1. Introduction
The calculus theory of time scales was introduced by Hilger [
1] in order to unify, extend and generalize ideas from discrete calculus,
quantum calculus and continuous calculus to arbitrary time scales calculus. A time scale \(\mathbb{T}\) is an arbitrary closed subset of the real numbers \(\mathbb{R}\).
For an introduction to time scales calculus and dynamic equations, see Bohner and Peterson books [
2,
3].
We are concerned with the oscillation behavior of all solutions of the second order nonlinear dynamic equation with damping
on a time sceles \(\mathbb{T}\) which is unbounded above
\begin{equation}\label{e1.1}
\begin{aligned}
(r(t)(x^{\Delta}(t))^{\alpha})^{\Delta}-p(t)(x^{\Delta}(t))^{\alpha}+q(t)f(x(t))=0,
\end{aligned}
\end{equation}
(1)
where \(t\in\mathbb{T}\), \(t\geqslant t_{0}>0\).
The equation will be studied under the following assumptions:
(H1) \(r(t),q(t)\) are positive real-valued rd-continuous functions on \(\mathbb{T}\), \(p(t)< 0,\) \(\frac{p(t)}{r(t)}\in\mathcal{R}^{+}\),
and \(\alpha\) is the quotient of two positive odd numbers;
(H2) \(f:\mathbb{R}\rightarrow \mathbb{R}\) is such that \(uf(u)>0\) for \(u\neq 0\);
(H3) \(f:\mathbb{R}\rightarrow \mathbb{R}\) is such that \(f(u)\geqslant ku^{\alpha}\) for \(u\neq 0\) and some \(k>0\);
(H4) \(\int^{\infty}_{t_{0}}\left(\frac{1}{r(t)}e_{\ominus\frac{p(s)}{r(s)}}(t,t_{0}\right)^{\frac{1}{\alpha}}\Delta t=\infty\).
We only consider these solutions of (1) which exist on some half-line \([t_0,\infty)_{\mathbb{T}}\) and
satisfy \(\sup\{|x(t)|:t_1\leqslant t0\), for any \(t_1 \geqslant t_0\).
If \(x(t)\) satisfies (1) on \([t_{1},\infty)_{\mathbb{T}}\) for some \(t_{1}\geqslant t_{0},\) then the function \(x(t)\) is called a solution of (1).
A solution \(x(t)\) of (1) is said to be oscillatory if it is neither eventually positive
nor eventually negative, otherwise it is called nonoscillatory.
The equation itself is called oscillatory if all of its solutions are oscillatory.
In the last decades, much interest has focused on obtaining
sufficient conditions for the oscillation of solutions of
different classes of dynamic equations on time scales, and we
refer the reader to the papers [
4,
5 ,
6,
7,
8,
9,
10,
11,
12,
13,
14]
In particular, much work has been done on the following dynamic equation
$$(p(t)x^{\Delta}(t))^{\Delta}+q(t)x(\sigma(t))=0,$$
$$(p(t)x^{\Delta}(t))^{\Delta}+q(t)(f\circ x(\sigma(t)))=0.$$
Erbe
et al. [
15] considered the second-order nonlinear damped dynamic equation
$$(r(t)(x^{\Delta}(t))^{\gamma})^{\Delta}+p(t)(x^{\Delta\sigma}(t))^{\gamma}+q(t)f(x(\tau(t)))=0,$$
and obtained some oscillation criteria.
Saker
et al. [
16] studied the oscillation criteria for difference equations with damping terms
$$\Delta(a_{n}(\Delta x_{n})^{\gamma})+p_{n}(\Delta x_{n})^{\gamma}+q_{n}f(x_{n+1})=0,$$
and obtained some oscillation criteria.
Deng
et al.
17 researched oscillation criteria for second order nonlinear delay dynamic equations
$$(r(t)|x^{\Delta}(t)|^{\gamma-1}x^{\Delta}(t))^{\Delta}+p(t)f(x(\tau(t)))=0,$$
Agwo
et al. [
18] considered the oscillation criteria of second order half linear delay dynamic equation
$$(r(t)g(x^{\Delta}(t)))^{\Delta}+p(t)f(x(\tau(t)))=0,$$
and obtained some oscillation criteria.
