1 Introduction

For the sake of our study, we will assume that the reader is familiar with the elementary notation and terminology used in the calculus of time scales, see, e.g., the book by Bohner and Peterson [8]. Here, our interest is to find some new sufficient condition for the oscillation of the second order dynamic equation of the form:

$$\begin{aligned}{}[r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }]^{\Delta } +\mathcal {P}(\theta )\mathcal {W}^{\beta }(\eta (\theta ))=0,\;\theta \in \mathbb {T}_{0}=[\theta _{0},\infty )\cap \mathbb {T}, \end{aligned}$$
(1.1)

where \(\mathbb {T}\subseteq \mathbb {R}\) is an arbitrary time scale unbounded above. A real valued function \(\mathcal {W}(\theta )\in C^{1}_{rd}[\theta _{0},\infty )_{\mathbb {T}}\) with \(r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }\in C^{1}_{rd}[\theta _{0},\infty )_{\mathbb {T}}\) is called the solution of (1.1) if it satisfies (1.1) on \([T_{\mathcal {W}},\infty )_{\mathbb {T}}\), where \(T_{\mathcal {W}}>\theta _{0}\) is chosen such that \(\eta (\theta )>\theta _{0}\) for \(\theta \ge T_{\mathcal {W}}\). Solutions vanishing identically in some neighbourhood of infinity will be excluded from our consideration.

Throughout, we suppose that

\((\mathcal {A}_{1})\):

\(\alpha \), \(\beta \) are quotient of odd positive integers, \(\eta \in C_{rd}(\mathbb {T}_{0},\mathbb {T})\) such that \(\eta (\theta )\le \theta \) and \(\eta (\theta )\rightarrow \infty \) as \(\theta \rightarrow \infty \);

\((\mathcal {A}_{2})\):

\(\mathcal {P}(\theta )\in C_{rd}(\mathbb {T}_{0}, \mathbb {R}_{+})\) with \(\mathcal {P}(\theta )\not \equiv 0\) and \(r(\theta )\in C_{rd}(\mathbb {T}_{0}, \mathbb {R}_{+})\) such that \(\int _{\theta _{0}}^{\infty }(r(s))^{-1/\alpha }\Delta s<\infty \).

We say that the Eq. (1.1) to be in non-canonical form (see [31]) if

$$\begin{aligned}\mathcal {A}(\theta )=\int _{\theta _{0}}^{\theta }(r(s))^{-1/\alpha }\Delta s<\infty \; \text {as}\; \theta \rightarrow \infty .\end{aligned}$$

Delay differential equations (DDEs) are commonly employed in mathematical modelling to induce oscillatory activity in physical and biological systems. Ottesen’s groundbreaking study [25] demonstrates how to solve a cardiovascular model with a discontinuous derivative using DDEs. This study also showed that complicated dynamic interactions between nonlinear behaviours and delays related to autonomic-cardiac control can lead to instability. Ataeea et al. [3] have studied a model-based method for stability analysis of autonomic-cardiac regulation, that is, it is vital to underline that the autonomic-cardiac regulation operates via the interaction between the autonomic nervous system (ANS) and cardiovascular system (CVS). Focusing on the oscillation theory developed by Sturm in 1836, there is substantial literature on the oscillation theory of (1.1), see, e.g., [1, 16, 21]. We know that discrete-time and continuous-time systems have the same theoretical and practical importance, and that discrete-time systems are more computer-friendly than their continuous-time counterparts. As a result, while studying a continuous-time system, it is important to research the corresponding discrete-time system. Fortunately, studying dynamic equations on time scales helps unify the study of continuous-time and discrete-time systems, as well as other applications. The study of dynamic equations on time scales has been a priority for many scholars and has yielded significant research findings since Hilger [19] introduced the notion of time-scale calculus. Especially, Erbe et al. [11] have initiated the study of nonlinear dynamic Eq. (1.1) with \(\alpha =\beta =1\) and present a single result for the non-canonical type equations. Since then, many scholars have studied various generalizations of Eq. (1.1) and improved the oscillation conditions by using methods like the Riccati transformation technique, comparison technique, inequality technique, etc., see [2, 4,5,6,7, 10, 12,13,14,15, 17, 18, 22,23,24, 27, 29, 30, 32, 33] and references cited therein.

Specially, for \(r(\theta )=1\), \(\alpha =1=\beta \), \(\eta (\theta )=\theta \) and \(\mathbb {T}=\mathbb {R}\), Philos [26] introduced the function class \(\mathcal {R}_{1}\) as follows: \(\mathcal {M}_{1}\in \mathcal {R}_{1}\) if \(\mathcal {M}_{1}:D\equiv \{(\theta ,s):\theta \ge s\ge \theta _{0}\}\) is continuous, satisfies \(\mathcal {M}_{1}(\theta ,\theta )=0\), \(\theta \ge \theta _{0}\), \(\mathcal {M}_{1}(\theta ,s)>0\), \(\theta >s\ge \theta _{0}\), and \(\frac{\partial \mathcal {M}_{1}}{\partial s}\le 0\) is continuous on D. Moreover, let \(m_{1}(\theta ,s)\in C(D,\mathbb {R})\) with

$$\begin{aligned}\frac{-\partial \mathcal {M}_{1}}{\partial s}(\theta ,s)=m_{1}(\theta ,s)\sqrt{\mathcal {M}_{1}(\theta ,s)},\;(\theta ,s)\in D.\end{aligned}$$

In this work, they showed that Eq. (1.1) is oscillatory if

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\frac{1}{\mathcal {M}_{1}(\theta ,\theta _{0})}\int _{\theta _{*}}^{\theta }\left[ \mathcal {M}_{1}(\theta ,\theta _{0})\mathcal {P}(s)-\frac{m^{2}_{1}(\theta ,s)}{4}\right] \Delta s=\infty . \end{aligned}$$
(1.2)

Saker et al. [27] extended Philos’s results to (1.1). In order to establish new oscillation results, they used the Riccati transformation and introduced the function class \(\mathcal {R}_{2}\) as follows: \(\mathcal {M}_{2}\in \mathcal {R}_{2}\) if \(\mathcal {M}_{2}\) is defined for \(\theta _{0}\le s\le \theta \), \((\theta ,s)\in [\theta _{0},\infty )_{\mathbb {T}}\), \(\mathcal {M}_{2}(\theta ,\theta )=0\), \(\mathcal {M}_{2}(\theta ,s)\ge 0\) for \(\theta \ge s\ge \theta _{0}\), and for each fixed \(\theta \), \(\mathcal {M}_{2}^{\Delta _{i}}(\theta ,s)\) is \(\Delta \)-integrable w.r.t variable \(i(i=1,2)\).

The following conclusions emerged from the results:

  1. 1.

    It appears that when \(\mathcal {A}(\theta )<\infty \), this paper contains only one results, namely, [27, Theorem 2.20]. But, this result guarantee that all solution of Eq. (1.1) oscillates or converges to zero;

  2. 2.

