\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations},
 Vol. 2007(2007), No. 169, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/169\hfil Oscillation criteria]
{Oscillation criteria for impulsive dynamic equations on time
scales}

\author[M. Huang, W. Feng\hfil EJDE-2007/169\hfilneg]
{Mugen Huang, Weizhen Feng}  

\address{Mugen Huang \newline
Institute of Mathematics and Information technology, 
Hanshan Normal University, Chaozhou 521041, China} 
\email{huangmugen@yahoo.cn}

\address{Weizhen Feng \newline
School of Mathematical Sciences, 
South China Normal University,
Guangzhou 510631, China}
\email{wsy@scnu.edu.cn}

\thanks{Submitted September 2, 2007. Published December 3, 2007.}
\subjclass[2000]{34A37, 34A60, 39A12, 34B37, 34K25}
\keywords{Oscillation; dynamic equations; time scales;
 impulsive; inequality}

\begin{abstract}
 Oscillation criteria for impulsive dynamic equations on time
 scales are obtained via impulsive inequality.
 An example is given to show that the impulses play a dominant
 part in the oscillations of dynamic equations on time scales.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}


\section{Introduction}

In this paper, we are interested in obtaining oscillation criteria
for  solutions of the second-order nonlinear impulsive dynamic
equation on time scales,
\begin{equation} \begin{gathered}
y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0, \quad t\in
\mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\;
t\neq t_k,\; k=1,2,\dots,\\
y(t^{+}_k)=g_k(y(t^{-}_k)),\quad
y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t^{-}_k)),
\quad k=1,2,\dots,\\
y(t^{+}_0)=y_0, \quad y^{\Delta}(t^{+}_0)=y^{\Delta}_0,
\end{gathered}
\label{e1.1}
\end{equation}
where $\mathbb{T}$ is a unbounded-above time scale , with $0\in
\mathbb{T}$, $t_k\in \mathbb{T}, 0\leq t_0<t_1<t_2<\dots <t_k<\dots$
and $\lim_{k\to \infty }t_k=\infty$.
\begin{equation}
y(t^{+}_k)=\lim_{h\to 0^{+}}y(t_k+h), \quad
y^{\Delta}(t^{+}_k)=\lim_{h\to 0^{+}}y^{\Delta}(t_k+h), \label{e1.2}
\end{equation}
which represent right limits of $y(t)$ at $t=t_k$ in the sense of
time scales, and in addition, if $t_k$ is right scattered, then
$y(t^{+}_k)=y(t_k), y^{\Delta}(t^{+}_k)=y^{\Delta}(t_k)$. We can
defined $y(t^{-}_k),y^{\Delta}(t^{-}_k)$ similar to \eqref{e1.2}.

We suppose that the following conditions hold:
\begin{itemize}
\item[(H1)] $ f\in C_{rd}(\mathbb{T}\times \mathbb{R}, \mathbb{R})$,
$xf(t, x)>0$  ($x\neq 0$) and
$f(t, x)/\varphi(x) \geq p(t)$
($x\neq 0$), where $p(t)\in C_{rd}(\mathbb{T}, \mathbb{R}_+)$ and
$x\varphi(x)>0$ ($x\neq 0$), $\varphi'(x)\geq 0$.

\item[(H2)]  $g_k, h_k\in C(\mathbb{R}, \mathbb{R})$ and there exist
positive constants $a_k, a^{*}_k, b_k, b^{*}_k$ such that
$$
 a^{*}_k\leq \frac{g_k(x)}{x}\leq a_k, \quad
 b^{*}_k\leq \frac{h_k(x)}{x}\leq b_k.
$$
\end{itemize}

We note that the theory of dynamic equations on time scales are an
adequate mathematical apparatus for the simulation of processes and
phenomena observed in biotechnology, chemical technology, economic,
neural networks, physics, social sciences etc. For further
applications and questions concerning solutions of dynamic equations
on time scales, see \cite{Aulbach,Bohner2, Bohner3}

Recently, impulsive dynamic equations on time scales have been
investigated by Agarwal et al. \cite{Agarwal2}, Belarbi et al.
\cite{Belarbi}, Benchohra et al. \cite{Benchohra1, Benchohra2,
Benchohra3, Benchohra4}, Chang et al. \cite{Chang} and so forth. In
\cite{Benchohra4}, Benchohra et al. considered the existence of
extremal solutions for a class of second order impulsive dynamic
equations on time scales, we can see that the existence of global
solutions can be guaranteed by some simple conditions.