Note that in the special case when \(\mathbb{T}=\mathbb{R}\), (1) becomes the second-order nonlinear damped differential equation
$$(r(t)(x'(t))^{\alpha})’+p(t)(x'(t))^{\alpha}+q(t)f(x(t))=0,\ \ t\in\mathbb{R},$$
and when \(\mathbb{T}=\mathbb{Z}\), (1) becomes the second-order nonlinear damped difference equation
$$\Delta(r(t)(\Delta x(t))^{\alpha})+p(t)(\Delta x(t))^{\alpha}+q(t)f(x(t))=0,\ \ t\in\mathbb{Z},$$
where \(\Delta x(t)=x(t+1)-x(t)\).
In this paper, we replace \(e_{p}(t,s)\) with \(e_{\ominus p}(t,s)\) and this is difference between our paper and other articles. Our result extend and improve some well-known oscillation results. The paper is organized as follows. In Section 2, we present some
basic definitions and useful results from the theory of calculus
on time scales on which we rely in the later section. In section
3, we intend to use the Riccati transformation technique, integral
averaging technique, and inequalities to obtain some sufficient
conditions for oscillation for oscillation of every solution of
(1). In section 4, we give a example to illustrate
our results. The last section is devoted to remarks and comments
concerning our results. We also formulate possible
new research directions.
2. Preliminaries
Lemma 2.1.[2]
We say that a function \(p:\mathbb{T}\rightarrow \mathbb{R}\) is regressive provided
$$1+\mu(t)p(t)\neq 0$$
for all \(t\in {{\mathbb{T}}^{\kappa }}\) holds.
We define the set \(\mathcal{R}^{+}\) of all positively regressive elements of \(\mathcal{R}\) by
$$\mathcal{R}^{+}=\left\{p\in\mathcal{R}: 1+\mu(t)p(t)>0\ for \ all \ t\in\mathbb{T}\right\}.$$
\(\mathcal{R}^{+}\) is a subgroup of \(\mathcal{R}.\)
Lemma 2.2.[2]
Show that if \(p\in\mathcal{R},\) then the function \(\ominus p\) defined by
$$(\ominus p)(t):=-\frac{p(t)}{1+\mu(t)p(t)}$$
for all \(t\in \mathbb{T}^{\kappa }\) are also elements of \(\mathcal{R}\). Since \(p\in\mathcal{R}^{+},\) we have \(\ominus p\in\mathcal{R}^{+}\) by Lemma 2.1.
If \(p\in\mathcal{R}^{+},\) then \(e_{p}(t,t_{0})>0\) for all \(t\in\mathbb{T}.\)
Lemma 2.3.[2]
If \(p\in\mathcal{R},\) then we have define the exponential function by
$$e_{p}(t,s)=\exp\left(\int^{t}_{s}\xi_{\mu(\tau)}(p(\tau))\Delta\tau\right)\ \ \ for \ \ s,t\in\mathbb{T},$$
where \(\xi_{h}(z)=\frac{1}{h}\log(1+zh),\ h>0.\)
Lemma 2.4.[2]
If \(p,q\in\mathcal{R},\) then
(1) \(e_{p}(t,t)\equiv 1;\)
(2) \(e_{p}(\sigma(t),s)=(1+\mu(t)p(t))e_{p}(t,s);\)
(3) \(\frac{1}{e_{p}(t,s)}=e_{\ominus p}(t,s).x\)
Lemma 2.5.[2]
Let \(y\in C_{rd}\) and \(p\in\mathcal{R}^{+},\) we have
\begin{equation}\label{e2.5}
\begin{aligned}
(ye_{\ominus p}(\cdot, t_{0}))^{\Delta}(t)&=y^{\Delta}(t)e_{\ominus p}(\sigma(t),t)+y(t)(\ominus p)(t)e_{\ominus p}(t,t_{0})\\
&=y^{\Delta}(t)e_{\ominus p}(\sigma(t),t)+y(t)\frac{(\ominus p)(t)}{1+\mu(t)(\ominus p)(t)}e_{\ominus p}(\sigma(t),t)\\
&=\left(y^{\Delta}(t)-(\ominus(\ominus p))(t)y(t)\right)e_{\ominus p}(\sigma(t),t)\\
&=(y^{\Delta}(t)-p(t)y(t))e_{\ominus p}(\sigma(t),t). \nonumber
\end{aligned}
\end{equation}
Lemma 2.6.[6]
Assume that \(\alpha>0\) is the ratio of positive odd integers and \(x^\alpha(t) \in C_{rd}^1(I,\mathbb{R})\). Then
\begin{equation}
(x^\alpha(t))^\Delta\geqslant
\begin{cases}
\alpha(x(\sigma(t)))^{\alpha-1}y^\Delta(t),\ \ \ 0< \alpha\leqslant 1,\\
\alpha(x(t))^{\alpha-1}x^\Delta(t),\ \ \ \ \ \ \ \alpha\geqslant 1.\nonumber
\end{cases}
\end{equation}
3. Main Results
Now, we are in a position to state and prove some new results which guarantee that every solution of (1) oscillates.
