

$$\begin{aligned} y> 0,\ \ L_1y<0,\ \ L_2y>0,\ \ L_3y<0,\ \ \ [t_1,\infty ). \end{aligned}$$

Because of (1.1) and \((H_{4b})\), we see that

$$\begin{aligned} \begin{aligned} 0\geqslant (L_2y(t))'+k_1 q(t)x(\tau (t)), \end{aligned} \end{aligned}$$
(2.15)

then

$$\begin{aligned} \begin{aligned} 0\geqslant \frac{p_0}{\sigma '(t)}(L_2y(\sigma (t)))'+p_0k_1 q(\sigma (t))x(\tau (\sigma (t))). \end{aligned} \end{aligned}$$
(2.16)

Combining (2.15) along with (2.16), we obtain

$$\begin{aligned} \begin{aligned} 0&\geqslant L_3y(t)+\frac{p_0}{\sigma '(t)}(L_2y(\sigma (t)))'+k_1q(t)x(\tau (t))+ k_1 p_0q(\sigma (t))x(\tau (\sigma (t)))\\&\geqslant L_3y(t)+\frac{p_0}{\sigma _0}(L_2y(\sigma (t)))'+k_1Q(t)(x(\tau (t))+ p_0x(\tau (\sigma (t)))), \end{aligned} \end{aligned}$$
(2.17)

where \(Q(t)=\min \{q(t),q(\sigma (t))\}\). Using \((H_3)\) in the definition of y, we get

$$\begin{aligned} \begin{aligned} y(t)=x(t)+p(t)x(\sigma (t))\leqslant x(t)+p_0x(\sigma (t)). \end{aligned} \end{aligned}$$
(2.18)

By virtue of (2.18), (2.17) becomes

$$\begin{aligned} L_3y(t)+\frac{p_0}{\sigma _0}(L_2y(\sigma (t)))'+k_1 Q(t)y(\tau (t))\leqslant 0, \end{aligned}$$

that is,

$$\begin{aligned} \begin{aligned} (L_2y(t)+\frac{p_0}{\sigma _0} L_2y(\sigma (t)))'+k_1Q(t)y(\tau (t))\leqslant 0. \end{aligned} \end{aligned}$$

Integrating this inequality from \(\eta (t)\) to t and using the fact that y is decreasing, we see that

$$\begin{aligned} \begin{aligned} L_2y(\eta (t))+\frac{p_0}{\sigma _0}L_2y(\sigma (\eta (t)))&\geqslant L_2y(t)+\frac{p_0}{\sigma _0}L_2y(\sigma (t))+k_1\int _{\eta (t)}^tQ(s)y(\tau (s))\mathrm{d}s\\&\geqslant k_1y(\tau (t))\int _{\eta (t)}^tQ(s)\mathrm{d}s. \end{aligned} \end{aligned}$$

Since \(t<\sigma (t)\), \(L_2y\) is decreasing, we have

$$\begin{aligned} \begin{aligned} (1+\frac{p_0}{\sigma _0})L_2y(\eta (t))\geqslant k_1y(\tau (t))\int _{\eta (t)}^tQ(s)\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.19)

Using (2.10) with \(v=\eta (t)\) and \(u =\tau (t)\) in (2.19), we arrive at

$$\begin{aligned} (1+\frac{p_0}{\sigma _0})L_2y(\eta (t)) \geqslant k_1L_2y(\eta (t))\pi (\tau (t),\eta (t))\int _{\eta (t)}^tQ(s)\mathrm{d}s. \end{aligned}$$

That is,

$$\begin{aligned} \frac{\sigma _0+p_0}{\sigma _0}\geqslant k_1\pi (\tau (t),\eta (t))\int _{\eta (t)}^tQ(s)\mathrm{d}s. \end{aligned}$$

Taking the limit inferior on both side of the above inequality, we obtain a contradiction to (2.14). The proof is complete. \(\square \)

Corollary 2.2

If all assumptions of Theorems 2.1 and 2.3 are satisfied, then (1.1) is oscillatory.