    Additionally, the main results are supported by [27, Eq.2.18]. But, this not true for \(B<0\) and \(u>0\). Indeed, if \(B<0\), then \(g(u)<0\). So, it’s maximum value is 0. As a result,

    $$\begin{aligned}\max _{u\in \mathbb {R}}g(u)=g(u^{*})=\frac{\alpha ^{\alpha }}{(\alpha +1)^{\alpha +1}}\frac{B^{\alpha +1}}{A^{\alpha }}>0\end{aligned}$$

    can not be possible.

Wu et al. [32] invastigated (1.1) with \(\alpha =\beta \) and \(f(\theta ,\mathcal {W})=\mathcal {P}(\theta )g(\mathcal {W}(\eta (\theta )))\), where \(\frac{g(u)}{u}\ge L>0\) and improved the work of Saker et al. [27] by relaxing some restriction of the coefficients. As in the previous work, we also raised the same questions.

Motivated by the above discussion, we derive some new oscillation conditions for Eq. (1.1), which simplify and improve the results reported in the literature. To begin, we improve the theorem by proving the main results with a new inequality. Secondly, we improve the theorem by establishing conditions that guarantee only oscillations of Eq. (1.1).

2 Auxiliary lemmas

Due to \((\mathcal {A}_{2})\), we may let

$$\begin{aligned} \hat{\mathcal {A}}(\theta )=\int _{\theta }^{\infty }\frac{\Delta s}{r^{1/\alpha }(s)}\;\text {for}\;\theta \in [\theta _{0},\infty )_{\mathbb {T}}, \end{aligned}$$
(2.1)

then \(\hat{\mathcal {A}}(\theta )\rightarrow 0\) as \(\theta \rightarrow \infty \).

Lemma 2.1

If X and Y are nonnegative and \(\gamma >1\), then

$$\begin{aligned}X^{\gamma }-\gamma Y^{\gamma -1}\ge (1-\gamma )Y^{\gamma },\end{aligned}$$

where the equality holds if and only if \(X=Y\).

Lemma 2.2

[15] Let \(\mathcal {W}(\theta )>0\) eventually be a solution of (1.1) for \(\theta \ge \theta _{0}\). Then \(\mathcal {W}(\theta )\) satisfies either:

$$\begin{aligned}&(L_{h})\;r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }>0,\,[r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }]^{\Delta }<0, \ \text{ or } \\&(L_{nh})\;r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }<0,\,[r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }]^{\Delta }<0 \end{aligned}$$

for \(\theta \in [\theta _{*},\infty )_{\mathbb {T}}\subset [\theta _{0},\infty )_{\mathbb {T}}\).

Lemma 2.3

Let \(\mathcal {W}(\theta )>0\) eventually be a positive solution of (1.1) for \(\theta \ge \theta _{0}\). If

$$\begin{aligned} \int _{\theta ^{*}}^{\infty }\mathcal {P}(\nu )\Delta \nu =\infty , \end{aligned}$$
(2.2)

then \(\mathcal {W}(\theta )\) satisfies \((L_{nh})\) of Lemma 2.2. Moreover, \(\left( \frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\right) \) is increasing.

Proof

Assume, on the other hand, that \(\mathcal {W}(\theta )\) meets the condition \((L_{h})\) of Lemma 2.2 for \(\theta \in [\theta _{1},\infty )_{\mathbb {T}}\). Then there exist \(\theta _{2}>\theta _{1}\) and \(\kappa >0\) such that \(\mathcal {W}(\theta )\ge \kappa \) and hence \(\mathcal {W}^{\beta }(\eta (\theta ))\ge \kappa ^{\beta }\) for \(\theta \in [\theta _{2},\infty )_{\mathbb {T}}\). Integrating (1.1) from \(\theta _{2}\) to \(\theta \), we get

$$\begin{aligned} r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }-r(\theta _{2})(\mathcal {W}^{\Delta }(\theta _{2}))^{\alpha }+\int _{\theta _{2}}^{\theta } \mathcal {P}(s)\mathcal {W}^{\beta }(\eta (s))\Delta s\le 0 \end{aligned}$$

implies that

$$\begin{aligned} \kappa ^{\beta }\int _{\theta _{2}}^{\theta }\mathcal {P}(s)\Delta s \le r(\theta _{2})(\mathcal {W}^{\Delta }(\theta _{2}))^{\alpha }, \end{aligned}$$

a contradiction to (2.2). Therefore, \(\mathcal {W}(\theta )\) satisfies \((L_{nh})\) for \(\theta \in [\theta _{1},\infty )_{\mathbb {T}}\).

Indeed, for \(s>\theta \ge \theta _{1}\) we have

$$\begin{aligned} r(s)(\mathcal {W}^{\Delta }(s))^{\alpha }\le r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }. \end{aligned}$$
(2.3)

Integrating (2.3) from \(\theta \) to s and letting \(s\rightarrow \infty \), we obtain

$$\begin{aligned} \mathcal {W}(\theta )-\mathcal {W}(s)\ge -r^{1/\alpha }(\theta )\mathcal {W}^{\Delta }(\theta )\int _{\theta }^{s}\frac{\Delta v}{r^{1/\alpha }(v)}, \end{aligned}$$

or,

$$\begin{aligned} -\mathcal {W}(\theta )\le r^{1/\alpha }(\theta )\mathcal {W}^{\Delta }(\theta )\int _{\theta }^{\infty }\frac{\Delta v}{r^{1/\alpha }(v)}= r^{1/\alpha }(\theta )\mathcal {W}^{\Delta }(\theta )\hat{\mathcal {A}}(\theta ). \end{aligned}$$

As a result,

$$\begin{aligned} \left( \frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\right) ^{\Delta } =\frac{\hat{\mathcal {A}}(\theta )\mathcal {W}^{\Delta }(\theta )+\mathcal {W}(\theta ) r^{-1/\alpha }(\theta )}{\hat{\mathcal {A}}(\theta )\hat{\mathcal {A}}(\sigma (\theta ))}\ge 0. \end{aligned}$$

Hence, proved. \(\square \)

Lemma 2.4

Let \(\mathcal {W}(\theta )>0\) eventually be a solution of (1.1) for \(\theta \ge \theta _{0}\). If

$$\begin{aligned} \int _{\theta _{*}}^{\infty }\frac{1}{r^{1/\alpha }(s)}\left[ \int _{\theta _{*}}^{s}\mathcal {P}(v)\Delta v\right] ^{1/\alpha }\Delta s=\infty , \end{aligned}$$
(2.4)

then \(\mathcal {W}(\theta )\rightarrow 0\) as \(\theta \rightarrow \infty \). In addition to (2.4), assume that

$$\begin{aligned} \int _{\theta _{*}}^{\infty }\mathcal {P}(\nu )\hat{\mathcal {A}}^{\beta }(\eta (\nu ))\Delta \nu =\infty \end{aligned}$$
(2.5)

hold, then \(\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\rightarrow \infty \) as \(\theta \rightarrow \infty \).

