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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 122, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/122\hfil Oscillation of solutions]
{Oscillation of solutions to impulsive dynamic equations on time scales}

\author[Q. Li, F. Guo \hfil EJDE-2009/122\hfilneg]
{Qiaoluan Li, Fang Guo} 

\address{Qiaoluan Li \newline
College of Mathematics and Information Science,
Hebei Normal University, \newline
 Shijiazhuang, 050016, China}
\email{qll71125@163.com}

\address{Fang Guo \newline
Department of Mathematics and Computer, Bao Ding University,
071000, China} 
\email{gf825@126.com}


\thanks{Submitted February 26, 2009. Published September 29, 2009.}
\thanks{Supported by grant 07M004 from the Natural Science Foundation of
Hebei Province, \hfill\break\indent and by the Main Foundation of
Hebei Normal University} 
\subjclass[2000]{34A60, 39A12, 34K25}
\keywords{Oscillation; time scales; impulsive dynamic equations}

\begin{abstract}
 In this article, we study the oscillation of second order
 impulsive dynamic equations on time scales. The effect of
 the moments of impulse are fixed. Using Riccati transformation
 techniques, we obtain some conditions for the oscillation of
 all solutions
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

 This paper concerns the oscillation of second-order
impulsive dynamic equations on time scales. We consider the
 system
\begin{equation}
\begin{gathered} \label{e1}
(a(t)x^\Delta (t))^\Delta +p(t)x(\sigma(t))=0,\quad
t\in\mathbb{J}_{ \mathbb{T}} :=[t_0,\infty)\cap \mathbb{T},t\neq
t_k,\; k=1,2,\dots,\\
x(t_{k}^{+})=b_kx(t_k),\quad
x^\Delta (t_{k}^{+})=c_kx^\Delta (t_k),\quad k=1,2,\dots,\\
x(t_{0}^{+})=x(t_0),\quad x^\Delta (t_{0}^{+})=x^\Delta (t_0),
\end{gathered}
\end{equation}
where $\mathbb{T}$ is a time scales, unbounded-above, with
$t_k\in \mathbb{T}$, $0\leq t_0<t_1<t_2<\dots<t_k<\dots$,
$\lim_{k\to \infty}t_k=\infty$
and $y(t_{k}^{+})=\lim_{h\to 0^+}y(t_k+h)$,
$y^\Delta (t_{k}^{+})=\lim_{h\to 0^+}y^\Delta (t_k+h)$,
which represent right limits of
$y(t),y^\Delta (t)$ at $t=t_k$ in the sense of time scales.
We can define $y(t_{k}^{-})$, $y^\Delta (t_{k}^{-})$ similarly.

In this paper, we assume that $a(t)\in C_{\rm rd}(\mathbb{T},
\mathbb{R^+})$, $p(t)\in C_{\rm rd}(\mathbb{T},
\mathbb{R^+})$, $b_k>0$, $c_k>0$, $d_k=\frac{c_k}{b_k}$, $ t_k$
are right dense, where $ C_{\rm rd}$ denotes
 the set of rd-continuous functions,
$\sigma(t):=\inf \{{s\in \mathbb{T} :s>t}\}$,
$\mathbb{R}^+ =\{x:x>0\}$.


\begin{definition} \label{def1} \rm 
 A function $x$ is  a solution of
\eqref{e1}, if it satisfies
$(a(t)x^\Delta (t))^\Delta +p(t)x(\sigma(t))=0$ a.e. on
$\mathbb{J}_{\mathbb{T}}\backslash\{t_k\}$, $k=1,2,\dots$, and
for each $k=1,2,\dots,x$ satisfies the impulsive condition
$x(t_{k}^{+})=b_kx(t_k)$, $x^\Delta (t_{k}^{+})=c_kx^\Delta (t_k)$
and the initial condition
$x(t_{0}^{+})=x(t_0)$, $x^\Delta (t_{0}^{+})=x^\Delta (t_0)$.
\end{definition}

\begin{definition} \label{def2} \rm
 A solution $x$ of \eqref{e1} is
oscillatory if it is neither eventually positive nor eventually
negative; otherwise it is called non-oscillatory. Equation \eqref{e1} is
called oscillatory if all solutions are oscillatory.
\end{definition}

