1 Introduction

Differential equations with deviating arguments can be considered as models of numerous practical applications, see [1].

Delay differential equations make it more accurate to describe models in which some depend only upon the past state and some depend upon the past state as well as the rate of change of the past state. Particularly, it is very important to consider the delay argument in the feedback mechanism; we can refer to [2, 3] and the reference therein.

Advanced differential equations can be used to simulate many processes in science in which evolution rate depends not only on the present, but also on the future. The appearance of advanced variables can reflect the influence of future factors, so it is very friendly to the decision-making processes. For instance, economic problems, mechanical control engineering and demographic dynamics are possible areas affected by future factors, see [4].

Differential equations with delay and advanced arguments occur in many applied problems, especially in mathematical economics, see [5, 6].

The oscillation theory of differential equations with deviating arguments is studied for the first time in document [7]. A developed theory has been formed on the oscillation of delay equations, see [8,9,10]. A few results have considered equations with advanced arguments, see [11,12,13]. In addition, for mixed differential equations with delay and advanced arguments (MDE), the oscillation theory is much less developed, see recent literature [14,15,16].

In 2016, Agarwal et al. [17] investigated the second-order neutral differential equation

$$\begin{aligned} \begin{aligned} (r(t)(x(t)+p(t)x(\tau (t)))^\alpha )'+q(t)x^\alpha (\sigma (t))=0,\ \ t\geqslant t_0, \end{aligned} \end{aligned}$$

where \(\tau (t)\leqslant t\), \(\sigma (t)\leqslant t\), \(\int _{t_0}^\infty \frac{1}{r^{1/\alpha }(s)}\mathrm{d}s<\infty \).

In 2016, Shi et al. [18] considered the second-order Emden–Fowler neutral delay dynamic equation

$$\begin{aligned} \begin{aligned} (r(t)|y^{\Delta }(t)|^{\alpha -1}y^{\Delta }(t))^\Delta +q(t)|x(\delta (t))|^{\beta -1}x(\delta (t))=0, \quad \ t\geqslant t_0, \end{aligned} \end{aligned}$$
(1.1)

where \(y(t)=x(t)+p(t)x(\tau (t))\), \(\int _{t_0}^\infty \frac{1}{r^{1/\alpha }(s)}\Delta s=\infty \), \(\tau (t)\leqslant t\), \(\delta (t)\leqslant t\). When \({\mathbb {T}}={\mathbb {R}}\), (1.1) becomes

$$\begin{aligned} \begin{aligned} (r(t)|y'(t)|^{\alpha -1}y'(t))'+q(t)|x(\delta (t))|^{\beta -1}x(\delta (t))=0, \quad \ t\geqslant t_0. \end{aligned} \end{aligned}$$

In 2017, Wu et al. [19] studied the second-order superlinear dynamic equation

$$\begin{aligned} \begin{aligned} (r(t)y^\Delta (t))^\Delta +q(t)y^\alpha (\sigma (t))=0,\ \quad t\geqslant t_0, \end{aligned} \end{aligned}$$
(1.2)

where \(\int _{t_0}^\infty \frac{1}{r(s)}\Delta s<\infty \), \(\sigma (t)\) is forward jump operator. They obtained oscillation theorems on an arbitrary time scale by using generalized Riccati transformations. When \({\mathbb {T}}={\mathbb {R}}\), (1.2) becomes

$$\begin{aligned} \begin{aligned} (r(t)y'(t))'+q(t)y^\alpha (t)=0, \quad t\geqslant t_0. \end{aligned} \end{aligned}$$
(1.3)

Equation (1.3) is systematically studied in monograph [20].

In 2019, Chatzarakis et al. [21] obtained sufficient conditions for the oscillation and asymptotic behavior of all solutions of the second-order half-linear differential equations with advanced argument of the form

$$\begin{aligned} \begin{aligned} (r(t)(y'(t))^\alpha )'+q(t)y^\alpha (\sigma (t))=0, \quad t\geqslant t_0, \end{aligned} \end{aligned}$$

where \(\sigma (t)>t\), \(\int _{t_0}^\infty \frac{1}{r^{1/\alpha }(s)}ds<\infty \).

Furthermore, regarding the oscillation of the second-order Emden–Fowler neutral differential equations with canonical operators, readers can refer to literature [22,23,24,25], and for noncanonical operators, readers can see [24, 26,27,28,29,30].