Proof

From \((\mathcal {A}_{2})\) and (2.4), we conclude that (2.2) holds. Therefore, from Lemma 2.3 it follows that \(\mathcal {W}(\theta )\) satisfies \((L_{nh})\) of Lemma 2.2. Indeed, \(\lim _{\theta \rightarrow \infty }\mathcal {W}(\theta )=\mathcal {C}\ge 0\) exists. We assert that \(\mathcal {C}=0\). If not, then we can find \(\theta _{2}>\theta _{1}\) such that \(\mathcal {W}(\theta )\ge \mathcal {C}>0\) and hence \(\mathcal {W}^{\beta }(\eta (\theta ))\ge \mathcal {C}^{\beta }\) for \(\theta \in [\theta _{2},\infty )_{\mathbb {T}}\). Upon using this relation in (1.1)and then integrating from \(\theta _{2}\) to \(\theta \), we get

$$\begin{aligned} \mathcal {C}^{\beta }\int _{\theta _{2}}^{\theta }\mathcal {P}(s)\Delta s\le -r(\theta ) (\mathcal {W}^{\Delta }(\theta ))^{\alpha }, \end{aligned}$$

which on integration again from from \(\theta _{2}\) to \(\theta \) gives

$$\begin{aligned} \mathcal {C}^{\beta /\alpha }\int _{\theta _{2}}^{\theta } \frac{1}{r^{1/\alpha }(s)}\left[ \int _{\theta _{2}}^{s}\mathcal {P}(v)\Delta v \right] ^{1/\alpha }\Delta s\le \mathcal {W}(\theta _{2}), \end{aligned}$$

a contradiction to (2.4). Therefore, \(\lim _{\theta \rightarrow \infty }\mathcal {W}(\theta )=0\).

From L’Hospital rule (see [8, Theorem 1. 119]), it follows that

$$\begin{aligned} \lim _{\theta \rightarrow \infty }\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}=\lim _{\theta \rightarrow \infty }\frac{\mathcal {W}^{\Delta }(\theta )}{\hat{\mathcal {A}}^{\Delta }(\theta )}= \lim _{\theta \rightarrow \infty }(-r^{-1/\alpha }(\theta )\mathcal {W}^{\Delta }(\theta ))=l, \end{aligned}$$

where \(0<l\le \infty \). We claim that \(l=\infty \). If not, then there exist \(\theta _{2}>\theta _{1}\) such that \(-r^{-1/\alpha }(\theta )\mathcal {W}^{\Delta }(\theta ) \le l<\infty \) for \(\theta \in [\theta _{2},\infty )_{\mathbb {T}}\). Using this inequality in (1.1) and then integrating from \(\theta _{2}\) to \(\theta \), we get

$$\begin{aligned} \int _{\theta _{2}}^{\theta }\mathcal {P}(s)\mathcal {W}^{\beta }(\eta (s))\Delta s\le -r(\theta ) (\mathcal {W}^{\Delta }(\theta ))^{\alpha }\le l^{\alpha }. \end{aligned}$$

Upon using the fact that \(\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\) is positive and increasing, we obtain

$$\begin{aligned} \frac{\mathcal {W}^{\beta }(\eta (\theta _{2}))}{\hat{\mathcal {A}}^{\beta }(\eta (\theta _{2}))} \int _{\theta _{2}}^{\theta }\mathcal {P}(s)\hat{\mathcal {A}}^{\beta }(\eta (s))\Delta s\le l^{\alpha }, \end{aligned}$$

a contradiction to (2.5). Therefore, \(\lim _{\theta \rightarrow \infty }\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}=\infty \). Hence, proved. \(\square \)

3 Main results

We’re now in a position to present our first result.

Theorem 3.1

Assume that \(\alpha >\beta \) and (2.4) hold. If

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\hat{\mathcal {A}}^{\alpha }(\theta )\int _{\theta _{*}}^{\infty }\mathcal {P}(\nu )\Delta \nu >0, \end{aligned}$$
(3.1)

then (1.1) oscillates.

Proof

Without loss of generality, let \(\mathcal {W}(\theta )>0\) eventually be a solution of (1.1) for \(\theta \ge \theta _{0}\). Indeed, (2.4) implies (2.2) and hence \(\mathcal {W}(\theta )\) belongs to the class \((L_{nh})\) for \(\theta \in [\theta _{1},\infty )_{\mathbb {T}}\). By (3.1), we can find points \(\theta _{2}>\theta _{1}\) and a positive constant \(b>0\) such that

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\hat{\mathcal {A}}^{\alpha }(\theta )\int _{\theta _{*}}^{\infty }\mathcal {P}(\nu )\Delta \nu >b^{-\alpha }. \end{aligned}$$
(3.2)

Now, integrating (1.1) from the point \(\theta _{2}\) to \(\theta \), we get

$$\begin{aligned} \int _{\theta _{2}}^{\theta } \mathcal {P}(s)\mathcal {W}^{\beta }(\eta (s))\Delta s\le -r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }, \end{aligned}$$

which further implies

$$\begin{aligned} \frac{1}{r^{1/\alpha }(\theta )} \left[ \int _{\theta _{2}}^{\theta } \mathcal {P}(s)\mathcal {W}^{\beta }(\eta (s))\Delta s\right] ^{1/\alpha }\le -\mathcal {W}^{\Delta }(\theta ). \end{aligned}$$

Integrating the last inequality again from \(\theta \) to u and letting \(u\rightarrow \infty \), we obtain

$$\begin{aligned} \int _{\theta }^{\infty }\frac{1}{r^{1/\alpha }(s)} \left[ \int _{\theta _{2}}^{s} \mathcal {P}(v)\mathcal {W}^{\beta }(\eta (v))\Delta v\right] ^{1/\alpha }\Delta s&\le \left( \mathcal {W}(\theta )-\lim _{u\rightarrow \infty }\mathcal {W}(u)\right) \le \mathcal {W}(\theta ), \end{aligned}$$

or,

$$\begin{aligned} \hat{\mathcal {A}}(\theta )\mathcal {W}^{\beta /\alpha }(\eta (\theta ))\left[ \int _{\theta _{2}}^{\theta } \mathcal {P}(v)\Delta v\right] ^{1/\alpha }\le \hat{\mathcal {A}}(\theta )\left[ \int _{\theta _{2}}^{\theta } \mathcal {P}(v)\mathcal {W}^{\beta }(\eta (v))\Delta v\right] ^{1/\alpha }\le \mathcal {W}(\theta ) \end{aligned}$$
(3.3)

due to decreasing and positive nature of \(\mathcal {W}(\theta )\).

