Introduction

In this work, we plan to study the oscillatory behaviour of all solutions to the nonlinear higher-order dynamic equations of the form:

$$\begin{aligned} \left( a(t)\left( x^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }=q(t)x^{\beta }(\tau (t))+p(t)x^{\gamma }(\omega (t)), \end{aligned}$$
(1)

where \(t \in [t_0,\infty )_{\mathbb {T}}=[t_0,\infty )\cap {\mathbb {T}}\) with \(t_0 \in {\mathbb {T}}\) and \(\sup {\mathbb {T}} =\infty\). A solution of (1) is a function x(t) continuous on \([T_x, \infty ), T_x\ge t_0\), which satisfies (1) on \([T_x,\infty )_{\mathbb {T}}\). Solutions vanishing identically in some neighborhood of infinity will be excluded from our consideration. Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative, and to be nonoscillatory otherwise.

We assume that

  1. (i)

    \(\alpha , \beta\) and \(\gamma\) are the ratios of odd positive integers and \(\alpha \ge 1\),

  2. (ii)

    a(t), p(t) and \(q(t) \in C_{rd}([t_0,\infty )_{{\mathbb {T}}},{\mathbb {R}}_+)\) and \(a^{\Delta }(t)\ge 0,\)

  3. (iii)

    \(\omega (t),\tau (t) \in C_{rd}([t_0,\infty )_{{\mathbb {T}}}, {\mathbb {R}})\), \(\tau (t)\le t\), \(\omega (t) \ge t\) and \(\tau (t)\rightarrow \infty\) as \(t \rightarrow \infty\).

We let

$$\begin{aligned} A(v,u)=\int _{u}^{v}\frac{1}{a^{\frac{1}{\alpha }}(s)} \Delta s \quad \text {and}\quad A(t,t_{0})=\int _{t_0}^{t}\frac{1}{a^{\frac{1}{\alpha }}(s)} \Delta s\rightarrow \infty \ as \ t \rightarrow \infty , \end{aligned}$$
(2)

The oscillation theory of dynamic equations has drawn a lot of interest recently since it has a wide range of applications in engineering and the natural sciences, see for e.g., the papers [2,3,4,5, 9, 10, 12,13,14,15,16,17, 19, 21, 22, 27,28,29] and the monographs [1, 11, 25, 26]. The motivation of this study comes from the following directions:

  1. 1.

    Grace and Chhatria [20] developed a novel method to derive the oscillation criterion for the aforementioned third-order dynamic equations by linearising the given equation

    $$\begin{aligned} \left( a(t)\left( x^{\Delta \Delta }(t)\right) ^{\alpha } \right) ^{\Delta }+ q(t) x^{\alpha } (\omega (t))=0, \end{aligned}$$

    and from that of its linear forms. They concluded by addressing problems on the investigation of higher orders dynamic equations.

  2. 2.

    Recently, Grace et al. [21] have examined a higher-order dynamic equation of the form

    $$\begin{aligned} \left( a(t)\left( y^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }+q(t)x^{\beta }(\tau (t))=0, \end{aligned}$$

    where \(y(t)=x(t)+p(t)x^{\gamma }(\delta (t))\) and n is even. With the help of comparison, integral averaging, Riccati transformation techniques, and some \(\limsup\) and \(\liminf\) type conditions, they obtained several results for the oscillation and asymptotic behaviour of the aforementioned equations.

  3. 3.

    One can observe that the findings for odd-order dynamic equations are incredibly few, see, e. g. [9, 16, 18, 22, 30, 31]

  4. 4.

    As far as we are aware, no paper has been published in the literature that addresses the oscillatory behaviour of solutions to the higher-order dynamic equation of the form (1), see, e. g. [7, 8, 23, 24].

In the light of the aforementioned information, the current research aims to initiate the investigation of the oscillation problem of (1) by using the methodology developed in [20]. The outcomes are novel, enhance, and correlate many of the well-known oscillation criteria that were mentioned in the literature for Eq. (1).