Inspired by the above articles, we study oscillation of a class of the second-order Emden–Fowler neutral differential equation with advanced and delay arguments

$$\begin{aligned} \begin{aligned} (r(t)(x(t)+p(t)x(\tau (t)))')'+q(t)x^\gamma (\sigma (t))=0, \ \quad t\geqslant t_0, \end{aligned} \end{aligned}$$
(1.4)

where \(\gamma >1\) is a quotient of odd positive integers, \(r(t),\ \tau (t),\ \sigma (t)\in C^1([t_0,\infty ), (0,\infty ))\), \(\ q(t)\in C([t_0,\infty ), [0,\infty ))\) and \(p(t)\in C^1([t_0,\infty ),(0,1))\), \(\inf \limits _{t\geqslant t_0}p(t)\ne 0\). We also suppose that for all \(t\geqslant t_0\), \(\sigma (t)\geqslant t\), \(\sigma '(t)\geqslant 0\), \(\tau (t)\leqslant t\), \(\lim \limits _{t\rightarrow \infty }\tau (t)=\infty \) and p(t), q(t) do not vanish identically on any half-line of form \([t_0,\infty )\). We set \( y(t)=x(t)+p(t)x(\tau (t)). \)

By a solution of (1.4), we mean a real-valued function \(x(t)\in C([T_x,\infty ),{\mathbb {R}}),\ \ T_x\geqslant t_0\) which has the property \(r(t)y'(t)\in C^1([T_x,\infty ),{\mathbb {R}})\) and satisfies (1.4) on \([T_x,\infty )\). We consider only those solutions of (1.4) which exist on some half-line \([T_x,\infty )\) and satisfy the condition \(\sup \{|x(t)|:T_y\leqslant t<\infty \}>0\), for any \(T_y\geqslant T_x\).

Following Trench [31], we shall say that (1.4) is in canonical form if

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty \frac{1}{r(s)}=\infty . \end{aligned} \end{aligned}$$

Conversely, we say that (1.4) is in noncanonical form if

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty \frac{1}{r(s)}\mathrm{d}s<\infty . \end{aligned} \end{aligned}$$
(1.5)

As is customary, a solution x(t) of (1.4) 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.

A neutral delay differential equation is a differential equation in which the highest-order derivative of the unknown function appears both with and without delay. During the last three decades, oscillation of neutral differential equations has become an important area of research, see [32, 33].

One of the traditional tools in the study of oscillation of neutral differential equations is to simplify neutral differential equation into differential inequality without delay in the highest-order derivative term. Under canonical condition, a very important inequality is

$$\begin{aligned} \begin{aligned} x(t)\geqslant (1-p(t))y(t), \end{aligned} \end{aligned}$$
(1.6)

where x(t) is the positive solution of neutral differential equation, \(y(t)=x(t)+p(t)x(\tau (t))\). But (1.6) does not hold under noncanonical conditions, and there is an obvious difference between noncanonical and canonical conditions.

A widely used technique which has been used to reduce neutral differential equation with noncanonical condition to a differential inequality without delay in the highest-order derivative term is to introduce a new function m(t) satisfying some conditions so that \(\frac{y(t)}{m(t)}\) is nondecreasing and satisfies \(x(t)\geqslant (1-p(t)\frac{m(\tau (t))}{m(t)})y(t)\), see [33].

By using the inequality principle, comparison principle and Riccati transform, we establish sufficient conditions for oscillation of (1.4). Our results are easy to verify, as we can see from examples.

2 Main Results

We are now in a position to state and prove our main results.

Through the rest of the paper, we will be using the following notation:

$$\begin{aligned} \pi (t)=\int _{t}^\infty {\frac{1}{r(s)}}\mathrm{d}s. \end{aligned}$$

Lemma 2.1

Assume that (1.5) holds and

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty q(s)\mathrm{d}s=\infty . \end{aligned} \end{aligned}$$
(2.1)

Furthermore, suppose that (1.4) has a positive solution x(t) on \([t_1,\infty )\), where \(t_1\in [t_0,\infty )\) is sufficiently large. Then,

$$\begin{aligned} \begin{aligned} y(t)>0,\ \ y'(t)<0,\ \quad (r(t)y'(t))'<0, \quad t\in [t_1,\infty ). \end{aligned} \end{aligned}$$
(2.2)

Moreover,

$$\begin{aligned} \begin{aligned} \left( \frac{y(t)}{\pi (t)}\right) '\geqslant 0,\ \quad t\in [t_1,\infty ). \end{aligned} \end{aligned}$$
(2.3)

Proof

Assume that x(t) is a nonoscillatory solution of (1.4) on \([t_0,\infty )\). Without loss of generality, we may assume that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\geqslant t_0\). From (1.4), we have

$$\begin{aligned} \begin{aligned} (r(t)y'(t))'=-q(t)x^\gamma (\sigma (t))\leqslant 0,\ \quad \ t\geqslant t_1. \end{aligned} \end{aligned}$$

Therefore, \(y'(t)\) is either eventually negative or eventually positive. Assume on the contrary that there exists \(t_2\geqslant t_1\) such that \(y'(t)>0\) on \([t_2,\infty )\). Then,

$$\begin{aligned} \begin{aligned} x(t)=y(t)-p(t)x(\tau (t))\geqslant y(t)-p(t)y(\tau (t))\geqslant (1-p(t))y(t), \quad t\geqslant t_2. \end{aligned} \end{aligned}$$
(2.4)