Theorem 3.1.
Assume that (H1)-(H4) hold. Furthermore, assume that there exists a positive real rd-continuous differentiable function \(v(t)\) such that
\begin{equation}\label{e3.1}
\begin{aligned}
\underset{t\rightarrow \infty}{\lim\sup}\int^{t}_{t_{0}}\left(kv(s)q(s)-\frac{\psi^{\alpha+1}(s)r(s)}{(\alpha+1)^{\alpha+1}v^{\alpha}(s)}\right)\Delta s=\infty,
\end{aligned}
\end{equation}
(2)
where
\begin{equation}\label{e3.2}
\begin{aligned}
\psi(t)=\frac{r(t)v^{\Delta}(t)+v(t)p(t)}{r(t)}
\end{aligned}
\end{equation}
(3)
Then every solution of (1) is oscillatory.
Proof.
Suppose to the contrary that \(x(t)\) is a nonoscillatory solution of (1).
With loss of generality, we may assume that \(x(t)>0\) for \(t\geqslant t_{1}>t_{0}\).
We shall consider only this cases, since in view of (H2), the proof of the case when \(x(t)\) is eventually negative is similar.
Now, we claim that \(x^{\Delta}(t)\) has a fixed sign on the interval \([t_{2},\infty)\) for some \(t_{2}\geqslant t_{1}\). From (1),
since \(q(t)>0\) and \(f(x(t))>0\), we have
$$(r(t)(x^{\Delta}(t))^{\alpha})^{\Delta}-p(t)(x^{\Delta}(t))^{\alpha}=-q(t)f(x(t))< 0,$$
i.e.,
$$(r(t)(x^{\Delta}(t))^{\alpha})^{\Delta}-p(t)(x^{\Delta}(t))^{\alpha}< 0.$$
By setting \(y(t)=r(t)(x^{\Delta}(t))^{\alpha}\), we immediately see that \(y^{\Delta}(t)-\frac{p(t)}{r(t)}y(t)< 0\),
by Lemma 2.2 and Lemma 2.3, we have \((y(t)e_{\ominus\frac{p}{r}})^{\Delta} 0\) on \([t_{2}, \infty)\) for some \(t_{2}\geqslant t_{1}\). Then in view (1) and (H1) we have
$$x(t)>0, x^{\Delta}(t)> 0, (r(t)(x^{\Delta}(t))^{\alpha})^{\Delta}< 0,t\geqslant t_{2}.$$
We see that for \(t\geqslant t_{3}=\sigma (t_{2})\)
\begin{equation}\label{e3.3}
\begin{aligned}
r(t)(x^{\Delta}(t))^{\alpha}> r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}, x^{\alpha}(\sigma(t))> x^{\alpha}(t).
\end{aligned}
\end{equation}
(4)
Define the function \(w(t)\) by the Riccati substitution
\begin{equation}\label{e3.4}
\begin{aligned}
w(t):= v(t)r(t)\left(\frac{x^{\Delta}(t)}{x(t)}\right)^{\alpha}, t\geqslant t_{2}.