Auxiliary Lemmas

To obtain our result, we need the following auxiliary lemmas:

Lemma 2.1

Let conditions (i)–(iii) and (2) hold. Then Eq.

$$\begin{aligned} \left( a(t)\left( x^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }= q(t)x^{\beta }(\tau (t)), \end{aligned}$$
(3)

or,

$$\begin{aligned} \left( a(t)\left( x^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }= p(t)x^{\gamma }(\omega (t)), \end{aligned}$$
(4)

has an eventually positive solution satisfying

  1. (I)

    \(x^{\Delta ^{n-1}}(t)<0\), \(x^{\Delta }(t)>0\), \(x(t)>0\) eventually,

  2. (II)

    \(x^{\Delta ^{n-1}}(t)>0,\ldots , x^{\Delta }(t)>0\) and \(x(t)>0\) eventually,

  3. (III)

    \(x^{\Delta ^{n-1}}(t)<0,\ldots , x^{\Delta }(t)<0\) and \(x(t)>0\) eventually, if n is even.

Proof

The proof immediately follows from the proof of [6, Theorem 5]. So, we omit the details. \(\square\)

The following lemma is immediate.

Lemma 2.2

Let x(t) be an eventually increasing solution of Eq. (1). Then \(x^{\beta -\alpha }(t)\ge \delta _{1}(t,t_{1})\) and \(x^{\gamma -\alpha }(t)\ge \delta _{2}(t,t_{1})\), where \(\delta _{1}(t,t_{1})\) and \(\delta _{2}(t,t_{1})\) are given by

$$\begin{aligned}&\delta _{1}(t,t_{1})= {\left\{ \begin{array}{ll} 1 \quad \text {if}\; \alpha =\beta \\ b \quad \text {if}\; \alpha<\beta , \end{array}\right. }&\delta _{2}(t,t_{1})= {\left\{ \begin{array}{ll} 1\quad \text {if}\; \alpha =\gamma \\ c\quad \text {if}\; \alpha <\gamma , \end{array}\right. } \end{aligned}$$

where b and c are positive constants and all large \(t\ge t_{1}\).

Proof

Since x(t) is an increasing solution of Eq. (1) for \(t\ge t_{0}\), then there exists a constant \(b_{1}>0\) such that \(x(t)\ge b_{1}\) for \(t\ge t_{1}\). Now, if \(\alpha =\beta\), then \(x^{\beta -\alpha }(t)=1\) and when \(\alpha <\beta\), \(x^{\beta -\alpha }(t)\ge b_{1}^{\beta -\alpha }=b\). Therefore,

$$\begin{aligned} x(t)\ge \delta _{1}(t,t_{1})= {\left\{ \begin{array}{ll} 1 \quad \text {if}\; \alpha = \beta ,\\ b \quad \text {if}\; \alpha <\beta . \end{array}\right. } \end{aligned}$$

In a similar manner, we can show that \(x^{\gamma -\alpha }(t)\ge \delta _{2}(t,t_{1})\). \(\square\)

Remark 2.3

Upon using Lemma 2.2, Eq. (1) takes the form of half-linear dynamic equation

$$\begin{aligned} \left( a(t)\left( x^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }\ge q(t)\delta _{1}(t,t_{1}) x^{\alpha }(\tau (t))+p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t)). \end{aligned}$$
(5)

Main Results

For simplicity in what follows, for \(\eta (t),\varphi (t)\in C_{rd}\left( [t_{0},\infty )_{{\mathbb {T}}}, {\mathbb {R}}\right)\), we define:

$$\begin{aligned} \eta _{1}(t)&=\eta (t),\;\eta _{i+1}(t)=\eta _{i}(\eta (t)),\\ I_{1}(t)&=\eta (t)-t,\;I_{i+1}(t)=\int _{t}^{\eta (t)}I_{i}(s) \Delta s\;\text {for}\;i=1,2,\ldots ,n-1, \end{aligned}$$

and

$$\begin{aligned} \varphi _{1}(t)&=\varphi (t),\;\varphi _{i+1}(t)=\varphi _{i}(\varphi (t)),\\ J_{1}(t)&=t-\varphi (t),\;J_{i+1}(t)=\int _{\varphi (t)}^{t}J_{i}(s) \Delta s\;\text {for}\;i=1,2,\ldots ,n-1. \end{aligned}$$

Now, we have the following main result:

Theorem 3.1

Let n be odd, conditions (i)–(iii) and (2) hold and assume that nondecreasing functions \(\varphi (t)\) and \(\xi (t)\in C_{rd}([t_{0},\infty )_{{\mathbb {T}}}, {\mathbb {R}})\) are such that

$$\begin{aligned} \varphi (t)<t, \quad \rho (t)=\varphi _{n-1}(\omega (t))>t\quad \text {and}\quad \tau (t)\le \xi (t)\; \text {for}\;t\ge t_{0}. \end{aligned}$$
(6)

If the first order advanced equation

$$\begin{aligned} W^{\Delta }(t)-\frac{1}{\alpha } \left( \frac{1}{a(\rho (t))}\right) p(t)\delta _{2}(t,t_{1})\left( J_{n-1}(\omega (t))\right) ^{\alpha } W(\rho (t))=0 \end{aligned}$$
(7)

and the first order delay equation

$$\begin{aligned} Y^{\Delta }+\frac{1}{\alpha }\left( \frac{\theta }{(n-2)!}\right) ^{\alpha } q(t)\delta _{1}(t,t_{1})\tau ^{(n-2)\alpha }(t)a^{\frac{1-\alpha }{\alpha }}(\xi (t)) \left( \frac{\xi (t)-\tau (t)}{a^{\frac{1}{\alpha }}(\xi (t))}\right) ^{\alpha } Y(\xi (t))=0 \end{aligned}$$
(8)

are oscillatory for \(t\ge t_{1}>t_{0}\), then Eq. (1) is oscillatory.

Proof

Let x(t) be a nonoscillatory solution of (1) such that \(x(t)>0\), \(x(\tau (t))>0\), and \(x(\omega (t))>0\) for \(t \ge t_{1}> t_{0}\). It follows from Eq. (5) that

$$\begin{aligned} \left( a(t)\left( x^{\Delta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\Delta }&\ge q(t)\delta _{1} (t,t_{1}) x^{\alpha }(\tau (t))+p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t))\\&\ge p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\tau (t))\ge 0. \end{aligned}$$

Hence, \(a(t)(x^{\Delta ^{n-1}}(t))^{\alpha }\) is nondecreasing and of one sign. Clearly, x(t) satisfies one of the two cases (I) and (II) of Lemma 2.1. Indeed,

$$\begin{aligned} \left( a(t)(x^{\Delta ^{n-1}}(t))^{\alpha }\right) ^{\Delta }= \left( (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{\alpha }\right) ^{\Delta }, \end{aligned}$$

which implies that

$$\begin{aligned} \left( a(t)(x^{\Delta ^{n-1}}(t))^{\alpha }\right) ^{\Delta }&=\alpha (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{\alpha -1} (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{\Delta }\\&\ge q(t)\delta _{1}(t,t_{1}) x^{\alpha }(\tau (t))+p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t)). \end{aligned}$$

As a result,

$$\begin{aligned} \left( a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t)\right) ^{\Delta }\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{1-\alpha }\left( q(t)\delta _{1}(t,t_{1}) x^{\alpha }(\tau (t))+p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t))\right) . \end{aligned}$$

Case-I Consider

$$\begin{aligned} \left( a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t)\right) ^{\Delta }\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{1-\alpha } p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t)). \end{aligned}$$
(9)

In this case, we have

$$\begin{aligned} x^{\Delta ^{m}}(t)>0\;\text {for}\;m=0,1,2,\ldots ,n-1. \end{aligned}$$

Consequently,

$$\begin{aligned} x^{\Delta ^{n-2}}(t)&\ge x^{\Delta ^{n-2}}(t)-x^{\Delta ^{n-2}}(\varphi (t))=\int _{\varphi (t)}^{t}x^{\Delta ^{n-1}}(s) \Delta s \ge J_{1}(t)x^{\Delta ^{n-1}}(\varphi (t)). \end{aligned}$$

The repeated integrations of this inequality from \(\varphi (t)\) to t yield

$$\begin{aligned} x(t)\ge J_{n-1}(t)x^{\Delta ^{n-1}}(\varphi _{n-1}(t)). \end{aligned}$$
(10)