Define

$$\begin{aligned} \begin{aligned} w(t):=\frac{r(t)y'(t)}{y^\gamma (\sigma (t))}>0. \end{aligned} \end{aligned}$$

Then, by (2.4) we see that w(t) satisfies the inequality

$$\begin{aligned} \begin{aligned} w'(t)&=\frac{(r(t)y'(t))'}{y^\gamma (\sigma (t))}-\gamma \frac{r(t)y'(t)y'(\sigma (t))\sigma '(t)}{y^{\gamma +1} (\sigma (t))}\\&=\frac{-q(t)x^\gamma (\sigma (t))}{y^\gamma (\sigma (t))}-\gamma w(t)\frac{y'(\sigma (t))\sigma '(t)}{y(\sigma (t))}\\&\leqslant -q(t)(1-p(\sigma (t)))^\gamma , \quad t\geqslant t_2. \end{aligned} \end{aligned}$$
(2.5)

Integrating (2.5) from \(t_2\) to t, we have

$$\begin{aligned} \begin{aligned}&w(t)\leqslant w(t_2)-\int _{t_2}^tq(s)(1-p(\sigma (s)))^\gamma ds\leqslant w(t_2)\\ {}&-\inf _{t\geqslant t_2}(1-p(\sigma (t)))^\gamma \int _{t_2}^tq(s)\mathrm{d}s,\ \ t\geqslant t_2. \end{aligned} \end{aligned}$$
(2.6)

In view of (2.1), it is obvious that (2.6) comes to contradiction with the positivity of w(t). Hence, the case \(y'(t)>0\) is impossible. Thus, y(t) satisfies (2.2) for \(t\geqslant t_1\). On the other hand, it follows from the monotonicity of \(r(t)y'(t)\) that for \(l\geqslant t\),

$$\begin{aligned} y(t)\geqslant -\int _t^lr^{-1}(s)r(s)y'(s)\mathrm{d}s\geqslant -r(t)y'(t)\int _t^lr^{-1}(s)\mathrm{d}s,\ \ t\geqslant t_1. \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} y(t)\geqslant -r(t)y'(t)\pi (t), \quad \ t\geqslant t_1. \end{aligned} \end{aligned}$$
(2.7)

From (2.7), we conclude that \(\frac{y(t)}{\pi (t)}\) is nondecreasing, since

$$\begin{aligned} \left( \frac{y(t)}{\pi (t)}\right) '=\frac{y'(t)\pi (t)-y(t)\pi '(t)}{\pi ^2(t)}=\frac{r(t)y'(t)\pi (t)+y(t)}{r(t) \pi ^2(t)} \geqslant 0, \quad t\geqslant t_1. \end{aligned}$$

The proof is complete. \(\square \)

Theorem 2.1

Assume that (1.5) holds. If

$$\begin{aligned} \begin{aligned} 0<1-p(t)\frac{\pi (\tau (t))}{\pi (t)}<1, \quad \ \inf _{t\geqslant t_0}(1-p(t)\frac{\pi (\tau (t))}{\pi (t)})>0 \end{aligned} \end{aligned}$$
(2.8)

and

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty \frac{1}{r(u)}\int _{t_0}^u\pi ^\gamma (\sigma (s))q(s)\mathrm{d}s\mathrm{d}u=\infty , \end{aligned} \end{aligned}$$
(2.9)

then (1.4) is oscillatory.

Proof

Suppose the contrary and assume that x(t) is a nonoscillatory solution of (1.4) on \([t_0,\infty )\). Without loss of generality, we may assume that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\geqslant t_0\). Note that (2.1) is necessary for (2.9) to be valid. In fact, since the function

$$\begin{aligned} \int _{t_0}^\infty \pi ^\gamma (\sigma (s))q(s)\mathrm{d}s \end{aligned}$$

is unbounded due to (1.5) and \(\pi '(t)<0\), (2.1) must hold. Then, by Lemma 2.1, y(t) satisfies (2.2) for \(t\geqslant t_1\). It follows from (2.3) that there is \(c>0\) such that

$$\begin{aligned} \begin{aligned} y(t)\geqslant c\pi (t), \quad t \geqslant t_1, \end{aligned} \end{aligned}$$
(2.10)

and

$$\begin{aligned} \begin{aligned} x(t)&\geqslant y(t)-p(t)x(\tau (t))\geqslant y(t)-p(t)y(\tau (t)) \geqslant y(t)-p(t)\frac{\pi (\tau (t))}{\pi (t)}y(t)\\&=(1-p(t)\frac{\pi (\tau (t))}{\pi (t)})y(t), \quad t\geqslant t_1. \end{aligned} \end{aligned}$$
(2.11)