Remark 2.1

Under condition \((H_{4b})\), we study (1.1) and obtain a sufficient condition containing lower limit, which is more applicable than Theorem 2.2.

3 Example

In this section, we will present an example to illustrate our main results.

Example 4.1

Consider the third-order neutral differential equation

$$\begin{aligned} \begin{aligned} \left\{ t^3\left[ t^{-2}(x(t)+\frac{1}{2}x(t+1))'\right] '\right\} '+4t^3x(t-2)=0,\ \ \ t>2, \end{aligned} \end{aligned}$$
(3.1)

where \(a(t)=t^3\), \(b(t)=t^{-2}\), \(p(t)=\frac{1}{2}e^{-(4t^3+4t^2+4t+1)}\), \(p_0=\frac{1}{2}e^{-1}\), \(q(t)=4t^3\), \(\sigma (t)=t+1\), \(\sigma _0=1\), \(\tau (t)=t-2\). Let \(m(t)=e^{-t^4}\), \(\eta (t)=t-1\), \(k_1=1\). We can conclude from (3.1) that \(Q(t)=4t^3\),

$$\begin{aligned} \pi (\tau (t),\eta (t))= & {} \int _{t-2}^{t-1}\frac{\int _{s}^{t-1}\frac{di}{a(i)}}{b(s)}\mathrm{d}s =\int _{t-2}^{t-1}\left\{ -\frac{1}{2}(t-1)^{-2}s^2+\frac{1}{2}\right\} \mathrm{d}s\\= & {} -\frac{1}{6}t+\frac{2}{3}+\frac{1}{6}\frac{(t-2)^3}{(t-1)^2}, \\&\pi (\tau (t),\eta (t))\int _{\eta (t)}^tQ(t)\mathrm{d}s=\left( -\frac{1}{6}t+\frac{2}{3}+\frac{1}{6}\frac{(t-2)^3}{(t-1)^2} \right) (4t^3-2t^2-1)\\> & {} \frac{\sigma _0+p_0}{k_1\sigma _0}=\frac{3}{2},\ \ \ t>2. \end{aligned}$$

And let \(d=8\), we have

$$\begin{aligned} \frac{d}{b(t)}+\frac{m'(t)}{m(t)}=dt^2-4t^3<0,\ \ t>2. \end{aligned}$$

It is easy to see

$$\begin{aligned} \inf _{t\geqslant t_0}\left\{ 1-p(t)\frac{m(t)}{m(\sigma (t))}\right\}>0,\ \ t>0. \end{aligned}$$

So, (3.1) satisfies all the conditions of Corollary 2.2; then, (3.1) oscillates.

However, letting \(\xi (t)=t+2\), we have

$$\begin{aligned} \pi (\tau (\sigma (t)),\xi (t))= & {} \int _{t-1}^{t+2}s^2\int _s^{t+2}\frac{1}{i^3}\mathrm{d}i\mathrm{d}s=\frac{1}{2}\int _{t-1}^{t+2}\{-(t+2)^{-2}s^2+1\}\mathrm{d}s\\= & {} \frac{1}{6}\left( \frac{(t-1)^3}{(t+2)^{2}}-t+7\right) . \end{aligned}$$

(2.7) becomes

$$\begin{aligned} \omega '(t)+\frac{1-e^{-1}}{6}t^3\left( \frac{(t-1)^3}{(t+2)^{2}}-t+7\right) \omega (t+2)=0. \end{aligned}$$
(3.2)

The oscillation of (3.2) cannot be simply judged. Therefore, the oscillation of (3.1) cannot be explained by Corollary 2.1.

4 Conclusion

In this paper, we present a simplified theorem for almost oscillation of (1.1) and sufficient conditions for nonexistence of Kneser solutions. Combining above two cases, we can get the result of oscillatory behavior for (1.1). The conclusion in this paper is also applicable to equation (1.2). Based on the above research, we find it very difficult to obtain sufficient conditions for the existence of Kneser solutions involving upper or lower limits under the condition of \((H_{4a})\). Readers can continue to study in depth.