In recently years, there has been an increasing interest in
studying the oscillation and non-oscillation of solutions of
various equations on time scales, and we refer the reader to
papers \cite{e1,s1,s2,z1} and references cited therein. The time
scales calculus has a tremendous potential for applications in
mathematical models of real processes. Impulsive dynamic equations
on time scales have been investigated by Agarwal \cite{a1},
Benchohra \cite{b1} and so forth. Benchohra \cite{b1} considered
the existence of extremal solutions for a class of second order
impulsive dynamic equations on time sales.

The oscillation of impulsive differential equations and difference
equations have been investigated by many authors and numerous
papers have been  published on this class of equations and  good
results were obtained  (see  \cite{g1,m1}  and the references
therein). But fewer papers are on the oscillation of impulsive
dynamic equations on time scales.

For example, Huang \cite{h1}  considered the equation
\begin{gather*}
y^{\Delta\Delta}(t)+f(t, y^{\sigma}(t))=0,\quad
t\in\mathbb{J}_{ \mathbb{T}}:=[0,\infty)\cap \mathbb{T},\quad
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(t_0),\quad y^\Delta (t_{0}^{+})=y^\Delta (t_0).
\end{gather*}
Using Riccati transformation techniques, they obtain sufficient
conditions for oscillations of all solutions.

\section{Results}

  In the following, we  assume the solutions of \eqref{e1}
exist in  $\mathbb{J}_{\mathbb{T}}$. To the best of our knowledge, the
  question of the oscillation for second order self-conjugate
  impulsive dynamic equations has not been yet considered.

\begin{lemma} \label{lem1}
 Suppose that $x(t)>0$, $t\geq t_{0}'\geq t_0$ is a solution
of \eqref{e1}. If
\begin{equation}
\int_{t_0}^{t_1}\frac{\Delta
s}{a(s)}+d_1\int_{t_1}^{t_2}\frac{\Delta
s}{a(s)}+d_1d_2\int_{t_2}^{t_3}\frac{\Delta
s}{a(s)}+\dots+d_1d_2\dots d_n\int_{t_n}^{t_{n+1}}\frac{\Delta
s}{a(s)}+\dots=\infty,
\label{e2}
\end{equation}
then $x^\Delta (t_{k}^{+})\geq 0$ and $x^\Delta (t)\geq 0$ for
$t\in (t_k,t_{k+1}]_\mathbb{T}$, where
$t_k \geq t_{0}'$
\end{lemma}

The proof is similar to that in  \cite[Lemma 2.1]{h1};
so we omit it. We remark that when $a(t)\equiv 1$,
Lemma \ref{lem1} reduces to \cite[Lemma 2.1]{h1}.

\begin{lemma} \label{lem2}
Assume that $ q(t)\in C_{\rm rd}(\mathbb{T},\mathbb{R}^+)$, if
$$
\omega ^{\Delta\Delta}(t)+q(t)\omega^\Delta (t)\leq 0
$$
has a positive solution, then
$$
\omega ^{\Delta\Delta}(t)+q(t)\omega^\Delta (t)= 0
$$
has a positive solution.
\end{lemma}

The proof is similar to that in  \cite[Lemma 4.1.2]{e2};
so we omit it.