Using (10) in the inequality \((a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{\Delta }\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{1-\alpha } p(t)\delta _{2}(t,t_{1}) x^{\alpha }(\omega (t))\), we have

$$\begin{aligned} W^{\Delta }(t)&\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{1-\alpha } p(t)\delta _{2}(t,t_{1}) \left( J_{n-1}(\omega (t))x^{\Delta ^{n-1}}(\rho (t))\right) ^{\alpha }\\&\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(\rho (t))x^{\Delta ^{n-1}}(\rho (t)))^{1-\alpha } p(t)\delta _{2}(t,t_{1}) \left( J_{n-1}(\omega (t))x^{\Delta ^{n-1}}(\rho (t))\right) ^{\alpha }\\&\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(\rho (t)))^{1-\alpha } p(t)\delta _{2}(t,t_{1}) \left( J_{n-1}(\omega (t))\right) ^{\alpha }x^{\Delta ^{n-1}}(\rho (t)), \end{aligned}$$

that is,

$$\begin{aligned} W^{\Delta }(t)-\frac{1}{\alpha } \left( \frac{1}{a(\rho (t))}\right) p(t)\delta _{2}(t,t_{1})\left( J_{n-1}(\omega (t))\right) ^{\alpha } W(\rho (t))\ge 0, \end{aligned}$$

where \(W(t)= a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t)\). Because of [21, Lemma 4(II)], the associated dynamic Eq. (7) also has a positive solution, a contradiction.

Case-II It is easy to see that there exists a constant \(\theta \in (0,1)\) such that

$$\begin{aligned} x(\tau (t)) \ge \frac{\theta }{(n-2)!} \tau ^{n-2}(t) x^{\Delta ^{n-2}}(\tau (t)) \text{ for } t \ge t_{1}. \end{aligned}$$

Using this inequality in Eq. (9), one can see that

$$\begin{aligned}&\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{\Delta } \ge \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{1-\alpha } q(t) \delta _{1}\left( t, t_{1}\right) x^{\alpha }(\tau (t))\\&\quad \ge \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{1-\alpha } q(t) \delta _{1}\left( t, t_{1}\right) \left( \frac{\theta }{(n-2)!}\right) ^{\alpha } \tau ^{(n-2) \alpha }(t)\left( x^{\Delta ^{n-2}}(\tau (t))\right) ^{\alpha }, \end{aligned}$$

or,

$$\begin{aligned} \left( a^{\frac{1}{\alpha }}(t) Z^{\Delta }(t)\right) ^{\Delta }&\ge \left( \frac{\theta }{(n-2)!}\right) ^{\alpha } \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{1-\alpha } q(t) \delta _{1}\left( t, t_{1}\right) \tau ^{(n-2) \alpha }(t) Z^{\alpha }(\tau (t))\nonumber \\&\ge \left( \frac{\theta }{(n-2)!}\right) ^{\alpha } \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(\tau (t)) Z^{\Delta }(\tau (t))\right) ^{1-\alpha } q(t) \delta _{1}\left( t, t_{1}\right) \tau ^{(n-2) \alpha }(t) Z^{\alpha }(\tau (t)), \end{aligned}$$
(11)

where \(Z(t)=x^{\Delta ^{n-2}}(t)\). Now, for \(v\ge u \ge t_{1}\), we see that

$$\begin{aligned} Z(u)-Z(v) \ge (v-u)\left( -Z^{\Delta }(v)\right) . \end{aligned}$$

Setting \(u=\tau (t)\) and \(v=\xi (t)\) we have

$$\begin{aligned} Z(\tau (t))\ge \left( \xi (t)-\tau (t)\right) \left( -Z^{\Delta }(\xi (t))\right) . \end{aligned}$$