\end{aligned}
\end{equation}
(5)
In view of (1), we have
\begin{equation}\label{e3.5}
\begin{aligned}
w^{\Delta}(t)=&r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}\left(\frac{v(t)}{x^{\alpha}(t)}\right)^{\Delta}+\frac{v(t)(r(t)(x^{\Delta}(t))^{\alpha})^{\Delta}}{x^{\alpha}(t)}\\
=&r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}\frac{v^{\Delta}(t)x^{\alpha}(t)-v(t)(x^{\alpha}(t))^{\Delta}}{x^{\alpha}(t)x^{\alpha}(\sigma(t))}\\
&+\frac{v(t)}{x^{\alpha}(t)}(p(t)(x^{\Delta}(t))^{\alpha}-q(t)f(x(t)))\\
=&-\frac{v(t)q(t)f(x(t))}{x^{\alpha}(t)}+v(t)\frac{p(t)(x^{\Delta}(t))^{\alpha}}{x^{\alpha}(t)}+\frac{v^{\Delta}(t)}{v(\sigma(t))}w(\sigma(t))\\
&-\frac{v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}(x^{\alpha}(t))^{\Delta}}{x^{\alpha}(t)x^{\alpha}(\sigma(t))},
\end{aligned}
\end{equation}
(6)
Using (4) in (6) and by (H3), we have
\begin{equation}
\begin{aligned}
w^{\Delta}(t)\leqslant&-kv(t)q(t)+v(t)\frac{p(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}}{r(t)x^{\alpha}(\sigma(t))}+\frac{v^{\Delta}(t)}{v(\sigma(t))}w(\sigma(t))\\
&-\frac{v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}(x^{\alpha}(t))^{\Delta}}{x^{\alpha}(t)x^{\alpha}(\sigma(t))}\\
=&-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}(x^{\alpha}(t))^{\Delta}}{x^{\alpha}(t)x^{\alpha}(\sigma(t))}, \nonumber
\end{aligned}
\end{equation}
where \(\psi(t)\) as defined as (3).
By Lemma 2.5, if \(0< \alpha\leqslant1\), we have
\begin{equation}\label{e3.6}
\begin{aligned}
w^{\Delta}(t)\leqslant&-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))
-\frac{\alpha v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}x^{\Delta}(t)}{x^{\alpha}(t)x(\sigma(t))}\\
\leqslant&-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{\alpha v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}x^{\Delta}(t)}{x^{\alpha+1}(\sigma(t))},
\end{aligned}
\end{equation}
(7)
if \(\alpha>1\), we have
\begin{equation}\label{e3.7}
\begin{aligned}
w^{\Delta}(t)\leqslant&-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))
-\frac{\alpha v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}x^{\Delta}(t)}{x(t)x^{\alpha}(\sigma(t))}\\
\leqslant&-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{\alpha v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}x^{\Delta}(t)}{x^{\alpha+1}(\sigma(t))}.
\end{aligned}
\end{equation}
(8)
Thus, by (7) and (8), we obtain
\begin{equation}\label{e3.8}
\begin{aligned}
w^{\Delta}(t)&\leqslant
-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{\alpha v(t)r(\sigma(t))(x^{\Delta}(\sigma(t)))^{\alpha}x^{\Delta}(t)}{x^{\alpha+1}(\sigma(t))}\\
&=-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{\alpha v(t)r^{\frac{1}{\alpha}}(t)x^{\Delta}(t)}{v(\sigma(t))x(\sigma(t))r^{\frac{1}{\alpha}}(t)}w(\sigma(t))\\
&\leqslant-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))
-\frac{\alpha v(t)r^{\frac{1}{\alpha}}(\sigma(t))x^{\Delta}(\sigma(t))}{v(\sigma(t))x(\sigma(t))r^{\frac{1}{\alpha}}(t)}w(\sigma(t))\\
&=-kv(t)q(t)+\frac{\psi(t)}{v(\sigma(t))}w(\sigma(t))-\frac{\alpha v(t)}{v^{\frac{\alpha+1}{\alpha}}(\sigma(t))r^{\frac{1}{\alpha}}(t)}w^{\frac{\alpha+1}{\alpha}}(\sigma(t))
\end{aligned}
\end{equation}
(9)
hold for all \(\alpha>0\).