\begin{theorem} \label{thm1}
Assume that $a(t)\equiv 1 $  and \eqref{e2} holds.
\begin{equation}
\Big(\prod_{T\leq t_{k}<t}d_{k}^{-1}\omega
^\Delta \Big)^\Delta +\prod_{T\leq
t_{k}<t}d_{k}^{-1}p(t)\omega(t)=0,\quad \text{a.e. }
t>T\geq t_0\label{e3}
\end{equation}
is oscillatory, then  \eqref{e1} is oscillatory.
\end{theorem}

\begin{proof}
 Suppose to the contrary that \eqref{e1} has a
non-oscillatory solution $x(t)$, we may assume that $x(t)$ is
eventually positive solution of \eqref{e1}; i.e.,
 $x(t)>0$, $t\geq T\geq t_0$.  From Lemma \ref{lem1}, we have
$x^\Delta (t)\geq 0$, $x^\Delta (t_{k}^{+})\geq 0$, $t\geq T$,
$t\in \mathbb{T}$. Let  $z(t)=\frac{x^\Delta (t)}{x(t)}\geq 0$.
For $t\neq t_k$, we get
\begin{gather*}
z^\Delta (t) =  \frac{x^{\Delta\Delta}(t)x(t)-(x^\Delta (t))^2}{x(t)
 x(\sigma(t))}
=-p(t)-\frac{z^2(t)}{1+\mu(t)z(t)},\\
z(t^{+}_{k})=\frac{x^\Delta (t_k^{+})}{x(t_k^+)}
=\frac{c_kx^\Delta (t_k)}{b_kx(t_k)}=d_kz(t_k).
\end{gather*}
Thus we arrive at
\begin{gather*}
z^\Delta (t)+\frac{z^2(t)}{1+\mu(t)z(t)}+p(t)=0,\quad
t\in [T,\infty)\cap\mathbb{T},\; t\neq t_k, \\
z(t_{k}^{+})=d_kz(t_k).
\end{gather*}
Now we define
 $v(t)=(\prod_{T\leq t_{k} <t}d_{k}^{-1})z(t)$,
$t>T$, $t\in \mathbb{T}$.
Then for $t_n>T$,
$$
v(t_{n}^{+})=(\prod_{T\leq t_k\leq t_n}d_{k}^{-1})z(t_n^{+})=
(\prod_{T\leq t_k\leq t_n}d_{k}^{-1})d_nz(t_n)=v(t_n),
$$
which implies that $v(t)$ is rd-continuous on
$(T,\infty)\cap\mathbb{T}$. For $t\neq t_n$, we have
\begin{align*}
v^\Delta (t) &=  \prod_{T\leq t_k<
t}d_{k}^{-1}z^\Delta (t)\\
&= \prod_{T\leq t_k<
t}d_{k}^{-1}[-p(t)-\frac{z^2(t)}{1+\mu(t)z(t)}]\\
&=\prod_{T\leq t_k<
t}d_{k}^{-1}[-p(t)-\frac{(\prod_{T\leq t_k<
t}d_{k})^{2}v^2(t)}{1+\prod_{T\leq t_k<
t}d_{k}\mu(t)v(t)}]\\
&= -\prod_{T\leq t_k<
t}d_{k}\frac{v^2(t)}{1+\prod_{T\leq t_k<
t}d_{k}\mu(t)v(t)}-\prod_{T\leq t_k< t}d_{k}^{-1}p(t).
\end{align*}
For $t=t_n$, the left-hand derivative of $v(t)$ at $t=t_n$ is
given by
\begin{align*}
v^\Delta (t_{n}^{-})
&= \prod_{T\leq t_k< t_n}d_{k}^{-1}z^\Delta (t_{n}^{-})\\
&=  \prod_{T\leq t_k<