Based on the oscillatory behavior of the impulsive dynamic equations
on time scales,  Benchohra et al. \cite{Benchohra1} discuss the
existence of oscillatory and nonoscillatory solutions by lower and
upper solutions method for the first order impulsive dynamic
equations on certain time scales
\begin{equation} \begin{gathered}
y^{\Delta}(t)=f(t,y(t)), \quad t\in
\mathbb{J}_{\mathbb{T}}:=[0,\infty)\bigcap \mathbb{T},\;
t\neq t_k,\; k=1,\dots,\\
y(t^{+}_k)=I_k(y(t^{-}_k)),\quad k=1,\dots.
\end{gathered}
\label{e1.3}
\end{equation}
On the other hand, Huang et al. \cite{Huang} considered the second
order nonlinear impulsive dynamic equations on time scales
\begin{equation} \begin{gathered}
y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0, \quad t\in
\mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\;
t\neq t_k,\; k=1,2,\dots,\\
y(t^{+}_k)=g_k(y(t^{-}_k)),
y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t^{-}_k)),
\quad k=1,2,\dots,\\
y(t^{+}_0)=y_0, \quad y^{\Delta}(t^{+}_0)=y^{\Delta}_0,
\end{gathered} \label{e1.4}
\end{equation}
extend the well-known results of Chen et al. \cite{Chen} for the
impulsive differential equations to \eqref{e1.4}.

Motivated by the ideas in \cite{Peng}, we establish the
sufficient conditions for the oscillation of all solutions of
\eqref{e1.1}, which utilize Riccati transformation techniques and
impulsive inequality. Those results extend some well-known impulsive
inequality on differential equations to impulsive dynamic equations.
Our method is different from most existing ones. An example is given
to show that though a dynamic equation on time scales is
nonoscillatory, it may become oscillatory if some impulses are added
to it. That is, in some cases, impulses play a dominating part in
oscillations of dynamic equations on time scales.

For the remainder of the paper, we assume that, for each
$k=1, 2, \dots,$ the points of impulses $t_k$ are right dense
(rd for short). In order to define the solutions of the problem \eqref{e1.1},
we introduce the two spaces:
\begin{align*}
AC^{i}&=\{y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R}
\text{ which is $i$-times $\Delta$-differentiable, and its  $i$-th }\\
&\quad \text{delta-derivative $y^{\Delta^{(i)}}$ is absolutely continuous}\};
\\
PC&=\{y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R}
\text{ which is rd-continuous expect at  $t_k$, for which }\\
&\quad\text{$y(t^{-}_k),y(t^{+}_k), y^{\Delta}(t^{-}_k), y^{\Delta}(t^{+}_k)$
exist with $y(t^{-}_k)=y(t_k)$, $y^{\Delta}(t^{-}_k)=y^{\Delta}(t_k)$}\}.
\end{align*}

A function $y\in PC\bigcap AC^{2}(\mathbb{J}_{\mathbb{T}}\backslash
\{t_1, \dots\}, \mathbb{R})$ is said to be a solution of \eqref{e1.1}, if
it satisfies $y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0$ a.e. on
$\mathbb{J}_{\mathbb{T}}\backslash \{t_k\}, k=1, 2, \dots$, and for
each $k=1, 2, \dots, y$ satisfies the impulsive condition
$y(t^{+}_k)=g_k(y(t_k)), y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t_k))$
and the initial conditions $y(t^{+}_0)=y_0,
y^{\Delta}(t^{+}_0)=y^{\Delta}_0$.


A solution $y$ of \eqref{e1.1} is called oscillatory if it is neither
eventually positive nor eventually negative; otherwise it is called
nonoscillatory. Equation \eqref{e1.1} is called oscillatory if all
solutions are oscillatory.

\section{Preliminary Results}

We will briefly recall some basic definitions and facts from the
time scales calculus that we will use in the sequel. For more
details see \cite{Agarwal1, Bohner2, Bohner3}.