Then, using the inequality [
19]
$$Bu-Cu^{\frac{1+\alpha}{\alpha}}\leqslant \frac{\alpha^{\alpha}}{(1+\alpha)^{\alpha+1}}\frac{B^{\alpha+1}}{C^{\alpha}},$$
let \(B=\frac{\psi(t)}{v(\sigma(t))},\) \(C=\frac{\alpha v(t)}{v^{\frac{\alpha+1}{\alpha}}(\sigma(t))r^{\frac{1}{\alpha}}(t)}\) and \(u=w(\sigma(t)),\)
we obtain
\begin{equation}\label{e3.9}
\begin{aligned}
w^{\Delta}(t)&\leqslant -kv(t)q(t)
+\frac{\alpha^{\alpha}}{(1+\alpha)^{1+\alpha}}\left(\frac{\psi(t)}{v(\sigma(t))}\right)^{\alpha+1}
\left(\frac{v^{\frac{\alpha+1}{\alpha}}(\sigma(t))r^{\frac{1}{\alpha}}(t)}{\alpha v(t)}\right)^{\alpha}\\
&=-kv(t)q(t)+\frac{\psi^{\alpha+1}(t)r(t)}{(\alpha+1)^{\alpha+1}v^{\alpha}(t)}.
\end{aligned}
\end{equation}
(10)
Integrating (10) from \(t_{3}\) to \(t\), we obtain
$$w(t)-w(t_{3})\leqslant-\int^{t}_{t_{3}}\left(kv(s)q(s)-\frac{\psi^{\alpha+1}(s)r(s)}{(\alpha+1)^{\alpha+1}v^{\alpha}(s)}\right)\Delta s,$$
which yields
$$\int^{t}_{t_{3}}\left(kv(s)q(s)-\frac{\psi^{\alpha+1}(s)r(s)}{(\alpha+1)^{\alpha+1}v^{\alpha}(s)}\right)\Delta s\leqslant w(t_{3})-w(t)< w(t_{3})$$
for all large \(t\). This is contrary to (2).
Next, we consider Case(2). Then exists \(t_{2}\geqslant t_{1}\) such that \((x^{\Delta}(t))^{\alpha}< 0\) for \(t\geqslant t_{2}\).
Define the function \(u(t)=-r(t)(x^{\Delta}(t))^{\alpha}\). Then from (1), we have
$$u^{\Delta}(t)-\frac{p(t)}{r(t)}u(t)\geqslant 0.$$
Thus
$$u(t)\geqslant u(t_{2})e_{\ominus\frac{p(t)}{r(t)}}(t,t_{2}),$$
so that
$$(x^{\Delta}(t))^{\alpha}\leqslant-u(t_{2})\left(\frac{1}{r(t)}e_{\ominus\frac{p(t)}{r(t)}}(t,t_{2})\right),$$
i.e.
\begin{equation}\label{e3.10}
\begin{aligned}
x^{\Delta}(t)\leqslant \left(-u(t_{2})\left(\frac{1}{r(t)}e_{\ominus\frac{p(t)}{r(t)}}(t,t_{2})\right)\right)^{\frac{1}{\alpha}}.
\end{aligned}
\end{equation}
(11)
Integrating (11) from \(t_{2}\) to \(t\), we get
$$x(t)-x(t_{2})\leqslant(r(t_{2}))^{\frac{1}{\alpha}}x^{\Delta}(t_{2})\int^{t}_{t_{2}}\left(\frac{1}{r(s)}e_{\ominus\frac{p(m)}{r(m)}}(s,t_{2})
\right)^{\frac{1}{\alpha}}\Delta s.$$
Condition (H4) implies that \(x(t)\) is eventually negative, which is a contradiction. The proof is complete.
Corollary 3.2.