t_n}d_{k}^{-1}\lim_{t\to  t_{n}^{-}}[-p(t)-\frac{z^2(t)}{1+\mu(t)z(t)}]\\
&= \prod_{T\leq t_k<
t_n}d_{k}^{-1}[-p(t_n)-\frac{z^2(t_n)}{1+\mu(t_n)z(t_n)}] \\
&=  \prod_{T\leq t_k<
t_n}d_{k}^{-1}[-p(t_n)-\frac{\prod_{T\leq t_k<
t_n}d_{k}^{2}v^2(t_n)}{1+\mu(t_n)\prod_{T\leq t_k<
t_n}d_{k}v(t_n)}] \\
&=  -\prod_{T\leq t_k<
t_n}d_{k}^{-1}p(t_n)-\prod_{T\leq t_k<
t_n}d_{k}\frac{v^2(t_n)}{1+\mu(t_n)\prod_{T\leq t_k<
t_n}d_{k}v(t_n)}].
\end{align*}
Similarly, we obtain
$$
v^\Delta (t_{n}^{+})=-\prod_{T\leq t_k\leq
t_n}d_{k}\frac{v^2(t_n)}{1+\prod_{T\leq t_k\leq
t_n}d_{k}\mu(t_n)v(t_n)}-\prod_{T\leq t_k\leq
t_n}d_{k}^{-1}p(t_n).
$$
 So for $t>T$,
\begin{equation}
v^\Delta (t)+\prod_{T\leq t_k<t}d_{k}\frac{v^2(t)}{1+\prod_{T\leq t_k<
t}d_{k}\mu(t)v(t)}+\prod_{T\leq t_k< t}d_{k}^{-1}p(t)=0,\quad
\text{a.e. }\label{e4}
\end{equation}
Now we define $q(t)=\prod_{T\leq t_k<t}d_{k}v(t)$,
$w(t)=e_q(t,\,t_0)>0$, $t>T$. Then
\begin{gather*}
w^\Delta (t) =  q(t)w(t)=\prod_{T\leq t_k<
t}d_{k}v(t)w(t),\\
\begin{aligned}
\Big(\prod_{T\leq t_k< t}d_{k}^{-1}w^\Delta \Big)^\Delta
&= (v(t)w(t))^\Delta =w^\Delta (t)v(t)+w(\sigma(t))v^\Delta (t)\\
&=  \prod_{T\leq t_k<
t}d_{k}v^{2}(t)w(t)+e_{q}(\sigma(t),\,t_0)v^\Delta (t).
\end{aligned}
\end{gather*}
Since
$$
e_q(\sigma(t),\,t_0)=(1+\mu(t)q(t))e_q(t,\,t_0)=(1+\mu(t)v(t)
\prod_{T\leq t_k< t}d_{k})w,
$$
 by \eqref{e4} we obtain
\begin{align*}
\Big(\prod_{T\leq t_k<
t}d_{k}^{-1}w^\Delta \Big)^\Delta
&= w[\prod_{T\leq t_k<
t}d_{k}v^{2}(t)+(1+\mu(t)v(t)\prod_{T\leq t_k<
t}d_{k})v^\Delta (t)]\\
&= -w[1+\mu(t)v(t)\prod_{T\leq t_k<
t}d_{k}]\prod_{T\leq t_k< t}d_{k}^{-1}p(t)\\
&\leq -w(t)\prod_{T\leq t_k<
t}d_{k}^{-1}p(t),\quad \text{a.e. }
\end{align*}
This implies
$$
\Big(\prod_{T\leq t_k<
t}d_{k}^{-1}w^\Delta \Big)^\Delta +\prod_{T\leq t_k<
t}d_{k}^{-1}p(t)w\leq 0,\quad \text{a.e. }
$$
has a positive solution. By Lemma \ref{lem2}, we obtain
$$
\Big(\prod_{T\leq t_{k}<t}d_{k}^{-1}\omega
^\Delta \Big)^\Delta +\prod_{T\leq
t_{k}<t}d_{k}^{-1}p(t)\omega =0, \quad \text{a.e.}
$$
has a positive solution, a
contradiction, and so, the proof is complete.
\end{proof}