Using this inequality in Eq. (11), we have

$$\begin{aligned}{} & {} \left( a^{\frac{1}{\alpha }}(t) Z^{\Delta }(t)\right) ^{\Delta } \ge \left( \frac{\theta }{(n-2) !}\right) ^{\alpha } \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(\xi (t)) Z^{\Delta }(\xi (t))\right) ^{1-\alpha } q(t) \delta _{1}\left( t, t_{1}\right) \tau ^{(n-2) \alpha }(t) \\{} & {} \quad \times \left( \frac{(\xi (t)-\tau (t)}{a^{\frac{1}{\alpha }}(\xi (t))}\right) \left( -a^{\frac{1}{\alpha }}(\xi (t)) Z^{\Delta }(\xi (t))\right) ^{\alpha }, \end{aligned}$$

or,

$$\begin{aligned} -Y^{\Delta }(t) \ge \frac{1}{\alpha }\left( \frac{\theta }{(\textrm{n}-2) !}\right) ^{\alpha } q(t) \delta _{1}\left( t, t_{1}\right) \tau ^{(n-2) \alpha }(t) a^{\frac{1-\alpha }{\alpha }}(\xi (t))\left( \frac{(\xi (t)-\tau (t))}{a^{\frac{1}{\alpha }(\xi (t))}}\right) ^{\alpha }(Y(\xi (t)), \end{aligned}$$

where \(Y(t)=-a^{\frac{1}{\alpha }}(t)Z^{\Delta }(t)\). Because of [21, Lemma 4(I)], the associated dynamic Eq. (8) also has a positive solution, which is a contradiction. This completes the proof. \(\square\)

Theorem 3.2

Let \(\alpha =\beta =\gamma\) and n be even. In addition to the hypotheses of Theorem 3.1, we assume that there exists a nondecreasing function \(\eta (t)\in C([t_{0},\infty )_{{\mathbb {T}}},{\mathbb {R}}_{+})\) such that

$$\begin{aligned} \eta (t)>t\quad \text {and}\quad \mu (t)=\eta _{n-1}(\tau (t))<t. \end{aligned}$$
(12)

If the delay equation

$$\begin{aligned} X^{\Delta }(t)+ \frac{1}{\alpha (a(\mu (t)))}q(t) \left( I_{n-1}(\tau (t))\right) ^{\alpha } X(\mu (t))=0 \end{aligned}$$
(13)

is oscillatory for \(t\ge t_{1}>t_{0}\), then Eq. (1) is oscillatory.

Proof

Let x(t) be a nonoscillatory solution of Eq. (5) such that \(x(t)>0\), \(x(\tau (t))>0\) and \(x(\omega (t))>0\) for \(t \ge t_{1}\) for some \(t_{1}>t_{0}\). As in the proof of Theorem 3.1, the cases (I) and (II) are the same and hence we omit the details.

Now, we consider the Case-III of Lemma 2.1 for \(t \ge t_{2}\). From the fact that \(a^{\Delta }(t) \ge 0\), we see that x(t) satisfies

$$\begin{aligned} (-1)^{m} x^{\Delta ^{i}}(t)>0, \quad m=1,2, \ldots , n. \end{aligned}$$

Consequently,

$$\begin{aligned} x^{\Delta ^{n-2}}(t) \ge -x^{\Delta ^{n-2}}(\eta (t))+x^{\Delta ^{n-2}}(t)&=\int _{t}^{\eta (t)} x^{\Delta ^{n-1}}(s) \Delta s \\&\ge (\eta (t)-t) x^{\Delta ^{n-1}}(\eta (t)) = -I(t) x^{\Delta ^{n-1}}(\eta (t)). \end{aligned}$$

The repeated integrations of this inequality from t to \(\eta (t)\) yield

$$\begin{aligned} x(t) \ge -I_{n-1}(t) x^{\Delta ^{n-1}}\left( \eta _{n-1}(t)\right) , \end{aligned}$$
(14)

which implies

$$\begin{aligned} x(\tau (t)) \ge -\left( I_{n-1}(\tau (t)) x^{\Delta ^{n-1}}(\mu (t))\right) \quad \text {for}\ t\ge t_{3}>t_{2}. \end{aligned}$$