On any time scale $\mathbb{T}$, we define the forward and backward
jump operators by
$$
\sigma(t)=\inf\{s\in \mathbb{T}, {s>t}\}, \quad
 \rho(t)=\sup\{s\in \mathbb{T}: s<t\},
$$
where $\inf \phi=\sup \mathbb{T}, \sup \phi=\inf \mathbb{T}$, and
$\phi$ denotes the empty set. A nonmaximal element $t\in \mathbb{T}$
is called right-dense if $\sigma(t)=t$ and right-scattered if
$\sigma(t)>t$. A nonminimal element $t\in \mathbb{T}$ is said to be
left-dense if $\rho(t)=t$ and left-scattered if $\rho(t)<t$. The
graininess $\mu$ of the time scale $\mathbb{T}$ is defined by
$\mu(t)=\sigma(t)-t$.

A mapping $f: \mathbb{T}\to \mathbb{X}$ is said to be differentiable
at $t\in \mathbb{T}$, if there exists $b\in \mathbb{X}$ such that
for any $\varepsilon>0$, there exists a neighborhood \textbf{U} of
$t$ satisfying $|[f(\sigma(t))-f(s)]-b[\sigma(t)-s]|\leq \varepsilon
|\sigma(t)-s|$, for all $s\in \textbf{U}$. We say that $f$ is delta
differentiable (or in short: differentiable) on $\mathbb{T}$
provided $f^{\Delta}(t)$ exist for all $t\in \mathbb{T}$.

A function $f: \mathbb{T}\to \mathbb{R}$ is called $rd-continuous$
provided it is continuous at right-dense points in $\mathbb{T}$ and
its left-sided limits exist (finite) at left-dense points in
$\mathbb{T}$. The set of rd-continuous functions $f: \mathbb{T}\to
\mathbb{R}$ will be denoted by $C_{rd}(\mathbb{T}, \mathbb{R})$.

The derivative and forward jump operator $\sigma$ are related by the
formula
\begin{equation}
f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t).
\label{e2.1}
\end{equation}

Let $f$ be a differentiable function on [a,b]. Then $f$ is
increasing, decreasing, nondecreasing and nonincreasing on $[a, b]$
if $f^{\Delta}>0, f^{\Delta}<0, f^{\Delta}\geq 0$ and
$f^{\Delta}\leq 0$ for all $t\in [a, b)$, respectively.

We will use  the following product  and quotient
 rules for derivative of two differentiable functions
$f$ and $g$:
\begin{gather}
(fg)^{\Delta}=f^{\Delta}g+f^{\sigma}g^{\Delta}=fg^{\Delta}+f^{\Delta}g^{\sigma},
\label{e2.2} \\
(\frac{f}{g})^{\Delta}=\frac{f^{\Delta}g-fg^{\Delta}}{gg^{\sigma}},
\label{e2.3}
\end{gather}
where $f^{\sigma}=f\circ \sigma, gg^{\sigma}\neq 0$.
The integration by parts formula reads
\begin{equation}
\int^{b}_{a}f^{\Delta}(t)g(t)\Delta t
=f(t)g(t)|^{b}_{a}-\int^{b}_{a}f^{\sigma}(t)g^{\Delta}(t)\Delta t.
\label{e2.4}
\end{equation}
Chain Rule: Assume $g: \mathbb{T}\to \mathbb{R}$ is
$\Delta-$differentiable on $\mathbb{T}$ and $f: \mathbb{R}\to
\mathbb{R}$ is continuously differentiable. Then $f\circ g:
\mathbb{T}\to \mathbb{R}$ is $\Delta-$differentiable and satisfies
\begin{equation}
(f\circ g)^{\Delta}(t)=\{\int^{1}_{0}f'(g(t) +h\mu
(t)g^{\Delta}(t))dh\}g^{\Delta}(t).
\label{e2.5}
\end{equation}
A function $p: \mathbb{T}\to \mathbb{R}$ is called regressive if for
all $t\in \mathbb{T}$
$$
1+\mu(t)p(t)\neq 0.
$$
The set of all $rd-continuous$ function $f$ which satisfy
$1+\mu(t)p(t)>0$ for all $t\in \mathbb{T}$ will be denoted by
$\mathcal {R}^{+}$. The generalized exponential function $e_{p}$ is
defined by
$$
e_{p}(t, s)=\exp\big\{\int^{t}_{s}\xi_{\mu(\tau)}(p(\tau))\Delta
\tau\big\},
$$
with $\xi_h(z)=\log(1+hz)/h$ if $h\neq 0$ and $\xi_h(z)=z$ if
$h=0$.