\begin{theorem} \label{thm2}
 Assume that $b_k=c_k,t_k$ are right
dense for all $k=1,2,\dots$. Then the oscillation of all solutions
of \eqref{e1} is equivalent to the oscillation of all solutions of
the  equation
\begin{equation}
(a(t)y^\Delta (t))^\Delta +p(t)y(\sigma(t))=0.\label{e5}
\end{equation}
\end{theorem}

\begin{proof}
Let $y(t)$ be any solution of \eqref{e5}. Set
 $x(t)=y(t)\prod_{t_0\leq t_k<t}b_k$ for $t>t_0$, then
 $$
x(t_{n}^{+})=y(t_{n}^{+})\prod_{t_0\leq t_k\leq t_n}b_k
=b_nx(t_n).
$$
Furthermore, for $t\neq t_n$, we have
\begin{gather*}
x^\Delta (t)=(\prod_{t_0\leq t_k<t}b_k)y^\Delta (t),\\
x^\Delta (t_{n}^{+})=(\prod_{t_0\leq t_k\leq
t_n}b_k)y^\Delta (t_{n}^{+})=b_nx^\Delta (t_n)=c_nx^\Delta (t_n).
\end{gather*}
For $t\neq t_n$,
\begin{align*}
(a(t)x^\Delta (t))^\Delta &=  [a(t)(\prod_{t_0\leq t_k<
t}b_k)y^\Delta (t)]^\Delta
 =\prod_{t_0\leq t_k<t}b_k(a(t)y^\Delta (t))^\Delta \\
&=  -p(t)\prod_{t_0\leq t_k<t}b_ky(\sigma(t))
 =-p(t)\prod_{t_0\leq t_k<\sigma(t)}b_ky(\sigma(t))\\
&=-p(t)x(\sigma(t)).
\end{align*}
Thus $x(t)$ is the solution of \eqref{e1}.

Conversely, if $x(t) $ is the solution of \eqref{e1}. Set
$y(t)=x(t)\prod_{t_0\leq t_k< t}b_k^{-1}$.
Thus we have
$$
y(t_n^{+})=x(t_n^{+})\prod_{t_0\leq t_k\leq
t_n}b_k^{-1}=y(t_n).$$ Furthermore for $t\neq t_n,\,n=1,2,\dots$,
we have
\begin{gather*}
y^\Delta (t)=x^\Delta (t)\prod_{t_0\leq t_k<t}b_k^{-1},\\
y^\Delta (t_{n}^{-})=x^\Delta (t_{n}^{-})\prod_{t_0\leq
t_k<t_n}b_k^{-1}=x^\Delta (t_{n})\prod_{t_0\leq
t_k<t_n}b_k^{-1},\\
y^\Delta (t_{n}^{+})=x^\Delta (t_{n}^{+})\prod_{t_0\leq
t_k\leq t_n}b_k^{-1}=x^\Delta (t_{n})\prod_{t_0\leq
t_k<t_n}b_k^{-1}.
\end{gather*}
 For  $t\neq t_n$, $n=1,2,\dots$,
\begin{align*}
(a(t)y^\Delta (t))^\Delta
&=  [a(t)(\prod_{t_0\leq t_k<t}b_k^{-1})x^\Delta (t)]^\Delta
 =\prod_{t_0\leq t_k< t}b_{k}^{-1}(a(t)x^\Delta (t))^\Delta \\
&=  -p(t)\prod_{t_0\leq t_k< t}b_k^{-1}x(\sigma(t))
 =-p(t)\prod_{t_0\leq t_k<\sigma(t)}b_k^{-1}x(\sigma(t))\\
& =-p(t)y(\sigma(t)).
\end{align*}
For $t=t_n$,
\begin{gather*}
\begin{aligned}
 (a(t_n^{-})y^\Delta (t_{n}^{-}))^\Delta
&=(a(t_n^{-})\prod_{t_0\leq t_k<
t_n}b_k^{-1}x^\Delta (t_{n}^{-}))^\Delta \\
&= -\prod_{t_0\leq t_k<
t_n}b_k^{-1}p(t_n)x(\sigma(t_{n}))=-p(t_n)y(\sigma(t_n)),
\end{aligned}
\\
 (a(t_n^{+})y^\Delta (t_{n}^{+}))^\Delta =
(a(t_n)y^\Delta (t_{n}^{+}))^\Delta =-p(t_n)y(\sigma(t_n)).
\end{gather*}
Thus we arrive at
$$
 (a(t)y^\Delta (t))^\Delta +p(t)y(\sigma(t))=0.
$$
This shows that $y(t)$ is a solution of
\eqref{e5}. The proof is complete.
\end{proof}