Assume that (H1)-(H4) hold. If
$$\limsup\limits_{t\rightarrow\infty}\int^{t}_{t_{0}}\left(kq(s)-\frac{p^{\alpha+1}(s)}{(\alpha+1)^{\alpha+1}r^{\alpha}(t)}\right)\Delta s=\infty,$$
the every solution of (1) is oscillatory.
Corollary 3.3.
Assume that (H1)-(H4) hold. If there is \(\lambda \geqslant1\) such that
$$\limsup\limits_{t\rightarrow\infty}\int^{t}_{t_{0}}\left(ks^{\lambda}q(s)-\frac{(r(s)(s^{\lambda})^{\Delta}
-s^{\lambda}p(s))^{\alpha+1}}{(\alpha+1)^{\alpha+1}(s^{\lambda})^{\alpha}r^{\alpha}(s)}\right)\Delta s=\infty,$$
the every solution of (1) is oscillatory.
Corollary 3.4.
Assume that (H1)-(H4) hold. If
$$\limsup\limits_{t\rightarrow\infty}\int^{t}_{t_{0}}\left(kR(s,t_{0})q(s)-\frac{(r(s)(R(s,t_{0}))^{\Delta}
-R(s,t_{0})p(s))^{\alpha+1}}{(\alpha+1)^{\alpha+1}(R(s,t_{0}))^{\alpha}r^{\alpha}(s)}\right)\Delta s=\infty,$$
where \(R(t,t_{0})=\int^{t}_{t_{0}}\frac{1}{r(s)}\Delta s,\) the every solution of (1) is oscillatory.
Theorem 3.5.
Assume that (H1)-(H4) hold. Furthermore, suppose that \(v(t)\) be as defined in Theorem 3.1 and a function
\(H\in C(\mathbb{D},\mathbb{T})\), where \(\mathbb{D}:=\{(t,s):t\geqslant s\geqslant t_{0}\},\) such that
\begin{equation}
\begin{aligned}
&H(t,t)=0, \ \ for \ t\geqslant t_{0},\\
&H(t,s)>0, \ \ for \ (t,s)\in\mathbb{D}_{0}, \nonumber
\end{aligned}
\end{equation}
where \(\mathbb{D}_{0}:=\{(t,s):t>s\geqslant t_{0}\}\), and \(H\) has a nonpositive continuous partial derivative \(H^{\Delta_{s}}(t,s):=\partial H(t,s)/\partial s\)
on \(\mathbb{D}_{0}\) with respect to the second variable and satisfies
\begin{equation}\label{e3.11}
\begin{aligned}
\limsup\limits_{t\rightarrow\infty}\frac{1}{H(t,t_{0})}\int^{t}_{t_{0}}\left(kH(t,s)v(s)q(s)
-\frac{v^{\alpha+1}(\sigma(t))r(s)A^{\alpha+1}(t,s)}{(1+\alpha)^{1+\alpha}H^{\alpha}(t,s)v^{\alpha}(s)}\right)\Delta s=\infty,
\end{aligned}
\end{equation}
(12)
where
$$A(t,s)=H(t,s)\frac{\psi(s)}{v(\sigma(t))}+H^{\Delta_{s}}(t,s).$$
Then every solution of (1) is oscillatory.
Proof.
Suppose to the contrary that \(x(t)\) is a nonoscillatory solution of (1)
and let \(t_{1}\geqslant t_{0}\) be such that \(x(t)\neq0\) for all \(t\geqslant t_{1}\), so with loss of generality,
we may assume that \(x(t)\) is an eventually positive solution of (1) with \(x(t)>0\) for \(t\geqslant t_{1}\) sufficiently large. In view of Theorem 3.1
we see that \(x^{\Delta}(t)\) is eventually negative or eventually positive. If \(x^{\Delta}(t)\) is eventually negative,
we are then back to Case (2) of Theorem 3.1 and we obtain a contradiction.