From Lemma 2.4, it follows that \(\lim _{\theta \rightarrow \infty }\mathcal {W}(\theta )=0\). Therefore, we can find \(\theta _{3}>\theta _{2}\) and \(b>0\) such that \(\mathcal {W}(\theta )\le b^{\frac{\alpha }{\beta -\alpha }}\) implies that \(\mathcal {W}^{\frac{\beta -\alpha }{\alpha }}(\theta )\ge b\) for \(\theta \in [\theta _{3},\infty )_{\mathbb {T}}\). Using this in (3.3), we have

$$\begin{aligned} \mathcal {W}(\theta )\ge b \hat{\mathcal {A}}(\theta )\mathcal {W}(\eta (\theta ))\left[ \int _{\theta _{2}}^{\theta } \mathcal {P}(v)\Delta v\right] ^{1/\alpha }. \end{aligned}$$

On simplification, we get

$$\begin{aligned} \hat{\mathcal {A}}^{\alpha }(\theta )\left[ \int _{\theta _{2}}^{\theta } \mathcal {P}(v)\Delta v\right] \le \frac{1}{b^{\alpha }}, \end{aligned}$$

a contradiction to (3.2). This completes the proof. \(\square \)

Theorem 3.2

Assume that \(\alpha <\beta \) and (2.4), (2.5) hold. If

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\hat{\mathcal {A}}^{\alpha }(\theta )\hat{\mathcal {A}}^{\beta -\alpha } (\eta (\theta ))\int _{\theta _{*}}^{\infty }\mathcal {P}(\nu )\Delta \nu >0, \end{aligned}$$
(3.4)

then (1.1) oscillates.

Proof

Without loss of generality, let \(\mathcal {W}(\theta )>0\) eventually be a solution of (1.1) for \(\theta \ge \theta _{0}\). Indeed, (2.4) implies (2.2). As a result, \(\mathcal {W}(\theta )\) belongs to the class \((L_{nh})\) for \(\theta \in [\theta _{1},\infty )_{\mathbb {T}}\). By (3.4), we can find points \(\theta _{2}>\theta _{1}\) and a positive constant \(b>0\) such that

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\hat{\mathcal {A}}^{\alpha }(\theta )\hat{\mathcal {A}}^{\beta -\alpha } (\eta (\theta ))\int _{\theta _{2}}^{\infty }\mathcal {P}(\nu )\Delta \nu > b^{-\alpha }. \end{aligned}$$
(3.5)

By Lemma 2.4, we have that \(\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\) is increasing positive functions and \(\lim _{\theta \rightarrow \infty }\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}=\infty \). Thus, we can find \(\theta _{3}>\theta _{2}\) and \(b>0\) such that

$$\begin{aligned}\frac{\mathcal {W}(\theta )}{\hat{\mathcal {A}}(\theta )}\ge b^{\frac{\alpha }{\beta -\alpha }},\end{aligned}$$

which implies that

$$\begin{aligned}\mathcal {W}^{\frac{\beta -\alpha }{\alpha }}(\eta (\theta ))\ge \hat{\mathcal {A}}^{\frac{\beta -\alpha }{\alpha }}(\eta (\theta )) b\;\text {for}\;\theta \in [\theta _{3},\infty )_{\mathbb {T}}.\end{aligned}$$

Using the preceding inequality in (3.3), we have

$$\begin{aligned} \mathcal {W}(\theta )\ge \hat{\mathcal {A}}(\theta )\mathcal {W}(\eta (\theta ))\hat{\mathcal {A}}^{\frac{\beta -\alpha }{\alpha }}(\eta (\theta ))b\left[ \int _{\theta _{3}}^{\infty }\mathcal {P}(v)\Delta v\right] ^{1/\alpha }. \end{aligned}$$

On simplification, we get

$$\begin{aligned} \hat{\mathcal {A}}^{\alpha }(\theta )\hat{\mathcal {A}}^{\beta -\alpha } (\eta (\theta ))\left[ \int _{\theta _{3}}^{\infty }\mathcal {P}(v)\Delta v\right] \le \frac{1}{b^{\alpha }}, \end{aligned}$$

a contradiction to (3.5). This completes the proof. \(\square \)

For the next result, we no not require Eq. (2.4).

Theorem 3.3

Let \(\alpha >\beta \) and (3.1) holds. Assume that there exists a function \(\Omega \in C_{rd}^{1}(\mathbb {T},\mathbb {R}_{+})\) such that

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\left\{ \hat{\mathcal {P}}(\theta )+ \int _{\theta _{0}}^{\theta }\left[ \Omega (s)\mathcal {P}(s)-\frac{(\alpha /\beta )^{\alpha }}{(\alpha +1)^{\alpha +1}}\frac{r^{*}(s)\Omega ^{\alpha +1}(\sigma (s))}{(\Omega (s)\eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s\right\} =\infty , \end{aligned}$$
(3.6)

where \(\hat{\mathcal {P}}(\theta )=\Omega (\theta )\int _{\theta }^{\infty }\mathcal {P}(s)\Delta s\), \(r^{*}(t)=\max \{r(s):\eta (t)\le s \le \eta (\sigma (t))\}\) and \(g(\theta )=\hat{c}\mathcal {A}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta )))\). Then every solutions of Eq. (1.1) oscillates.

Proof

Let \(\mathcal {W}(\theta )\) be a non-oscillatory solution of (1.1). Without loss of generality, there exists \(\theta _{1}>\theta _{0}\) such that \(\mathcal {W}(\theta )>0\) for \(\theta \ge \theta _{1}\). From Lemma 2.2, if follows that \(r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }\) is decreasing and \(\mathcal {W}(\theta )\) is monotonic for \(\theta \ge \theta _{2}>\theta _{1}\). Ultimately, we have the following possibilities:

Case-a (\(\mathcal {W}(\theta )>0,\,r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }>0\)). Integrating (1.1) from \(\theta \) to u and letting \(u\rightarrow \infty \), we get

$$\begin{aligned} r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }\ge \int _{\theta }^{\infty }\mathcal {P}(s)\mathcal {W}^{\beta }(\eta (s))\Delta s\ge \mathcal {W}^{\beta }(\eta (\theta ))\int _{\theta }^{\infty }\mathcal {P}(s)\Delta s. \end{aligned}$$

Define

$$\begin{aligned} \mathcal {V}(\theta )&=\Omega (\theta )\frac{r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }}{\mathcal {W}^{\beta }(\eta (\theta ))}\ge \Omega (\theta )\int _{\theta }^{\infty }\mathcal {P}(s)\Delta s >0 \end{aligned}$$
(3.7)

for \(\theta \ge \theta _{3}\). Using the product rule, we have

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&= (r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha })^{\Delta }\left( \frac{\Omega (\theta )}{\mathcal {W}^{\beta }(\eta (\theta ))}\right) + r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }\left( \frac{\Omega (\theta )}{\mathcal {W}^{\beta }(\eta (\theta ))}\right) ^{\Delta }\\&= \frac{\Omega (\theta )(r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha })^{\Delta }}{\mathcal {W}^{\beta }(\eta (\theta ))}+r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }\\&\quad \times \left[ \frac{\Omega ^{\Delta }(\theta )\mathcal {W}^{\beta }(\eta (\theta ))-\Omega (\theta )(\mathcal {W}^{\beta }(\eta (\theta )))^{\Delta })}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta }(\eta (\sigma (\theta )))}\right] \\&\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }}{\mathcal {W}^{\beta }(\eta (\sigma (\theta )))}\\&\qquad -\frac{\Omega (\theta )r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }(\mathcal {W}^{\beta }(\eta (\theta )))^{\Delta }}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta }(\eta (\sigma (\theta )))}, \end{aligned}$$

that is,

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))-\Omega (\theta )\frac{r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }(\mathcal {W}^{\beta }(\eta (\theta )))^{\Delta }}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta }(\eta (\sigma (\theta )))}. \end{aligned}$$
(3.8)