\begin{example} \label{exa1} \rm
 Consider the equation
\begin{equation} \label{e6}
\begin{gathered}
x^{\Delta \Delta}+p(t)x(\sigma(t))=0,\quad
t\in \mathbb{T}=\mathbb{P}_{\frac{1}{2},\frac{1}{2}},\;
t\neq k+\frac{1}{5},\; k=0,1,2,\dots,
\\
x((k+\frac{1}{5})^{+})=b_kx(k+\frac{1}{5}),\;
x^\Delta ((k+\frac{1}{5})^{+})=b_kx^\Delta (k+\frac{1}{5}),\quad
b_k>0,\; k=1,2,\dots, \\
x((\frac{1}{5})^{+})=x(\frac{1}{5}),\quad
x^\Delta ((\frac{1}{5})^{+})=x^\Delta (\frac{1}{5}),
\end{gathered}
\end{equation}
where $p(t)\in C_{\rm rd}(\mathbb{T},\mathbb{R}^+)$,
$\mathbb{P}_{\frac{1}{2},\frac{1}{2}}=\bigcup_{k=0}^{\infty}
[k,k+\frac{1}{2}]$.
Assume that for each $t_0\geq 0$ there exists $k_0\in N$ and
$l_0\in N$ such that $k_0\geq t_0$ and
$$
\sum_{j=1}^{l_0}\int_{k_0+j}^{k_0+j+\frac{1}{2}}p(t)dt
+\frac{1}{2}\sum_{j=0}^{l_0-1}p(k_0+j+\frac{1}{2})\geq 4.
$$
 From \cite[Theorem 4.46]{b2}, we know that
$$
x^{\Delta \Delta}+p(t)x(\sigma(t))=0
$$
is oscillatory on $\mathbb{T}$. By
Theorem \ref{thm2},  \eqref{e6} is oscillatory.
\end{example}

\begin{example} \label{exa2}\rm
Consider the equation
\begin{equation} \label{e7}
\begin{gathered}
(\frac{\sigma(t)}{t}x^\Delta )^\Delta +tx(\sigma(t))=0,\quad
t\geq 1,\; t\neq k,\; k=1,2,\dots,\\
x(t_k^+)=b_kx(t_k),\quad x^\Delta (t_k^+)=b_kx^\Delta (t_k),\quad
b_k>0,\;k=1,2,\dots\\
x(1^+)=x(1),\quad x^\Delta (1^+)=x^\Delta (1),
\end{gathered}
\end{equation}
where $\mu(t)=\sigma(t)-t\leq ct$,  $\,\,c$ is a positive
constant. Since $\mu(t) \leq ct, $ we get
$$
\frac{t}{\sigma(t)}=\frac{t}{t+\mu (t)}\geq \frac{1}{1+c}.
$$
It is easy to see that
\begin{gather*}
\int_{1}^{\infty}\frac{t}{\sigma(t)}\Delta t\geq
\frac{1}{1+c}\int_{1}^{\infty}\Delta t=\infty, \\
 \int_{1}^{\infty}t\Delta t=\infty.
\end{gather*}
By \cite[Theorem 3.2]{b3}, we see that
$$
(\frac{\sigma(t)}{t}x^\Delta )^\Delta +tx(\sigma(t))=0,
$$
is oscillatory. So \eqref{e7} is oscillatory.
\end{example}

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\end{document}