Using (14) in \((a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{\Delta }\ge \frac{1}{\alpha } (a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t))^{1-\alpha } q(t)\delta _{1}(t,t_{1}) x^{\alpha }(\tau (t))\), we get

$$\begin{aligned} \left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{\Delta }&\ge \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{1-\alpha } q(t) x^{\alpha }(\tau (t)) \\&\ge \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(t) x^{\Delta ^{n-1}}(t)\right) ^{1-\alpha } q(t)\left( -I_{n-1}(\tau (t)) x^{\Delta ^{n-1}}(\mu (t))\right) ^{\alpha } \\&\ge \frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(\mu (t)) x^{\Delta ^{n-1}}(\mu (t))\right) ^{1-\alpha } q(t)\left( -I_{n-1}(\tau (t)) x^{\Delta ^{n-1}}(\mu (t))\right) ^{\alpha }\\&= -\frac{1}{\alpha }\left( a^{\frac{1}{\alpha }}(\mu (t))\right) ^{1-\alpha } q(t)\left( I_{n-1}(\tau (t))\right) ^{\alpha } x^{\Delta ^{n-1}}(\mu (t)), \end{aligned}$$

or,

$$\begin{aligned} X^{\Delta }(t)+\frac{1}{\alpha } \frac{1}{a(\mu (t)} q(t)\left( I_{n-1}(\tau (t))\right) ^{\alpha } X(\mu (t)) \le 0, \end{aligned}$$

where \(X(t)=a^{\frac{1}{\alpha }}(t)x^{\Delta ^{n-1}}(t)\). The rest of the proof is similar to that of Theorem 3.1- Case-II and hence is omitted. This completes the proof. \(\square\)

The following corollary follows immediate:

Corollary 3.3

Let the hypotheses of Theorems 3.1 and 3.2 hold and replace the Eqs. (7), (8) and (13) respectively by the following integral conditions

$$\begin{aligned}&\limsup _{t \rightarrow \infty } \int _{t}^{\rho (t)}\frac{1}{\alpha }\left( \frac{1}{a(\rho (s))}\right) p(s)\left( J_{n-1}(\omega (s))\right) ^{\alpha } \Delta s=\infty , \end{aligned}$$
(15)
$$\begin{aligned}&\limsup _{t \rightarrow \infty }\int _{\xi (t)}^{t}q(s)\tau ^{(n-2)\alpha }(s)a^{\frac{1-\alpha }{\alpha }}(\xi (s)) \left( \frac{\xi (s)-\tau (s)}{a^{\frac{1}{\alpha }}(\xi (s))}\right) ^{\alpha }\Delta s =\infty , \end{aligned}$$
(16)

and for \(\alpha =\beta =\gamma\),

$$\begin{aligned} \limsup _{t \rightarrow \infty }\int _{\mu (t)}^{t}\frac{1}{\alpha (a(\mu (s)))}q(s) \left( I_{n-1}(\tau (s))\right) ^{\alpha } \Delta s=\infty \; \text {when}\; n\; \text {is even}, \end{aligned}$$
(17)

then Eq. (1) is oscillatory.

Proof

Let x(t) be a nonoscillatory solution of (1) for \(t\ge t_{0}\). Then proceeding as in the proof of Theorem 3.1 and 3.2, we get Eqs. (7), (8) and (13) for \(t\ge t_{1}>t_{0}\). Now, integrating (7) from t to \(\rho (t)\), we get

$$\begin{aligned} \int _{t}^{\rho (t)}\frac{1}{\alpha } \left( \frac{1}{a(\rho (s))}\right) p(s)\delta _{2}(s,t_{1})\left( J_{n-1}(\omega (s))\right) ^{\alpha } W(\rho (s)) \Delta s= W(\rho (t))-W(t), \end{aligned}$$

that is,

$$\begin{aligned} \int _{t}^{\rho (t)}\frac{1}{\alpha } \left( \frac{1}{a(\rho (s))}\right) p(s)\delta _{2}(s,t_{1})\left( J_{n-1}(\omega (s))\right) ^{\alpha } W(\rho (s))\Delta s\le W(\rho (t)). \end{aligned}$$

Since W(t) is increasing, then for \(t\le s\), we get

$$\begin{aligned} W(\rho (t))\int _{t}^{\rho (t)}\frac{1}{\alpha } \left( \frac{1}{a(\rho (s))}\right) p(s)\delta _{2}(s,t_{1})\left( J_{n-1}(\omega (s))\right) ^{\alpha }\Delta s\le W(\rho (t)), \end{aligned}$$

that is,

$$\begin{aligned} \int _{t}^{\rho (t)}\frac{1}{\alpha } \left( \frac{1}{a(\rho (s))}\right) p(s)\delta _{2}(s,t_{1})\left( J_{n-1}(\omega (s))\right) ^{\alpha }\Delta s\le 1, \end{aligned}$$

a contradiction to (15).