From [32, Lemma 2.1], we have

$$\begin{aligned} (\mathcal {W}^{\beta }(\eta (\theta )))^{\Delta }\ge {\left\{ \begin{array}{ll} \beta \mathcal {W}^{\beta -1}(\eta (\sigma (\theta )))(\mathcal {W}(\eta (\theta )))^{\Delta },\;0\le \beta <1\\ \beta \mathcal {W}^{\beta -1}(\eta (\theta ))(\mathcal {W}(\eta (\theta )))^{\Delta },\;\beta \ge 1. \end{array}\right. } \end{aligned}$$
(3.9)

If \(\sigma (\theta )>\theta \), then by [9, Theorem 1.14] we have

$$\begin{aligned} (\mathcal {W}(\eta (\theta )))^{\Delta }&=\frac{\mathcal {W}(\eta (\sigma (\theta ))) -\mathcal {W}(\eta (\theta ))}{\sigma (\theta )-\theta }= \frac{\mathcal {W}(\eta (\sigma (\theta ))) -\mathcal {W}(\eta (\theta ))}{\eta (\sigma (\theta ))-\eta (\theta )}\eta ^{\Delta }(\theta )\nonumber \\&\ge \mathcal {W}^{\Delta }(\xi )\eta ^{\Delta }(\theta ), \end{aligned}$$
(3.10)

where \(\xi \in [\eta (\theta ), \eta (\sigma (\theta )))\). If \(\sigma (\theta )=\theta \), then we have \(\eta (\sigma (\theta ))=\sigma (\eta (\theta ))=\eta (\theta )\) and

$$\begin{aligned} (\mathcal {W}(\eta (\theta )))^{\Delta }=\mathcal {W}'(\eta (\theta ))\eta '(\theta ). \end{aligned}$$
(3.11)

Using (3.8), (3.10) and (3.11) in (3.7), we get

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))\nonumber \\&-{\left\{ \begin{array}{ll} \frac{\beta \Omega (\theta )r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta }(\eta (\sigma (\theta )))}\mathcal {W}^{\beta -1} (\eta (\sigma (\theta )))\mathcal {W}^{\Delta }(\xi )\eta ^{\Delta }(\theta ),\;0\le \beta<1\\ \frac{\beta \Omega (\theta )r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta }(\eta (\sigma (\theta )))} \mathcal {W}^{\beta -1}(\eta (\theta ))\mathcal {W}^{\Delta }(\xi )\eta ^{\Delta }(\theta ),\;\beta \ge 1. \end{array}\right. }\nonumber \\&=-\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))\nonumber \\&-{\left\{ \begin{array}{ll} \beta \Omega (\theta )\eta ^{\Delta }(\theta )\frac{r(\sigma (\theta )) (\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }\mathcal {W}^{\beta }(\eta (\sigma (\theta ))) \mathcal {W}^{\Delta }(\xi )}{\mathcal {W}^{\beta }(\eta (\theta ))\mathcal {W}^{\beta +1}(\eta (\sigma (\theta )))},\;0\le \beta <1\\ \beta \Omega (\theta )\eta ^{\Delta }(\theta )\frac{r(\sigma (\theta )) (\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha }}{\mathcal {W}^{\beta +1}(\eta (\sigma (\theta )))} \frac{\mathcal {W}(\eta (\sigma (\theta )))}{\mathcal {W}(\eta (\theta ))}\mathcal {W}^{\Delta }(\xi ),\;\beta \ge 1. \end{array}\right. } \end{aligned}$$
(3.12)

Using the fact that \(\eta (\theta )\) and \(\mathcal {W}(\theta )\) is nondecreasing, we have \(\mathcal {W}(\eta (\sigma (\theta )))\ge \mathcal {W}(\eta (\theta ))\). Therefore, from (3.12) we get

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))-\beta \Omega (\theta )\eta ^{\Delta }(\theta )\frac{r(\sigma (\theta ))(\mathcal {W}^{\Delta } (\sigma (\theta )))^{\alpha }}{\mathcal {W}^{\beta +1}(\eta (\sigma (\theta )))}\mathcal {W}^{\Delta }(\xi ) \end{aligned}$$
(3.13)

for \(\beta >0\). Again for \(\xi \in [\eta (\theta ), \eta (\sigma (\theta )))\), we have

$$\begin{aligned}r(\xi )(\mathcal {W}^{\Delta }(\xi ))^{\alpha }\ge r(\eta (\sigma (\theta ))) (\mathcal {W}^{\Delta }(\eta (\sigma (\theta ))))^{\alpha }\ge r(\sigma (\theta ))(\mathcal {W}^{\Delta }(\sigma (\theta )))^{\alpha },\end{aligned}$$

that is,

$$\begin{aligned}\mathcal {W}^{\Delta }(\xi )\ge (r(\sigma (\theta ))(\mathcal {W}^{\Delta } (\sigma (\theta )))^{\alpha })^{1/\alpha }(r^{*}(\theta ))^{-1/\alpha }.\end{aligned}$$

Using this in (3.13), we get

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&\le -\Omega (\theta )\mathcal {P}(\theta ) +\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta )) -\beta \Omega (\theta )\eta ^{\Delta }(\theta )\frac{r(\sigma (\theta ))(\mathcal {W}^{\Delta } (\sigma (\theta )))^{\frac{\alpha +1}{\alpha }}}{\mathcal {W}^{\beta +1}(\eta (\sigma (\theta )))(r^{*}(\theta ))^{1/\alpha }}\\&= -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))-\beta \Omega (\theta )\eta ^{\Delta }(\theta )\frac{\mathcal {W}^{\frac{\beta - \alpha }{\alpha }}(\eta (\sigma (\theta )))}{(r^{*}(\theta ))^{1/\alpha }} \left[ \frac{\mathcal {V}(\sigma (\theta ))}{\Omega (\sigma (\theta ))}\right] ^{\frac{\alpha +1}{\alpha }}. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))\\&\quad -\frac{\beta \Omega (\theta )\eta ^{\Delta }(\theta )}{(r^{*}(\theta ))^{1/\alpha }\Omega ^{\frac{\alpha +1}{\alpha }}(\sigma (\theta ))}\mathcal {V}^{\frac{\alpha +1}{\alpha }}(\sigma (\theta )) \mathcal {W}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta ))). \end{aligned}$$