\begin{lemma}[5, p. 255] \label{lem1}
Let $y, f\in C_{rd}$ and $p\in \mathcal {R}^{+}$. Then
$$
y^{\Delta}(t)\leq p(t)y(t)+f(t),
$$
implies that for all $t\in \mathbb{T}$,
$$
y(t)\leq y(t_0) e_{p}(t, t_0)+\int^{t}_{t_0}e_{p}(t,
\sigma(s))f(s)\Delta s\,.
$$
\end{lemma}

\section{Main results}

Next, we  prove some lemmas, which will be useful for
establishing oscillation criteria for \eqref{e1.1}.

\begin{lemma}\label{lem2}
Assume that $m\in PC^{1}[\mathbb{T}, \mathbb{R}]$ and
\begin{equation} \begin{gathered}
m^{\Delta}(t)\leq p(t)m(t)+q(t), \quad
t\in \mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\;
 t\neq t_k,\; k=1,2,\dots,\\
m(t^{+}_k)\leq d_k m(t^{-}_k)+b_k, \quad k=1,2,\dots,\\
\end{gathered} \label{e3.1}
\end{equation}
then for $t\geq t_0$,
\begin{equation} \begin{aligned}
m(t) &\leq m(t_0)\prod_{t_0<t_k<t}d_k e_{p}(t, t_0)
+\sum_{t_0<t_k<t}\Big(\prod_{t_k<t_j<t}d_{j}e_{p}(t, t_k)\Big)b_k\\
&\quad +\int^{t}_{t_0}\prod_{s<t_k<t}d_k e_{p}(t, \sigma(s))q(s)\Delta s.
\end{aligned} \label{e3.2}
\end{equation}
\end{lemma}

\begin{proof}
 Let $t\in [t_0, t_1]_{\mathbb{T}}$. then use Lemma \ref{lem1} to
obtain
$$
m(t)\leq m(t_0)e_{p}(t, t_0)+\int^{t}_{t_0}e_{p}(t,
\sigma(s))q(s)\Delta s, \quad t\in [t_0, t_1]_{\mathbb{T}}.
$$
Hence \eqref{e3.2} is true for $t\in [t_0, t_1]_{\mathbb{T}}$. Now assume
that \eqref{e3.2} holds for $t\in [t_0, t_n]_{\mathbb{T}}$ for some integer
$n>1$. Then for $t\in (t_n, t_{n+1}]_{\mathbb{T}}$, it follows from
\eqref{e3.1} and Lemma \ref{lem1}, we get
$$
m(t)\leq m(t^{+}_n)e_{p}(t, t_n)+\int^{t}_{t_n}e_{p}(t,
\sigma(s))q(s)\Delta s\,.
$$
Using \eqref{e3.1}, we obtain, from \eqref{e3.2},
\begin{align*}
m(t)&\leq [d_n m(t^{-}_n)+b_n]e_{p}(t, t_n)
+\int^{t}_{t_n}e_{p}(t, \sigma(s))q(s)\Delta s\\
&\leq d_n e_{p}(t, t_n)\Big[m(t_0)\prod_{t_0<t_k<t_n}d_k e_{p}(t_n, t_0)
+\sum_{t_0<t_k<t_n}\Big(\prod_{t_k<t_j<t_n}d_{j}e_{p}(t_n, t_k)\Big)b_k\\
&\quad +\int^{t_n}_{t_0}\prod_{s<t_k<t_n}d_k e_{p}(t_n,
\sigma(s))q(s)\Delta s\Big]
+b_n e_{p}(t, t_n)+\int^{t}_{t_n}e_{p}(t, \sigma(s))q(s)\Delta s\\
&\leq m(t_0)\prod_{t_0<t_k<t}d_k e_{p}(t, t_0)
+\sum_{t_0<t_k<t}\Big(\prod_{t_k<t_j<t}d_{j}e_{p}(t, t_k)\Big)b_k\\
&\quad +\int^{t}_{t_0}\prod_{s<t_k<t}d_k e_{p}(t, \sigma(s))q(s)\Delta s,
\end{align*}
which on simplification gives the estimate \eqref{e3.2} for $t\in [t_0,
t_{n+1}]_{\mathbb{T}}$, by induction, we get \eqref{e3.2} holds for
$t\geq t_0$.
\end{proof}