The rest of the proof is simple, so we omit the details. This completes the proof. \(\square\)

Example 3.4

Consider

$$\begin{aligned} \left[ \frac{1}{t^3}\left( x''\left( t\right) \right) ^3\right] '= \frac{1}{t^5} x^3\left( \frac{t}{8}\right) + \frac{1}{t^8} x^3\left( 8t\right) . \end{aligned}$$
(E1)

Here, \(\alpha =3\), \(\tau (t)=\frac{t}{8}\), \(\omega (t)=8t\), \(p(t)={\frac{1}{t^8}}\) and \(q(t)={\frac{1}{t^5}}\). Let \(\frac{t}{2}=\varphi (t)<t\), then \(\varphi (\omega (t))= 4t< \omega (t)\) and \(\rho (t)= \varphi (\varphi (\omega (t)))= 2t>t\) for \(t\ge t_{0}\). For \(a\left( t\right) =\frac{1}{t^3}\), we have

$$\begin{aligned} A(t,t_0)=\int _{t_0}^{t}\frac{ds}{a^{\frac{1}{\alpha }}(s)} = \int _{t_0}^{t} s^{\frac{2}{3}}ds= \frac{t^{5/3}-t_{0}^{5/3}}{2^{5/3}-1} \rightarrow \infty \ as \ t \rightarrow \infty . \end{aligned}$$

Also, condition (15) is satisfied. Indeed,

$$\begin{aligned} \limsup _{t \rightarrow \infty } \int _{t}^{\rho (t)}\frac{1}{\alpha }\left( \frac{1}{a(\rho (s))}\right) p(s)\left( J_{n-1}(\omega (s))\right) ^{\alpha }ds&= \limsup _{t \rightarrow \infty } \int _{t}^{2t} \frac{1}{3} 2s p(s) \left( \int _{4s}^{8s}\left( \frac{u}{2}\right) du\right) ^{3}ds \\&= \limsup _{t \rightarrow \infty } \int _{t}^{2t}\frac{1}{3}\frac{1}{s^8}2(24)^{3}s^{7}ds \\&= \infty . \end{aligned}$$

Let \(\xi (t)=\frac{t}{4}\), then from (16) we have

$$\begin{aligned} \limsup _{t\rightarrow \infty }\int _{\xi (t)}^{t}q(s)\tau ^{(n-2)\alpha }(s)a^{\frac{1-\alpha }{\alpha }}(\xi (s))\left( \frac{\xi (s)-\tau (s)}{a^{\frac{1}{\alpha }}(\xi (s))}\right) ^{\alpha }ds&= \int _{t/4}^{t}\frac{1}{s^{5}}s\left( \frac{s}{2}\right) ^{3}\left( \frac{s}{8}\right) ds\\&= \infty . \end{aligned}$$

Therefore, all the conditions of Corollary 3.3 are satisfied and hence (E1) is oscillatory.

Example 3.5

Consider

$$\begin{aligned} \Delta \left[ t\Delta ^{2} x(t)\right] = t x\left( \frac{t}{8}\right) + (1+t^2) x\left( 4t\right) \;\text {for}\;t\ge t_{0}=1. \end{aligned}$$
(E2)

Here, \(\alpha =1=\beta=\gamma\), \(\tau (t)=\frac{t}{8}\), \(\omega (t)=4t\), \(p(t)=1+t^2\) and \(q(t)= t\). Let \(\frac{t}{2}=\varphi (t)<t\), then \(\rho (t)= \varphi (\omega (t))= 2t>t\) for \(t\ge t_{0}\). For \(a\left( t\right) =t\), we have

$$\begin{aligned} A(t,t_0)=A(t,1)=\sum _{s=1}^{t}\frac{1}{a^{\frac{1}{\alpha }}(s)} = \sum _{s=1}^{t} \frac{1}{s} \rightarrow \infty \ as \ t \rightarrow \infty . \end{aligned}$$