Substituting (2.10) and (2.11) in (1.4), we see that

$$\begin{aligned} \begin{aligned} -(r(t)y'(t))'&=q(t)x^\gamma (\sigma (t))\geqslant q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma y(\sigma (t))^\gamma \\&\geqslant q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma c^\gamma \pi ^\gamma (\sigma (t)),\ \ t\geqslant t_1. \end{aligned} \end{aligned}$$
(2.12)

Integrating (2.12) from \(t_1\) to t, we have

$$\begin{aligned} -r(t)y'(t)\geqslant c^\gamma \int _{t_1}^tq(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma \pi ^\gamma (\sigma (s))\mathrm{d}s. \end{aligned}$$

That is,

$$\begin{aligned} -y'(t)\geqslant \frac{c^\gamma }{r(t)}\int _{t_1}^tq(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma \pi ^\gamma (\sigma (s))\mathrm{d}s. \end{aligned}$$

Integrating the latter inequality again from \(t_1\) to t and taking (2.8) and (2.9) into account, we get

$$\begin{aligned} \begin{aligned} y(t)&\leqslant y(t_1)-\int _{t_1}^t \frac{c^\gamma }{r(u)}\int _{t_1}^uq(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma \pi ^\gamma (\sigma (s))\mathrm{d}s\mathrm{d}u\\&\leqslant y(t_1)-c^\gamma \inf \limits _{t\geqslant t_1}(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma \\ {}&\int _{t_1}^t \frac{1}{r(u)}\int _{t_1}^uq(s)\pi ^\gamma (\sigma (s))\mathrm{d}s\mathrm{d}u\rightarrow -\infty , \quad t\rightarrow \infty , \end{aligned} \end{aligned}$$

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

Theorem 2.2

Assume that (1.5) and (2.1) hold. If

$$\begin{aligned} \begin{aligned} 0<1-p(t)\frac{\pi (\tau (t))}{\pi (t)}<1, \end{aligned} \end{aligned}$$
(2.13)

and

$$\begin{aligned} \begin{aligned} z'(t)\geqslant q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma M^\gamma (\sigma (t))z^{\gamma ^2}(\sigma (t)) \end{aligned} \end{aligned}$$
(2.14)

is oscillatory, where

$$\begin{aligned} M(t):=\int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma \pi (s)\pi ^\gamma (\sigma (s))\mathrm{d}s, \end{aligned}$$

then (1.4) is oscillatory.

Proof

Suppose the contrary and assume that x(t) is a nonoscillatory solution of (1.4) on \(t\geqslant t_1\geqslant t_0\). Without loss of generality, we may assume that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\). Because of (2.1), we can conclude that y(t) satisfies (2.2) for \(t\geqslant t_1\).

Considering the identity

$$\begin{aligned} \begin{aligned} (y(t)+r(t)y'(t)\pi (t))'=y'(t)+(r(t)y'(t))'\pi (t)+r(t)y'(t)\pi '(t)=(r(t)y'(t))'\pi (t), \end{aligned} \end{aligned}$$
(2.15)

by virtue of (1.4) and (2.11), (2.15) becomes

$$\begin{aligned} \begin{aligned} (y(t)+r(t)y'(t)\pi (t))'\leqslant -q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma y^\gamma (\sigma (t))\pi (t)\leqslant 0. \end{aligned} \end{aligned}$$
(2.16)

Thus, we see that \(\phi (t):=y(t)+r(t)y'(t)\pi (t)\geqslant 0\) is nonincreasing. Integrating (2.16) from t to \(\infty \) and using (2.7), we get

$$\begin{aligned} \begin{aligned} \phi (t)&=\phi (\infty )+\int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma y^\gamma (\sigma (s))\pi (s)\mathrm{d}s\\&\geqslant \int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma y^\gamma (\sigma (s))\pi (s)\mathrm{d}s\\&\geqslant \int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma (-r(\sigma (s))y'(\sigma (s)))^\gamma \pi (s)\pi ^\gamma (\sigma (s))\mathrm{d}s\\&\geqslant \int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma (-r(s)y'(s))^\gamma \pi (s)\pi ^\gamma (\sigma (s))\mathrm{d}s\\&\geqslant (-r(t)y'(t))^\gamma \int _{t}^\infty q(s)(1-p(\sigma (s))\frac{\pi (\tau (\sigma (s)))}{\pi (\sigma (s))})^\gamma \pi (s)\pi ^\gamma (\sigma (s))\mathrm{d}s, \end{aligned} \end{aligned}$$

since \(r(t)y'(t)\pi (t)<0\), we have

$$\begin{aligned} \begin{aligned} y(t)\geqslant M(t)(-r(t)y'(t))^\gamma . \end{aligned} \end{aligned}$$
(2.17)