\begin{lemma}\label{lem3}
Suppose that {\rm (H1), (H2)} hold and $y(t)>0$, $t\geq t'_0\geq t_0$ is
a nonoscillatory solution of \eqref{e1.1}. If
\begin{itemize}
\item[(H3)]
$\int^{\infty}_{t_j}\prod_{t_j<t_k<s}\frac{b^{*}_{k}}{a_k}\Delta s=\infty$
for some $t_j\geq t_0$.
\end{itemize}
Then $y^{\Delta}(t^{+}_k)\geq 0$ and
$y^{\Delta}(t)\geq 0$ for $t\in (t_k, t_{k+1}]_{\mathbb{T}}$, where
$t_k\geq t'_0$.
\end{lemma}

\begin{proof}
At first, we prove that $y^{\Delta}(t^{-}_k)\geq 0$ for
$t_k\geq t'_0$, otherwise, there exists some $j$ such that $t_j\geq
t'_0$ and $y^{\Delta}(t^{-}_j)<0$, hence
$$
y^{\Delta}(t^{+}_j)=h_j\left(y^{\Delta}(t^{-}_j)\right)\leq b^{*}_j
y^{\Delta}(t^{-}_j)<0.
$$
Let $y^{\Delta}(t^{+}_j)=-\alpha$ ($\alpha>0$). From \eqref{e1.1} and
(H1), for $t\in(t_{j+i-1}, t_{j+i}]_{\mathbb{T}}, i=1, 2,
\dots$, we obtain
$$
y^{\Delta \Delta}(t)=-f(t, y^{\sigma}(t))\leq
-p(t)\varphi(y^{\sigma}(t))\leq 0;
$$
i.e., $y^{\Delta}(t)$ is nonincreasing in
$(t_{j+i-1}, t_{j+i}]_{\mathbb{T}}$, $i=1, 2, \dots$, then
\begin{equation} \begin{gathered}
y^{\Delta}(t^{-}_{j+1})\leq y^{\Delta}(t^{+}_j)=-\alpha<0,\\
y^{\Delta}(t^{-}_{j+2})\leq
y^{\Delta}(t^{+}_{j+1})=h_{j+1}\left(y^{\Delta}(t^{-}_{j+1})\right)
\leq b^{*}_{j+1}y^{\Delta}(t^{-}_{j+1}) \leq -b^{*}_{j+1}\alpha<0.
\end{gathered}
\label{e3.3}
\end{equation}
By induction, we obtain
\begin{equation}
y^{\Delta}(t)\leq -\alpha \prod_{t_j<t_k<t}b^{*}_{k}<0 \quad t\in
(t_{j+n}, t_{j+n+1}]_{\mathbb{T}}. \label{e3.4}
\end{equation}
In view of (H2), we have $y(t^{+}_k)\leq a_k y(t^{-}_k)$.
Applying Lemma \ref{lem2}, we obtain for $t>t_j$
\begin{equation} \begin{aligned}
y(t)&\leq y(t^{+}_j)\prod_{t_j<t_k<t}a_k
-\alpha\int^{t}_{t_j}\prod_{s<t_k<t}a_k\prod_{t_j<t_k<s}b^{*}_k\Delta s\\
&=\prod_{t_j<t_k<t}a_k\Big[y(t^{+}_j)
-\alpha\int^{t}_{t_j}\prod_{t_j<t_k<s}\frac{b^{*}_k}{a_k}\Big]\Delta s.
\end{aligned}
\label{e3.5}
\end{equation}
Since $y(t^{+}_j)>0$, one can find that \eqref{e3.5} contradicts
(H3) as $t\to \infty$. Therefore, $y^{\Delta}(t^{-}_k)\geq 0$
 ($t_k\geq t'_0$). By condition (H2), we obtain, for any $t_k\geq t'_0$,
$$
y^{\Delta}(t^{+}_k)\geq b^{*}_k y^{\Delta}(t^{-}_k)\geq 0.
$$
Since $y^{\Delta}(t)$ is decreasing in $(t_k, t_{k+1}]_{\mathbb{T}}$,
$t_k\geq t'_0$, we have $y^{\Delta}(t)\geq y^{\Delta}(t^{-}_k)\geq 0$,
$t\in (t_k, t_{k+1}]_{\mathbb{T}}$, $t_k\geq t'_0$. The proof of
Lemma \ref{lem3} is complete.
\end{proof}