If \(x^{\Delta}(t)\) is eventually positive, we assume that there exists \(t_{2}> t_{1}\) such that \(x^{\Delta}(t)\geqslant 0\)
for \(t_{2}\geqslant t_{1}\) and proceed as in the proof of Case (1) of Theorem 3.1 and get
(9). From (9), it follows that
\begin{equation}\label{e3.12}
\begin{aligned}
\int^{t}_{t_{2}}kH(t,s)v(s)q(s)\Delta s\leqslant &-\int^{t}_{t_{2}}H(t,s)w^{\Delta}(s)\Delta s+\int^{t}_{t_{2}}H(t,s)\frac{\psi(s)}{v(\sigma(s))}w(\sigma(s))\Delta s\\
&-\int^{t}_{t_{2}}H(t,s)\frac{\alpha v(s)}{ v^{\frac{\alpha+1}{\alpha}}(\sigma(s))r^{\frac{1}{\alpha}}(s)}w^{\frac{\alpha+1}{\alpha}}(\sigma(s))\Delta s,
\end{aligned}
\end{equation}
(13)
Using the integration by parts formula, we have
\begin{equation}\label{e3.13}
\begin{aligned}
\int^{t}_{t_{2}}H(t,s)w^{\Delta}(s)\Delta s&=H(t,s)w(s)|^{t}_{t_{2}}-\int^{t}_{t_{2}}H^{\Delta_{s}}(t,s)w(\sigma(s))\Delta s\\
&=-H(t,t_{2})w(t_{2})-\int^{t}_{t_{2}}H^{\Delta_{s}}(t,s)w(\sigma(s))\Delta s,
\end{aligned}
\end{equation}
(14)
where \(H(t,t)=0.\) Substituting (14) into (13), we obtain
\begin{equation}
\begin{aligned}
\int^{t}_{t_{2}}kH(t,s)v(s)q(s)\Delta s\leqslant &H(t,t_{2})w(t_{2})+\int^{t}_{t_{2}}H^{\Delta_{s}}(t,s)w(\sigma(s))\Delta s
+\int^{t}_{t_{2}}H(t,s)\frac{\psi(s)}{v(\sigma(s))}w(\sigma(s))\Delta s\\
&-\int^{t}_{t_{2}}H(t,s)\frac{\alpha v(s)}{ v^{\frac{\alpha+1}{\alpha}}(\sigma(s))r^{\frac{1}{\alpha}}(s)}w^{\frac{\alpha+1}{\alpha}}(\sigma(s))\Delta s. \nonumber
\end{aligned}
\end{equation}
Hence,
\begin{equation}
\begin{aligned}
\int^{t}_{t_{2}}kH(t,s)v(s)q(s)\Delta s\leqslant &H(t,t_{2})w(t_{2})+\int^{t}_{t_{2}}\left(H(t,s)\frac{\psi(s)}{v(\sigma(s))}+H^{\Delta_{s}}(t,s)\right)w(\sigma(t))\Delta s\\
&-\int^{t}_{t_{2}}H(t,s)\frac{\alpha v(s)}{ v^{\frac{\alpha+1}{\alpha}}(\sigma(s))r^{\frac{1}{\alpha}}(s)}w^{\frac{\alpha+1}{\alpha}}(\sigma(t))\Delta s. \nonumber
\end{aligned}
\end{equation}
Then, using the inequality [
19]
$$Bu-Cu^{\frac{1+\alpha}{\alpha}}\leqslant \frac{\alpha^{\alpha}}{(1+\alpha)^{\alpha+1}}\frac{B^{\alpha+1}}{C^{\alpha}},$$
let \(B=H(t,s)\frac{\psi(t)}{v(\sigma(t))},\) \(C=H(t,s)\frac{\alpha v(t)}{v^{\frac{\alpha+1}{\alpha}}(\sigma(t))r^{\frac{1}{\alpha}}(t)}\) and \(u=w(\sigma(t)),\)
we obtain
\begin{equation}
\begin{aligned}
\int^{t}_{t_{2}}kH(t,s)v(s)q(s)\Delta s\leqslant &H(t,t_{2})w(t_{2})+\int^{t}_{t_{2}}\frac{v^{\alpha+1}(\sigma(t))r(s)
A^{\alpha+1}(t,s)}{(1+\alpha)^{1+\alpha}H^{\alpha}(t,s)v^{\alpha}(s)}\Delta s. \nonumber
\end{aligned}
\end{equation}
Then for all \(t\geqslant t_{2}\), we have
$$\int^{t}_{t_{2}}\left(kH(t,s)v(s)q(s)-\frac{v^{\alpha+1}(\sigma(t))r(s)A^{\alpha+1}(t,s)}{(1+\alpha)^{1+\alpha}H^{\alpha}(t,s)v^{\alpha}(s)}\right)
\Delta s\leqslant H(t,t_{2})w(t_{2}),$$
and this implies that
$$\frac{1}{H(t,t_{2})}\int^{t}_{t_{2}}\left(kH(t,s)v(s)q(s)-\frac{v^{\alpha+1}(\sigma(t))r(s)A^{\alpha+1}(t,s)}{(1+\alpha)^{1+\alpha}
H^{\alpha}(t,s)v^{\alpha}(s)}\right)\Delta s\leqslant w(t_{2})$$
for all large \(t\), which contradicts (12). The proof is complete.