From (2.12), we have

$$\begin{aligned} \begin{aligned} -(r(t)y'(t))'\geqslant q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma y^\gamma (\sigma (t)). \end{aligned} \end{aligned}$$
(2.18)

Substituting (2.17) in (2.18), we see that \(z(t)=-r(t)y'(t)\) is a positive solution of the first-order advanced differential inequality

$$\begin{aligned} \begin{aligned} z'(t)\geqslant q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma M^\gamma (\sigma (t))z^{\gamma ^2}(\sigma (t)), \end{aligned} \end{aligned}$$
(2.19)

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

Theorem 2.3

Assume that (1.5), (2.1) and (2.13) hold. Assume that there exists a function \(\varphi (t)>0\) satisfying \(\varphi '(t)\leqslant 0\) and \((r(t)\varphi '(t))'\geqslant 0\) such that

$$\begin{aligned} \begin{aligned} \int _{t_0}^\infty \varphi ^{\gamma }(s)Q(s)\mathrm{d}s=\infty \ \ \text {and}\ \ \int _{t_0}^\infty \frac{1}{r(s)\varphi (s)}\mathrm{d}s=\infty , \end{aligned} \end{aligned}$$
(2.20)

where \(Q(t):=-q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma \frac{ \pi (\sigma (t))^\gamma }{\pi ^\gamma (t)}\). Then, (1.4) is oscillatory.

Proof

Suppose the contrary and assume that x(t) is a nonoscillatory solution of (1.4). Without loss of generality, we may assume that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\sigma (t))>0\) for \(t\geqslant t_1\). Because of (2.1), we can conclude that y(t) satisfies (2.2) for \(t\geqslant t_1\). Let us make the substitution \(y(t)=\varphi (t)u(t)\) and define the following generalized Riccati transformation \(w(t):=\frac{r(t)y'(t)}{y^\gamma (t)}\) for \(t\geqslant t_1\). By the derivative product rule and quotient rule, we get

$$\begin{aligned} \begin{aligned} w'(t)=\frac{(r(t)y'(t))'}{y^\gamma (t)}-r(t)y'(t)\frac{(y^\gamma (t))'}{y^{2\gamma }(t)}. \end{aligned} \end{aligned}$$
(2.21)

From (2.18) and (2.21), we have

$$\begin{aligned} \begin{aligned} w'(t)\leqslant \frac{-q(t)(1-p(\sigma (t))\frac{\pi (\tau (\sigma (t)))}{\pi (\sigma (t))})^\gamma y(\sigma (t))^\gamma }{y^\gamma (t)}-r(t)y'(t)\frac{(y^\gamma (t))'}{y^{2\gamma }(t)}. \end{aligned} \end{aligned}$$

Because of (2.3), we conclude

$$\begin{aligned} w'(t)\leqslant Q(t)-r(t)y'(t)\frac{(y^\gamma (t))'}{y^{2\gamma }(t)}. \end{aligned}$$
(2.22)

By the substitution \(y(t)=\varphi (t)u(t)\) and (2.22), we get

$$\begin{aligned} w'(t)\leqslant & {} Q(t)-r(t)(\varphi (t)u(t))'\frac{(\varphi ^\gamma (t)u^\gamma (t))'}{\varphi ^{2\gamma }(t)u^{2\gamma }(t)}\nonumber \\= & {} Q(t)-r(t)\frac{(\varphi '(t)u(t)+\varphi (t)u'(t))[(\varphi ^\gamma (t))'u^\gamma (t)+\varphi ^\gamma (t)(u^\gamma (t)) ']}{\varphi ^{2\gamma }(t)u^{2\gamma }(t)}\nonumber \\= & {} Q(t)-r(t)\frac{(\varphi '(\varphi ^\gamma )'u u^\gamma )(t)+(\varphi (\varphi ^\gamma )'u'u^\gamma )(t)+ (\varphi '\varphi ^\gamma u(u^\gamma )')(t)+(\varphi \varphi ^\gamma u'(u^\gamma )')(t)}{\varphi ^{2\gamma }(t)u^{2\gamma }(t)}.\nonumber \\ \end{aligned}$$
(2.23)

Multiplying both sides of (2.23) by \(\varphi ^{\gamma }(t)\), we have

$$\begin{aligned} \begin{aligned} \varphi ^{\gamma }(t)w'(t)&\leqslant \varphi ^{\gamma }(t)Q(t)\\&-r(t)\frac{(\varphi '(\varphi ^\gamma )'u u^\gamma )(t) +(\varphi (\varphi ^\gamma )'u'u^\gamma )(t)+ (\varphi '\varphi ^\gamma u(u^\gamma )')(t)+(\varphi \varphi ^\gamma u'(u^\gamma )')(t)}{\varphi ^{\gamma }(t)u^{2\gamma }(t)}. \end{aligned} \end{aligned}$$
(2.24)