We remark that when $y$ is eventually negative, under the hypothesis
(H1)-(H3), it can be proved similarly that
$y^{\Delta}(t^{+}_k)\leq 0$ and for $t\in (t_k,t_{k+1}]_{\mathbb{T}}$,
$y^{\Delta}(t)\leq 0$ for $t_k\geq t'_0\geq t_0$.

\begin{theorem}\label{th1}
Suppose that {\rm (H1)-(H3)} hold and there exists a positive
integer $k_0$ such that $a^{*}_k\geq 1$ for $k\geq k_0$. If
\begin{equation}
\int^{\infty}_{t_0}\prod_{t_0<t_k<t}\frac{1}{b_k}p(t)\Delta
t=\infty, \label{e3.6}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 Suppose to the contrary that Eq.\eqref{e1.1} has a nonoscillatory
solution $y$, without loss of generality, we may assume that $y$ is
eventually positive solution of \eqref{e1.1}; i.e., $y(t)>0, t\geq t_0$ and
$k_0=1$. From lemma \ref{lem3}, we have $y^{\Delta}(t)\geq 0$,
$t\in (t_k, t_{k+1}]_{\mathbb{T}}$, $k=1, 2, \dots$. Let
\begin{equation}
w(t)=\frac{y^{\Delta}(t)}{\varphi(y(t))}\,. \label{e3.7}
\end{equation}
Then $w(t^{+}_k)\geq 0$, $k=1, 2, \dots$, and $w(t)>0, t\geq t_0$.
Using (H1) and \eqref{e1.1}, when $t\neq t_k$,
\begin{equation} \begin{aligned}
w^{\Delta}(t)
&=-\frac{f(t, y^{\sigma}(t))}{\varphi(y^{\sigma}(t))}
-\frac{y^{\Delta}(t)}{\varphi(y(t))\varphi(y^{\sigma}(t))}
\int^1_0\varphi'\left(y(t)+h\mu(t)y^{\Delta}(t)\right)dh y^{\Delta}(t)\\
&\leq -p(t)-\frac{\varphi(y(t))}{\varphi(y^{\sigma}(t))}
\Big(\frac{y^{\Delta}(t)}{\varphi(y(t))}\Big)^2
\int^1_0\varphi'\left(y(t)+h\mu(t)y^{\Delta}(t)\right)dh\\
&\leq -p(t).
\end{aligned}
\label{e3.8}
\end{equation}
Since $\varphi'(y(t))\geq 0$ and $\varphi(y(t))>0$, from (H2)
and $a^{*}_k\geq 1$, we obtain
\begin{equation}
w(t^{+}_k)=\frac{y^{\Delta}(t^{+}_k)}{\varphi(y(t^{+}_k))} \leq
\frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_k y(t^{-}_k))} \leq
\frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(y(t^{-}_k))} =b_k w(t^{-}_k),
\quad k=1, 2, \dots. \label{e3.9}
\end{equation}
Applying Lemma \ref{lem2}, we obtain from \eqref{e3.8} and \eqref{e3.9},
\begin{equation} \begin{aligned}
w(t)&\leq w(t_0)\prod_{t_0<t_k<t}b_k
-\int^{t}_{t_0}\prod_{s<t_k<t}b_k p(s)\Delta s\\
&=\prod_{t_0<t_k<t}b_k\Big[w(t_0)
-\int^{t}_{t_0}\prod_{t_0<t_k<s}\frac{1}{b_k}p(s)\Delta s\Big].
\end{aligned} \label{e3.10}
\end{equation}
In view of \eqref{e3.6} and \eqref{e3.10}, we get a contradiction as
$t\to \infty$. Then every solution of \eqref{e1.1} is oscillatory.
\end{proof}