4. Examples
Example 4.1.
Consider the equation
\begin{equation}\label{e4.1}
\begin{aligned}
((x^{\Delta}(t))^{\alpha})^{\Delta}+\frac{1}{t}(x^{\Delta}(t))^{\alpha}+tx^{\alpha}(t)=0,
\end{aligned}
\end{equation}
(15)
where \(r(t)=1\), \(p(t)=-\frac{1}{t}\), \(q(t)=t\), \(\mu(t)=\frac{t}{2},\) \(f(x(t))=x^{\alpha}(t)\) with \(k=1\), and \(\alpha>0\).
It is clear that conditions (H1)-(H4) are satisfied. Letting \(v(t)=t\), \(\mathbb{T}=[1,\infty)\), by Lemma 2.3 and Lemma 2.4 we have
\begin{equation}
\begin{aligned}
e_{\frac{p}{r}}(t,t_{0})&=e_{-\frac{1}{t}}(t,1)=\exp\left(\int_{1}^{t}\xi_{0}(-\frac{1}{\tau})d\tau\right)\\
&=\exp\left(\int_{1}^{t}(-\frac{1}{\tau})d\tau\right)
=\frac{1}{t}, \nonumber
\end{aligned}
\end{equation}
$$e_{\ominus\frac{p}{r}}(t,t_{0})=\frac{1}{e_{\frac{p}{r}}(t,t_{0})}=t,$$
$$\int_{t_{0}}^{\infty}\left(\frac{1}{r(t)}e_{\ominus\frac{p}{r}}(t,t_{0})\right)^{\frac{1}{\alpha}}\Delta t
=\int_{t_{0}}^{\infty}\left(t\right)^{\frac{1}{\alpha}}\Delta t=\infty,$$
$$\psi(t)=v^{\Delta}(t)+v(t)p(t)=0.$$
Hence,
$$\underset{t\rightarrow \infty}{\lim\sup}\int^{t}_{t_{0}}\left(kv(s)q(s)
-\frac{\psi^{\alpha+1}(s)r(s)}{(\alpha+1)^{\alpha+1}v^{\alpha}(s)}\right)\Delta s=\underset{t\rightarrow \infty}{\lim\sup}\int^{t}_{t_{0}}(t^{2})\Delta s=\infty.$$
That is (2) holds. By Theorem 3.1 we see that (15) is oscillatory.
5. Conclusion
The results of this article are presented in a form which is essentially new and of high degree of generalize. In this article, using generalized Riccati transformation and inequality technique, we offer some new sufficient conditions which insure that any solution of dynamic equation (1) oscillates. In addition, we can try to get some oscillation behavior of dynamic equation (1) if \(q(t)< 0\) or \(\int^{\infty}_{t_{0}}\left(\frac{1}{r(t)}e_{\ominus\frac{p(s)}{r(s)}}(t,t_{0}\right)^{\frac{1}{\alpha}}\Delta t< \infty\) in the future work.
Acknowledgments
This research is supported by Shandong Provincial Natural Science Foundation ( ZR2017MA043).
Competing Interests
The authors declare that they have no competing interests.