Integrating (2.24) from \(t_1\) to t, and using an integration by parts formula, we get

$$\begin{aligned} \begin{aligned} \varphi ^{\gamma }(t)w(t)&\leqslant \varphi ^{\gamma }(t_1)w(t_1)+\int _{t_1}^t(\varphi ^{\gamma })'(s)w(s)\mathrm{d}s -\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\frac{(\varphi '(\varphi ^\gamma )'u u^\gamma )(s) +(\varphi (\varphi ^\gamma )'u'u^\gamma )(s)+ (\varphi '\varphi ^\gamma u(u^\gamma )')(s)+(\varphi \varphi ^\gamma u'(u^\gamma )')(s)}{\varphi ^{\gamma }(t)u^{2\gamma }(s)}\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.25)

Because of \(y(t)=\varphi (t)u(t)\), we conclude

$$\begin{aligned} \begin{aligned} w(t)=\frac{r(t)(\varphi (t)u(t))'}{(\varphi (t)u(t))^\gamma }=\frac{r(t)\varphi '(t)u(t)}{(\varphi (t)u(t))^\gamma } +\frac{r(t) \varphi (t)u'(t)}{(\varphi (t)u(t))^\gamma }. \end{aligned} \end{aligned}$$
(2.26)

From (2.26) and (2.25), we obtain

$$\begin{aligned} \begin{aligned} \varphi ^{\gamma }(t)w(t)&=\varphi ^{\gamma }(t)\left( \frac{r(t)\varphi '(t)u(t)}{(\varphi (t)u(t))^\gamma }+ \frac{r(t)\varphi (t)u'(t)}{(\varphi (t)u(t))^\gamma }\right) =\frac{r(t)\varphi '(t)u(t)}{u^\gamma (t)}+ \frac{r(t)\varphi (t)u'(t)}{u^\gamma (t)}\\&\leqslant \varphi ^{\gamma }(t_1)w(t_1)+\int _{t_1}^t(\varphi ^{\gamma })'(s)\left( \frac{r(s)\varphi '(s)u(s)}{(\varphi (s)u(s))^\gamma } +\frac{r(s)\varphi (s)u'(s)}{(\varphi (s)u(s))^\gamma }\right) \mathrm{d}s\\&\quad -\,\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\frac{(\varphi '(\varphi ^\gamma )'u u^\gamma )(s)}{\varphi ^{\gamma }(t)u^{2\gamma }(s)}\mathrm{d}s -\,\int _{t_1}^tr(s)\frac{(\varphi (\varphi ^\gamma )'u'u^\gamma )(s)}{\varphi ^{\gamma }(t)u^{2\gamma }(s)}\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\frac{(\varphi '\varphi ^\gamma u(u^\gamma )')(s)}{\varphi ^{\gamma }(t)u^{2\gamma }(s)}\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\frac{(\varphi \varphi ^\gamma u'(u^\gamma )')(s)}{\varphi ^{\gamma }(t)u^{2\gamma }(s)}\mathrm{d}s\\&=\varphi ^{\gamma }(t_1)w(t_1) -\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s -\int _{t_1}^tr(s)\varphi '(s)\frac{u(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\varphi (s)\frac{ u'(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.27)

On the other hand, we have

$$\begin{aligned} \begin{aligned}&-\int _{t_1}^tr(s)\varphi '(s)\frac{ u(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s\\&\quad =\int _{t_1}^tr(s)\varphi '(s)\left( \left( \frac{u(s)}{u^\gamma (s)}\right) '-\frac{u'(s)}{u^\gamma (s)}\right) \mathrm{d}s\\&\quad =r(t)\varphi '(t)\frac{u(t)}{u^\gamma (t)}-r(t_1)\varphi '(t_1)\frac{u(t_1)}{u^\gamma (t_1)}-\int _{t_1}^t(r(s) \varphi '(s))'\frac{u(s)}{u^\gamma (s)}\mathrm{d}s-\int _{t_1}^tr(s)\varphi '(s)\frac{u'(s)}{u^\gamma (s)}\mathrm{d}s\\&\quad \leqslant r(t)\varphi '(t)\frac{u(t)}{u^\gamma (t)}-r(t_1)\varphi '(t_1)\frac{u(t_1)}{u^\gamma (t_1)} -\int _{t_1}^tr(s)\varphi '(s)\frac{u'(s)}{u^\gamma (s)}\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.28)

Thus, from (2.27), (2.28) and mean value theorem of integrals, we get for \(t\in [t_1,\infty )\),

$$\begin{aligned} \begin{aligned} \frac{r(t)\varphi (t)u'(t)}{u^\gamma (t)}&\leqslant r(t_1)\varphi (t_1)\frac{u'(t_1)}{u^\gamma (t_1)} -\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s-\int _{t_1}^tr(s)\varphi '(s)\frac{u'(s)}{u^\gamma (s)}\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\varphi (s)\frac{ u'(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s\\&=r(t_1)\varphi (t_1)\frac{u'(t_1)}{u^\gamma (t_1)} -\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s-r(t_1)\varphi '(t_1)\int _{t_1}^\xi \frac{u'(s)}{u^\gamma (s)}\mathrm{d}s\\&\quad -\,\int _{t_1}^tr(s)\varphi (s)\frac{ u'(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.29)