Clearly, \(J_{1}(t)=\frac{t}{2}\), then \(J_{2}(t)=\frac{3t^{2}}{16}\). Also, from condition (15) we have

$$\begin{aligned} \limsup _{t \rightarrow \infty } \sum _{s=t}^{\rho (t)}\frac{1}{\alpha }\left( \frac{1}{a(\rho (s))}\right) p(s)\left( J_{n-1}(\omega (s))\right) ^{\alpha }&= \limsup _{t \rightarrow \infty } \sum _{s=t}^{2t} \frac{1}{2s} (1+s^{2}) 3s^{2} \\&= \frac{3}{2}\limsup _{t \rightarrow \infty } \sum _{s=t}^{2t} s(1+s^{2}) \\&= \infty . \end{aligned}$$

Let \(\xi (t)=\frac{t}{3}\), then from (16) we have

$$\begin{aligned} \limsup _{t\rightarrow \infty }\sum _{s=\xi (t)}^{t}q(s)\tau ^{(n-2)\alpha }(s)a^{\frac{1-\alpha }{\alpha }}(\xi (s)) \left( \frac{\xi (s)-\tau (s)}{a^{\frac{1}{\alpha }}(\xi (s))}\right) ^{\alpha }&= \sum _{s=t/3}^{t} s \left( \frac{\frac{s}{3}-\frac{s}{8}}{\frac{s}{3}}\right) \\&= \frac{5}{8}\sum _{s=t/3}^{t} s = \infty . \end{aligned}$$

Similarly, using \(\eta (t)=4t\)(let) and \(\mu (t)=\eta (\tau (t))=\frac{t}{2}<t\), it is not difficult to check that condition (17) is also satisfied. Therefore, all the conditions of Corollary 3.3 are satisfied and hence (E2) is oscillatory.

Remark 3.6

We highlight some of the significance of our findings below:

  1. 1.

    Graef [24] studied the higher order nonliner differential equations with an advanced arguments of the form

    $$\begin{aligned} x^{(n)}=p(t)x^{\gamma }(\omega (t)),\;t\ge t_{0}>0, \end{aligned}$$
    (18)

    where \(n\ge 2\) is even or odd, \(p(t)>0\), \(\omega (t)\ge t\) and \(\gamma\) is the ratio of odd positive integers. Although (18) is equivalent to (1) when \({\mathbb {T}}={\mathbb {R}}\), \(a(t)=1\), \(\alpha =1\) and \(q(t)=0\), the results reported in [24] are not applicable to (1), because it is a Half-linear equations. Therefore, the oscillation criteria presented in this paper generalize that of in [24].

  2. 2.

    Similarly, the results reported in [7, 8] cannot be applied to (1).

Conclusion

In the present work, with the help of a comparison technique with the behaviour of first order delay and/or advanced dynamic equations as well as an integral criterion, several results for the oscillatory behaviour of solutions of Eq. (1) are presented. The main results are formulated by the Theorems 3.1, and 3.2. As an application of the main results, a Corollary 3.3 as well as two examples are then presented. Articles [4, 5, 20, 21] are concerned with the asymptotic behaviour and oscillation of solutions to higher-order dynamic equations, which is a topic very close to our investigations, and our results complement the results reported in these papers. Our obtained theorems not only generalise the existing results in the literature but can also be used to plan future research papers in a variety of directions. For example:

  1. (Q1)

    One can consider Eq. (1) with

    $$\begin{aligned} y(t)=x(t)+ p_{1}(t)x^{\delta _{1}}(\sigma (t))- p_{2}(t)x^{\delta _{2}}(\sigma (t)) \end{aligned}$$

    with \(\sigma (t)\le t\), and \(\delta _{1}\) and \(\delta _{2}\) are the ratios of positive odd integers.

  2. (Q2)

    One can consider Eq. (1) under the case

    $$\begin{aligned} \lim _{t\rightarrow \infty }{\mathcal {A}}(t,t_{0})= \lim _{t\rightarrow \infty }\int _{t_{0}}^{t}\frac{\Delta s}{r^{1/\alpha }(s)}<\infty . \end{aligned}$$