\begin{theorem}\label{th2}
Assume that {\rm (H1)-(H3)} hold and
$\varphi(ab)\geq \varphi(a)\varphi(b)$ for any $ab>0$. If
\begin{equation}
\int^{\infty}_{t_0}\prod_{t_0<t_k<t}\frac{\varphi(a^{*}_k)}{b_k}p(t)\Delta
t=\infty, \label{e3.11}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 As before, we may suppose $y(t)>0$, $t\geq t_0$ be a
nonoscillatory solution of \eqref{e1.1}, Lemma \ref{lem3} yields
$y^{\Delta}(t)\geq 0, t\geq t_0$, define $w(t)$ as in \eqref{e3.7}, we get
$w(t)\geq 0, t\geq t_0, w(t^{+}_k)\geq 0, k=1, 2, \dots$, \eqref{e3.8}
holds for $t\neq t_k$ and
\begin{equation}
w(t^{+}_k)=\frac{y^{\Delta}(t^{+}_k)}{\varphi(y(t^{+}_k))} \leq
\frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_ky(t^{-}_k))} \leq
\frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_k)\varphi(y(t^{-}_k))}
=\frac{b_k}{\varphi(a^{*}_k)}w(t^{-}_k). \label{e3.12}
\end{equation}
Using Lemma \ref{lem2}, we get from \eqref{e3.8} and \eqref{e3.12}
\begin{align*}
w(t)&\leq w(t_0)\prod_{t_0<t_k<t}\frac{b_k}{\varphi(a^{*}_k)}
-\int^{t}_{t_0}\prod_{s<t_k<t}\frac{b_k}{\varphi(a^{*}_k)}p(s)\Delta s\\
&= \prod_{t_0<t_k<t}\frac{b_k}{\varphi(a^{*}_k)}\Big[w(t_0)
-\int^{t}_{t_0}\prod_{t_0<t_k<s}\frac{\varphi(a^{*}_k)}{b_k}p(s)\Delta
s\Big].
\end{align*}
Letting $t\to \infty$, the above inequality contradicts to \eqref{e3.11}.
Then every solution of \eqref{e1.1} is oscillatory.
\end{proof}

 From Theorems \ref{th1} and \ref{th2}, we have the following
corollaries.

\begin{corollary}\label{cor1}
Suppose that {\rm (H1)-(H3)} hold and there exists a positive integer
$k_0$ such that $a^{*}_k\geq 1, b_k\leq 1$ for $k\geq k_0$. If
$\int^{\infty}p(t)\Delta t=\infty$, then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
 Without loss of generality, let $k_0=1$. By $b_k\leq 1$, we
get $\frac{1}{b_k}\geq 1$, therefore
$$
\int^{t}_{t_0}\prod_{t_0<t_k<s}\frac{1}{b_k}p(s)\Delta s\geq
\int^{t}_{t_0}p(s)\Delta s.
$$
Let $t\to \infty$ and using $\int^{\infty}p(t)\Delta t=\infty$, we
obtain from Theorem \ref{th1} that \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary}\label{cor2}
Suppose that {\rm (H1)-(H3)} hold and there exist a positive integer
$k_0$ and a constant $\alpha>0$ such that
\begin{equation}
a^{*}_k\geq 1, \quad \frac{1}{b_k}\geq
\Big(\frac{t_{k+1}}{t_k}\Big)^{\alpha},\quad \text{for } k\geq
k_0. \label{e3.13}
\end{equation}
If
\begin{equation}
\int^{\infty}t^{\alpha}p(t)\Delta t=\infty. \label{e3.14}
\end{equation}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
 Without loss of generality let $k_0=1$. Then \eqref{e3.6} yields
\begin{equation} \begin{aligned}
&\int^{t}_{t_0}\prod_{t_0<t_k<s}\frac{1}{b_k}p(s)\Delta s\\
&=\int^{t_1}_{t_0}p(t)\Delta
t+\frac{1}{b_1}\int^{t_2}_{t_1}p(t)\Delta t
+\dots+\frac{1}{b_1b_2\dots b_n}\int^{t}_{t_n}p(t)\Delta t\\
&\geq \frac{1}{t^{\alpha}_1}\Big[\int^{t_2}_{t_1}t^{\alpha}_2p(t)\Delta
t +\int^{t_3}_{t_2}t^{\alpha}_3p(t)\Delta t+\dots
+\int^{t}_{t_n}t^{\alpha}_{n+1}p(t)\Delta t\Big]\\
&\geq \frac{1}{t^{\alpha}_1}\Big[\int^{t_2}_{t_1}s^{\alpha}p(s)\Delta s
+\int^{t_3}_{t_2}s^{\alpha}p(s)\Delta s+\dots
+\int^{t}_{t_n}s^{\alpha}p(s)\Delta s\Big]\\
&=\frac{1}{t^{\alpha}_1}\int^{t}_{t_1}s^{\alpha}p(s)\Delta s,
\end{aligned}
\label{e3.15}
\end{equation}
for $t\in (t_n, t_{n+1}]_{\mathbb{T}}$. Let $t\to \infty$ and use
\eqref{e3.15}, \eqref{e3.14} yields \eqref{e3.6} holds. According to
Theorem \ref{th1},
we obtain \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary}\label{cor3}
Assume that {\rm (H1)-(H3)} hold and $\varphi(ab)\geq
\varphi(a)\varphi(b)$ for any $ab>0$. Suppose there exist a positive
integer $k_0$ and a constant $\alpha>0$ such that
$$
\frac{\varphi(a^{*}_k)}{b_k}\geq
\Big(\frac{t_{k+1}}{t_k}\Big)^{\alpha}, \quad \text{for } k\geq
k_0.
$$
If $\int^{\infty}t^{\alpha}p(t)\Delta t=\infty$, then \eqref{e1.1} is
oscillatory.
\end{corollary}