From transformation of integral, we have

$$\begin{aligned} \begin{aligned} \int _{t_1}^\xi \frac{u'(s)}{u^\gamma (s)}\mathrm{d}s=\int _{u(t_1)}^{u(\xi )}\frac{1}{v^\gamma }\mathrm{d}v =\frac{1}{1-\gamma }u^{1-\gamma }(\xi )-\frac{1}{1-\gamma }u^{1-\gamma }(t_1)\leqslant \frac{1}{\gamma -1}u^{1-\gamma }(t_1). \end{aligned} \end{aligned}$$
(2.30)

From (2.29) and (2.30), we obtain

$$\begin{aligned} \begin{aligned} \frac{r(t)\varphi (t)u'(t)}{u^\gamma (t)}&\leqslant r(t_1)\varphi (t_1)\frac{u'(t_1)}{u^\gamma (t_1)} -\int _{t_1}^t\varphi ^{\gamma }(s)Q(s)\mathrm{d}s-r(t_1)\varphi '(t_1)\frac{1}{\gamma -1}u^{1-\gamma }(t_1)\\&\quad -\,\int _{t_1}^tr(s)\varphi (s)\frac{ u'(s)(u^\gamma )'(s)}{u^{2\gamma }(s)}\mathrm{d}s. \end{aligned} \end{aligned}$$
(2.31)

Since \(\int _{t_1}^\infty \varphi ^{\gamma }(s)Q(s)\mathrm{d}s=\infty \), from (2.31), there exists a sufficiently large \(t_2\geqslant t_1\), such that

$$\begin{aligned} \begin{aligned} \frac{r(t)\varphi (t)u'(t)}{u^\gamma (t)}\leqslant -1-\int _{t_1}^t\frac{r(s)\varphi (s)u'(s)}{u^{\gamma }(s)}\frac{ (u^\gamma )'(s)}{u^{\gamma }(s)}\mathrm{d}s,\ \ t\geqslant t_2. \end{aligned} \end{aligned}$$
(2.32)

It follows that

$$\begin{aligned} \begin{aligned} u'(t)(u^\gamma )'(t)=\gamma u^{\gamma -1}(t)(u'(t))^2\geqslant 0. \end{aligned} \end{aligned}$$
(2.33)

Let \(\Omega (t)=\frac{r(t)\varphi (t)u'(t)}{u^\gamma (t)}\) and \(f(t)=u^\gamma (t)\). From (2.32), (2.33) and Gronwall’s inequality, we get

$$\begin{aligned} \begin{aligned} r(t)\varphi (t)u'(t)\leqslant -u^\gamma (t_2). \end{aligned} \end{aligned}$$
(2.34)

Multiplying both sides of (2.34) by \(\frac{1}{r(t)\varphi (t)}\), we obtain \(u'(t)\leqslant -\frac{u^\gamma (t_2)}{r(t)\varphi (t)}\). Integrating this equation from \(t_2\) to t, we obtain

$$\begin{aligned} u(t)\leqslant u(t_2)-u^\gamma (t_2)\int _{t_2}^t\frac{1}{r(s)\varphi (s)}\mathrm{d}s,\ \ t\geqslant t_2. \end{aligned}$$

That is a contradiction with \(u(t)>0\). This completes the proof. \(\square \)

3 Examples

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

Example 3.1

Consider the following advanced–delay differential equation

$$\begin{aligned} \begin{aligned} (t^\alpha (x(t)+p(t)x(t-1))')'+q(t) x^\gamma (t+1)=0, \quad t\geqslant 0, \end{aligned} \end{aligned}$$
(3.1)

where \(\gamma >1\), is a quotient of odd positive integers and \(\alpha >1\). Set \(r(t)=t^\alpha \), \(\tau (t)=t-1\), \(\sigma (t)=t+1\), \(q(t)=(\alpha -1)^\gamma \alpha (t+1)^{(\alpha -1)\gamma }t^{\alpha -1}\), \(p(t)=\frac{t^{1-\alpha }}{2(t-1)^{1-\alpha }}\), \(t_0=0\). Then, we have