Since \(\beta <\alpha \), then using the fact that \(r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }\) is nonincreasing, there exists a constant \(c>0\) such that \(r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }\le c\), that is,

$$\begin{aligned}\mathcal {W}(\theta )\le \mathcal {W}(\theta _{2})+\int _{\theta _{2}}^{\theta } \frac{\Delta s}{r^{1/\alpha }(s)}= \mathcal {W}(\theta _{2})+\mathcal {A}(\theta )\end{aligned}$$

and hence

$$\begin{aligned}\mathcal {W}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta )))\ge c^{\frac{\beta -\alpha }{\alpha }}\mathcal {A}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta )))=\hat{c}\mathcal {A}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta ))),\end{aligned}$$

where \(\hat{c}=c^{\frac{\beta -\alpha }{\alpha }}\). Using both the cases in the last inequality, we have

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))-\frac{\beta \Omega (\theta )\eta ^{\Delta }(\theta )g(\theta )}{(r^{*}(\theta ))^{1/\alpha } \Omega ^{\frac{\alpha +1}{\alpha }}(\sigma (\theta ))}\mathcal {V}^{\frac{\alpha +1}{\alpha }}(\sigma (\theta )). \end{aligned}$$
(3.14)

By choosing

$$\begin{aligned}X=(\beta \Omega (\theta )\eta ^{\Delta }(\theta )g(\theta ))^{\frac{\alpha }{1+\alpha }}(r^{*}(\theta ))^{\frac{-1}{1+\alpha }}\frac{\mathcal {V}(\sigma (\theta ))}{\Omega (\sigma (\theta ))}\end{aligned}$$

and

$$\begin{aligned}Y= \left( \frac{\alpha }{1+\alpha }\right) ^{\alpha }\left( \frac{\Omega ^{\Delta } (\theta )}{\Omega (\sigma (\theta ))}\right) ^{\alpha }\left[ (\beta \Omega (\theta )\eta ^{\Delta }(\theta )g (\theta ))^{\frac{-\alpha }{1+\alpha }}\Omega (\sigma (\theta ))(r^{*}(\theta ))^{\frac{1}{1+\alpha }}\right] ^{\alpha },\end{aligned}$$

Lemma 2.1 with \(\gamma =\frac{1+\alpha }{\alpha }\) gives

$$\begin{aligned}\beta \frac{\Omega (\theta )\eta ^{\Delta }(\theta )g(\theta )}{(r^{*}(\theta ) )^{1/\alpha }\Omega ^{\frac{\alpha +1}{\alpha }}(\sigma (\theta ))}\mathcal {V}^{\frac{\alpha +1}{\alpha }} (\sigma (\theta ))-\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta )) \ge \frac{(\alpha /\beta )^{\alpha }}{(1+\alpha )^{\alpha +1}}\frac{r^{*}(\theta )\Omega ^{\alpha +1} (\sigma (\theta ))}{(\Omega (\theta )\eta ^{\Delta }(\theta )g(\theta ))^{\alpha }}.\end{aligned}$$

Therefore, the inequality (3.14) becomes

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )&\le -\Omega (\theta )\mathcal {P}(\theta )+\frac{(\alpha /\beta )^{\alpha }}{(\alpha +1)^{\alpha +1}} \frac{r^{*}(\theta )\Omega ^{\alpha +1}(\sigma (\theta ))}{( \Omega (\theta )\eta ^{\Delta }(\theta )g(\theta ))^{\alpha }}. \end{aligned}$$

Integrating the preceding inequality from \(\theta _{3}\) to \(\theta \), we have

$$\begin{aligned} \mathcal {V}(\theta )&\le \mathcal {V}(\theta _{3})-\int _{\theta _{3}}^{\theta }\left[ \Omega (s)\mathcal {P}(s)- \frac{(\alpha /\beta )^{\alpha }}{(\alpha +1)^{\alpha +1}}\frac{r^{*}(s)\Omega ^{\alpha +1} (\sigma (s))}{(\Omega (s)\eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s. \end{aligned}$$

Upon using (3.7), we obtain

$$\begin{aligned} \Omega (\theta )\int _{\theta }^{\infty }\mathcal {P}(s)\Delta s + \int _{\theta _{3}}^{\theta }\left[ \Omega (s)\mathcal {P}(s)-\frac{(\alpha /\beta )^{\alpha }}{(\alpha +1)^{\alpha +1}}\frac{r^{*}(s)\Omega ^{\alpha +1}(\sigma (s))}{(\Omega (s)\eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s\le \mathcal {V}(\theta _{3}). \end{aligned}$$

Taking the limsup of both sides of this inequality as \(\theta \rightarrow \infty \), we get at a contradiction to (3.6).

Case-b (\(\mathcal {W}(\theta )>0, r(\theta )(\mathcal {W}^{\Delta }(\theta ))^{\alpha }<0\)). This case follows from the proof of Theorem 3.1. This completes the proof. \(\square \)

Remark 3.4

We note that Theorem 3.3 holds when \(\int _{\theta }^{\infty }\mathcal {P}(s)\Delta s<\infty \). In the case when the integral does not exists as \(\theta \rightarrow \infty \), we see that condition (3.6) can be replaced by

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\int _{\theta _{0}}^{\theta } \left[ \Omega (s)\mathcal {P}(s)-\frac{(\alpha /\beta )^{\alpha }}{(\alpha +1)^{\alpha +1}}\frac{r^{*}(s)\Omega ^{\alpha +1}(\sigma (s))}{(\Omega (s)\eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s =\infty , \end{aligned}$$
(3.15)

and the conclusion of Theorem 2.1 holds.

In the next theorem, we present some Philos’s type oscillation criteria for the oscillation of Eq. (1.1). Let \(D\equiv \{(\theta , s)\in \mathbb {T}^{2}:\theta \ge s\ge \theta _{0}\}\) and \(D_{0}\equiv \{(\theta ,s)\in \mathbb {T}^{2}:\theta >s\ge \theta _{0}\}\). We define the function class \(\mathcal {R}\) as follows: \(\mathcal {M}_{1}\in C_{rd}(D,\mathbb {R})\) is said to be belongs to the class \(\mathcal {R}\) if

  1. 1.

    \(\mathcal {M}_{1}(\theta ,\theta )=0\) on D & \(\mathcal {M}_{1}(\theta ,s)>0\) on \(D_{0}\);

  2. 2.

    For each fixed \(\theta \), \(\mathcal {M}_{1}^{\Delta _{s}}(\theta ,s)\) is \(\Delta \)-integrable with respect to s;

  3. 3.

    \(m_{1}\in C_{rd}(D_{0},\mathbb {R}_{+})\) such that \(-[\mathcal {M}_{1}(\theta ,s)\phi (s)]^{\Delta _{s}} = m_{1}(\theta ,s)[\mathcal {M}_{1}(\theta ,s)\phi (s)]^{\frac{\alpha }{\alpha +1}}\) for all \((\theta ,s)\in D_{0}\) and

    $$\begin{aligned}\int _{\theta _{0}}^{\theta }m_{1}^{\frac{\alpha +1}{\alpha }}(\theta ,s)\Delta s<\infty \;\text {for each fixed value}\;\theta \ge \theta _{0},\end{aligned}$$

    where \(\phi (\theta )\in C_{rd}^{1}(\mathbb {T},\mathbb {R}_{+})\).