The above corollary  can be deduced from Theorem \ref{th2}. Its
proof is similar to that of Corollary \ref{cor2}; so we omit it.

\section{Example}

  Consider the  second-order impulsive
dynamic equation
\begin{equation} \begin{gathered}
y^{\Delta \Delta}(t)+\frac{1}{t\sigma^2(t)}y^{\gamma}(\sigma(t))=0, \quad
t\geq 1,\; t\neq k,\; k=1, 2, \dots,\\
y(k^{+})=\frac{k+1}{k}y(k^{-}),\quad
y^{\Delta}(k^{+})=y^{\Delta}(k^{-}), \quad k=1, 2, \dots,\\
y(1)=y_0, \quad y^{\Delta}(1)=y^{\Delta}_0.
\end{gathered}
\label{e4.1}
\end{equation}
where $\gamma\geq 3$ and $\mu(t)\leq ct$, where $c$ is a positive
constant.

Since $a_k=a^{*}_k=(k+1)/k$, $b_k=b^{*}_k=1$,
$p(t)=1/(t\sigma^2(t))$, $t_k=k$ and $\varphi(y)=y^{\gamma}$. It
is easy to see that (H1)-(H3) hold. Let $k_0=1$, $\alpha=3$, hence
$$
\frac{\varphi(a^{*}_k)}{b_k}=(k+1)/ k^{\gamma}
=\big(\frac{t_{k+1}}{t_k}\big)^{\gamma} \geq
\Big(\frac{t_{k+1}}{t_k}\Big)^{3},
$$
and
$$
\int^{\infty}t^{\alpha}p(t)\Delta
t=\int^{\infty}t^3\frac{1}{t\sigma^2(t)}\Delta t
=\int^{\infty}\Big(\frac{t}{\sigma(t)}\Big)^{2}\Delta t.
$$
Since $\mu(t)\leq ct$, we get
$$
\frac{t}{\sigma(t)}=\frac{t}{t+\mu(t)}\geq \frac{1}{1+c},
$$
hence
$$
\int^{\infty}\Big(\frac{t}{\sigma(t)}\Big)^{2}\Delta t \geq
\frac{1}{(1+c)^2}\int^{\infty}\Delta t=\infty.
$$
By Corollary \ref{cor3}, we obtain that \eqref{e4.1} is oscillatory. But by
\cite{Bohner1} we know that the dynamic equation $y^{\Delta
\Delta}(t)+\frac{1}{t\sigma^2(t)}y^{\gamma}(\sigma(t))=0$ is
nonoscillatory.

In the above example, it is interesting that the dynamic equation
without impulses is nonoscillatory, but when some impulses are added
to it, it becomes oscillatory. Therefore, this example shows that
impulses play an important part in the oscillations of dynamic
equations on time scales.

\subsection*{Acknowledgements}
The authors are very grateful to the anonymous referee for
his/her careful reading of the original manuscript, and for
the helpful suggestions.


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\end{document}