$$\begin{aligned}&\pi (t)=\int _{t}^\infty r^{-1}(s)ds=\int _{t}^\infty s^{-\alpha }\mathrm{d}s=\frac{1}{\alpha -1}t^{-\alpha +1}<\infty , \\&1-\,p(t)\frac{\pi (\tau (t))}{\pi (t)}=1-\frac{t^{1-\alpha }}{2(t-1)^{1-\alpha }}\frac{(t-1)^{-\alpha +1}}{t^{-\alpha +1}}= \frac{1}{2}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\ \ \ \ \int _{t_0}^\infty \frac{1}{r(u)}\int _{t_0}^u\pi ^\gamma (\sigma (s))q(s)\mathrm{d}s\mathrm{d}u\\&\quad =\int _{0}^\infty \frac{1}{u^\alpha }\int _{0}^u\left( \frac{1}{\alpha -1}(s+1)^{-\alpha +1}\right) ^\gamma (\alpha -1)^\gamma \alpha (s+1)^{(\alpha -1)\gamma }s^{\alpha -1}\mathrm{d}s\mathrm{d}u\\&\quad =\int _{0}^\infty \frac{1}{u^\alpha }\int _{0}^u\alpha s^{\alpha -1}\mathrm{d}s\mathrm{d}u =\int _{0}^\infty 1\mathrm{d}u =\infty . \end{aligned} \end{aligned}$$

Therefore, (3.1) satisfies all the conditions of Theorem 2.1. From Theorem 2.1, we can conclude that (3.1) is oscillatory.

Example 3.2

Consider the following advanced–delay differential equation

$$\begin{aligned} \begin{aligned} (t^\alpha (x(t)+p(t)x(t-1))')'+t^\beta x^\gamma (t+1)=0,\quad \ t\geqslant 0, \end{aligned} \end{aligned}$$
(3.2)

where \(\gamma >1\), is a quotient of odd positive integers, \(\beta \geqslant 0\) and \(1<\alpha <\frac{\gamma +\beta }{\gamma }\). Set \(r(t)=t^\alpha \), \(\tau (t)=t-1\), \(\sigma (t)=t+1\), \(q(t)=t^\beta \), \(p(t)=\frac{t^{1-\alpha }}{2(t-1)^{1-\alpha }}\), \(t_0=0\). Then, we have

$$\begin{aligned}&\lim _{t\rightarrow \infty }\int _{t_0}^t r^{-1}(s)\mathrm{d}s=\lim _{t\rightarrow \infty }\int _{t_0}^t s^{-\alpha }\mathrm{d}s=\frac{1}{-\alpha +1}t^{-\alpha +1}<\infty , \\&\lim _{t\rightarrow \infty }\int _{t_0}^t q(s)\mathrm{d}s=\lim _{t\rightarrow \infty }\int _{t_0}^t s^\beta \mathrm{d}s=\frac{1}{\beta +1}t^{\beta +1}=\infty , \end{aligned}$$

and

$$\begin{aligned}&1-\,p(t)\frac{\pi (\tau (t))}{\pi (t)}=1-\frac{t^{1-\alpha }}{2(t-1)^{1-\alpha }}\frac{(t-1)^{-\alpha +1}}{t^{-\alpha +1}}= \frac{1}{2}. \end{aligned}$$

Let \(\varphi (t)=t^{-\frac{\beta }{\gamma }}\). Obviously, \(\varphi '(t)\leqslant 0\) and

$$\begin{aligned} (r(t)\varphi '(t))'=-\frac{\beta }{\gamma }(\alpha -\frac{\gamma +\beta }{\gamma })t^{(\alpha -\frac{2\gamma +\beta }{\gamma })}\geqslant 0. \end{aligned}$$

Since

$$\begin{aligned} \lim _{t\rightarrow \infty }\int _{t_0}^t\varphi ^\gamma (s)Q(s)\mathrm{d}s=\int _{t_0}^t s^{-\beta }s^\beta \frac{1}{2}\frac{(t+1)^{1-\alpha }}{t^{1-\alpha }}\mathrm{d}s=\frac{1}{2}\int _{t_0}^t \frac{(t+1)^{1-\alpha }}{t^{1-\alpha }}\mathrm{d}s=\infty , \end{aligned}$$

and

$$\begin{aligned} \lim _{t\rightarrow \infty }\int _{t_0}^t\frac{1}{r(s)\varphi (s)} =\int _{t_0}^t s^{\frac{\beta }{\gamma }-\alpha }\mathrm{d}s=(\frac{\beta +\gamma }{\gamma }-\alpha )t^{(\frac{\beta +\gamma }{\gamma }-\alpha )} =\infty . \end{aligned}$$

Therefore, (3.2) satisfies all the conditions of Theorem 2.3. From Theorem 2.3, we can conclude that (3.2) is oscillatory.

4 Conclusion

Neutral delay differential equations with canonical operator can easily be transformed into differential inequality without delay in the highest-order derivative term from (1.6). So we only study the noncanonical form in this paper. In this paper, we study the case when the exponent of \((r(t)y'(t))\) is 1. Furthermore, we may study the case when the exponent of \((r(t)y'(t))\) is the same as the exponent \(\gamma \) of \(y(\sigma (t))\), even for \(\beta \ne \gamma \).