Theorem 3.5

Let \(\alpha >\beta \) and (3.1) holds. Assume that there exists a function \(\Omega \in C_{rd}^{1}(\mathbb {T},\mathbb {R}_{+})\), and \(\mathcal {M}\in \mathcal {R}\) such that

$$\begin{aligned} \limsup _{\theta \rightarrow \infty }\frac{1}{\mathcal {M}(\theta ,\theta _{0})}\int _{\theta _{3}}^{\theta } \left[ \Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s)-\frac{(\alpha /\beta )^{\alpha }}{\alpha ^{\alpha +1}} \frac{(-m_{1}(\theta ,s))^{\alpha +1}r^{*}(s)\phi ^{\alpha }(s)}{( \eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s=\infty , \end{aligned}$$
(3.16)

where \(g(\theta )=\hat{c}\mathcal {A}^{\frac{\beta -\alpha }{\alpha }}(\eta (\sigma (\theta )))\). Then every solution of Eq. (1.1) oscillates.

Proof

Suppose on the contrary that \(\mathcal {W}(\theta )\) is a nonoscillatory solution of Eq. (1.1). Define \(\mathcal {V}(\theta )\) as in (3.7), where \(\Omega (\theta )\in C_{rd}^{1}(\mathbb {T},\mathbb {R}_{+})\). Multiplying (3.14) by \(\mathcal {M}(\theta ,s)\), we get

$$\begin{aligned} \mathcal {V}^{\Delta }(\theta )\mathcal {M}(\theta ,s)\le -\Omega (\theta ) \mathcal {P}(\theta )\mathcal {M}(\theta ,s)&+\frac{\Omega ^{\Delta }(\theta )}{\Omega (\sigma (\theta ))}\mathcal {V}(\sigma (\theta ))\mathcal {M}(\theta ,s)\\&-\frac{\beta \Omega (\theta )\eta ^{\Delta }(\theta )g(\theta )}{(r^{*}(\theta ))^{1/\alpha } \Omega ^{\frac{\alpha +1}{\alpha }}(\sigma (\theta ))}\mathcal {V}^{\frac{\alpha +1}{\alpha }}(\sigma (\theta ))\mathcal {M}(\theta ,s). \end{aligned}$$

Integrating the above inequality with respect to s from \(\theta _{3}\) to \(\theta \) for \(\theta \ge \theta _{3}\), we get

$$\begin{aligned} \int _{\theta _{3}}^{\theta }\Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s) \Delta s&\le -\int _{\theta _{3}}^{\theta }\mathcal {V}^{\Delta }(s)\mathcal {M} (\theta ,s)\Delta s +\int _{\theta _{3}}^{\theta }\mathcal {M}(\theta ,s) \frac{\Omega ^{\Delta }(s)}{\Omega (\sigma (s))}\mathcal {V}(\sigma (s))\Delta s\\&-\int _{\theta _{3}}^{\theta } \mathcal {M}(\theta ,s) \frac{\beta \Omega (s)\eta ^{\Delta }(s)g(s)}{(r^{*}(s))^{1/\alpha }} \left( \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}\right) ^{\frac{\alpha +1}{\alpha }}\Delta s. \end{aligned}$$

Upon using the integration by parts formula, we obtain

$$\begin{aligned} \int _{\theta _{3}}^{\theta }\Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s)\Delta s&\le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})+\int _{\theta _{3}}^{\theta }\mathcal {M}^{\Delta }(\theta ,s)\mathcal {V}(\sigma (s))\Delta s \\&+\int _{\theta _{3}}^{\theta }\mathcal {M}(\theta ,s)\frac{\Omega ^{\Delta }(s)}{\Omega (\sigma (s))}\mathcal {V}(\sigma (s))\Delta s\\&-\int _{\theta _{3}}^{\theta } \mathcal {M}(\theta ,s) \frac{\beta \Omega (s)\eta ^{\Delta }(s)g(s)}{(r^{*}(s))^{1/\alpha }}\left( \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}\right) ^{\frac{\alpha +1}{\alpha }}\Delta s \end{aligned}$$
$$\begin{aligned} \int _{\theta _{3}}^{\theta }\Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s)\Delta s&\le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})+\int _{\theta _{3}}^{\theta } \Big [\left( \mathcal {M}^{\Delta }(\theta ,s)+ \mathcal {M}(\theta ,s)\frac{\Omega ^{\Delta }(s)}{\Omega (\sigma (s))}\right) \mathcal {V}(\sigma (s))\nonumber \\&-\mathcal {M}(\theta ,s) \frac{\beta \Omega (s)\eta ^{\Delta }(s)g(s)}{(r^{*}(s))^{1/\alpha }} \left( \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}\right) ^{\frac{\alpha +1}{\alpha }}\Big ]\Delta s\nonumber \\&\le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})+\int _{\theta _{3}}^{\theta } \Big [\frac{-m_{1}(\theta ,s)}{\Omega (\sigma (s))}\left( \mathcal {M}(\theta ,s)\phi (s)\right) ^{\frac{\alpha +1}{\alpha }}\mathcal {V}(\sigma (s))\nonumber \\&-\mathcal {M}(\theta ,s) \frac{\beta \Omega (s)\eta ^{\Delta }(s)g(s)}{(r^{*}(s))^{1/\alpha }}\left( \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}\right) ^{\frac{\alpha +1}{\alpha }}\Big ]\Delta s. \end{aligned}$$
(3.17)

By choosing

$$\begin{aligned} X= \left( \beta \mathcal {M}(\theta ,s) \Omega (s)\eta ^{\Delta }(s)g(s)\right) ^{\frac{\alpha }{\alpha +1}}(r^{*} (s))^{\frac{-1}{1+\alpha }}\frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))} \end{aligned}$$

and

$$\begin{aligned} Y{} & {} = \left( \frac{\alpha }{1+\alpha }\right) ^{\alpha }\left( -m_{1}(\theta ,s)\left( \mathcal {M}(\theta ,s)\phi (s)\right) ^{\frac{\alpha +1}{\alpha }}\right) ^{\alpha }\\{} & {} \quad \times \left[ (\beta \mathcal {M}(\theta ,s) \Omega (\theta )\eta ^{\Delta }(\theta )g(\theta ))^{\frac{-\alpha }{1+\alpha }} (r^{*}(\theta ))^{\frac{1}{1+\alpha }}\right] ^{\alpha }, \end{aligned}$$

Lemma 2.1 with \(\lambda =\frac{1+\alpha }{\alpha }\) gives

$$\begin{aligned}&-m_{1}(\theta ,s)\left( \mathcal {M}(\theta ,s)\phi (s)\right) ^{\frac{\alpha +1}{\alpha }} \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}-\mathcal {M}(\theta ,s) \frac{\beta \Omega (s)\eta ^{\Delta }(s)g(s)}{(r^{*}(s))^{1/\alpha }}\left( \frac{\mathcal {V}(\sigma (s))}{\Omega (\sigma (s))}\right) ^{\frac{\alpha +1}{\alpha }}\\&\ge \frac{(\alpha /\beta )^{\alpha }}{(1+\alpha )^{\alpha +1}}\frac{r^{*}(s) \phi ^{\alpha }(s)(-m_{1}(\theta ,s))^{\alpha +1}}{(\Omega (s)\eta ^{\Delta }(s)g(s))^{\alpha }}. \end{aligned}$$

Therefore, inequality (3.17) becomes

$$\begin{aligned} \int _{\theta _{3}}^{\theta }\Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s)\Delta s&\le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})+\int _{\theta _{3}}^{\theta } \left( \frac{\alpha ^{\alpha }}{\alpha ^{\alpha +1}}\frac{(-m_{1} (\theta ,s))^{\alpha +1}r^{*}(s)\phi ^{\alpha }(s)}{(\beta \eta ^{\Delta }(s)g(s))^{\alpha }}\right) \Delta s, \end{aligned}$$

which implies that,

$$\begin{aligned} \int _{\theta _{3}}^{\theta }\left[ \Omega (s)\mathcal {P}(s)\mathcal {M}(\theta ,s) -\left( \frac{\alpha ^{\alpha }}{\alpha ^{\alpha +1}}\frac{(-m_{1}(\theta ,s))^{\alpha +1}r^{*}(s) \phi ^{\alpha }(s)}{(\beta \eta ^{\Delta }(s)g(s))^{\alpha }}\right) \right] \Delta s&\le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})\\&\le \mathcal {M}(\theta ,\theta _{0}) \mathcal {V}(\theta _{3}), \end{aligned}$$

that is,

$$\begin{aligned} \frac{1}{\mathcal {M}(\theta ,\theta _{0})}\int _{\theta _{3}}^{\theta }&\left[ \Omega (s) \mathcal {P}(s)\mathcal {M}(\theta ,s)-\frac{\alpha ^{\alpha }}{\alpha ^{\alpha +1}} \frac{(-m_{1}(\theta ,s))^{\alpha +1}r^{*}(s)\phi ^{\alpha }(s)}{(\beta \eta ^{\Delta }(s)g(s))^{\alpha }}\right] \Delta s \\&\qquad \le \mathcal {M}(\theta ,\theta _{3}) \mathcal {V}(\theta _{3})\le \mathcal {V}(\theta _{3})<\infty . \end{aligned}$$

Taking limit supremum both sides, we get a contradiction to (3.16). This completes the proof. \(\square \)

Remark 3.6

Choose \(\phi (\theta )=1\), \(\alpha =1\), and \(\mathcal {M}(\theta ,s)=(\theta -s)^{\mu }\), then

$$\begin{aligned} m(t,s)=\frac{-[\mathcal {M}(\theta ,s)]^{\Delta _{s}}}{\sqrt{\mathcal {M}(\theta ,s)}}\le \mu (\theta -s)^{\frac{\mu -2}{2}}. \end{aligned}$$

Indeed,

$$\begin{aligned} \int _{\theta _{0}}^{\theta }m^{2}(\theta ,s)\Delta s\le \frac{\mu }{\mu -1}(\theta -s)^{\mu -1}<\infty \end{aligned}$$

for \(\mu >1\) and for each fixed value \(\theta >\theta _{0}\). We may note that the above integral does not exists for \(0\le \mu \le 1\). As a result, in this case, we can not apply Lemma 2.1 to prove the theorem. Therefore, conditions (3) in the definition of function class \(\mathcal {R}\) is necessary. Thus, Theorem 3.5 improved or generalised the corresponding result obtained by Saker et al. [27] and Wu et al. [32].

4 Conclusion and examples

For the sake of completeness, the following examples are presented.

Example 4.1

Consider the Euler dynamic equation

$$\begin{aligned}{}[r(\theta )( \mathcal {W}^{\Delta }(\theta ))^{\alpha }]^{\Delta }+\mathcal {P} (\theta )\mathcal {W}^{\beta }(\theta /2)=0,\,\theta \ge \theta _{0}=1, \end{aligned}$$
(4.1)

where \(\lambda >0\), \(\alpha = 5\), \(\beta = 3\), \(\eta (\theta )=\frac{\theta }{2}\), \(\mathcal {P}(\theta )=\frac{\lambda }{\theta \sigma (\theta )}\) and \(r(\theta )=\theta ^{6}\). Indeed, \(\int _{\theta _{0}}^{\infty }r^{-1/\alpha }(s)\Delta s=5\) and \(\hat{\mathcal {A}}(\theta )=\frac{5}{\root 5 \of {\theta }}\). It is not difficult to see that all the conditions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1 Eq. (4.1) oscillates for suitable \(\lambda \).

Example 4.2

Let \(\mathbb {T}=\mathbb {R}\). Consider

$$\begin{aligned}{}[r(\theta )(\mathcal {W}'(\theta ))^{\alpha }]'+\mathcal {P} (\theta )\mathcal {W}^{\beta }(\theta /5)=0,\,\theta \ge \theta _{0}=1 \end{aligned}$$
(4.2)

with \(\alpha =1\), \(\beta =3\), \(\eta (\theta )=\theta /5\), \(\mathcal {P}(\theta )=(1+\theta )e^{2t}\), \(r(\theta )=e^{\theta }\) and hence \(\hat{\mathcal {A}}(\theta )=e^{-\theta }\). It is easy to verify that (2.4), (2.5) and (3.4) hold. Then by Theorem 3.2, Eq. (4.2) oscillates.

The oscillatory behaviour of Eq. (1.1) is the focus of this work when \((\mathcal {A}_{2})\) holds. To our best knowledge, the first work on oscillations of a general class of dynamic equations is [12]. This work, on the other hand, is about Eq.(1.1) with \(\eta (\theta )=\theta \). An exciting topic is how to construct oscillation criteria for Eq. (1.1). Hence, in this work, we find sufficient conditions to ensure that all solutions only oscillate. Our findings generalised or enhanced the findings of [2, 10,11,12, 15, 32]. Even for the discrete situation \(\mathbb {T} = \mathbb {Z}\), our findings are novel.

Finally, we would like to point out that the results reported in this paper can be extended to second order general neutral dynamic equations of the form

$$\begin{aligned} (r(\theta )(y^{\Delta }(\theta ))^{\alpha })^{\Delta }+\mathcal {P} (\theta )\mathcal {W}^{\beta }(\eta (\theta ))=0,\;\theta \in \mathbb {T}_{0}, \end{aligned}$$

where \(y(\theta )=\mathcal {W}(\theta )+q(\theta )\mathcal {W}(\tau (\theta ))\), \(q\in C_{rd}(\mathbb {T}_{0},\mathbb {R})\), \(\tau (\theta )\in C_{rd}(\mathbb {T}_{0},\mathbb {T})\) with \(\tau (\theta )\le \theta \), \(\tau (\theta )\rightarrow \infty \) as \(\theta \rightarrow \infty \), and \(\mathcal {P}(\theta ), \eta (\theta ), \alpha , \beta \) are as in (1.1).