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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 72, pp. 1--13.\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-2007/72\hfil Oscillation of second-order impulsive equations]
{Oscillation of second-order nonlinear impulsive dynamic equations
on time scales}

\author[M.. Huang,\hfil EJDE-2007/72\hfilneg]
{Mugen Huang, Weizhen Feng}

\address{Mugen Huang \newline
Institute of Mathematics and Information Technology\\
Hanshan Normal University\\
Chaozhou 521041, China} 
\email{huangmugen@sohu.com}

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


\thanks{Submitted November 27, 2006. Published May 15, 2007.}
\subjclass[2000]{39A10, 39A11, 39A99, 34C10, 34K11}
\keywords{Oscillation; impulsive dynamic equations; time scales}

\begin{abstract}
 In this article, we study the oscillation of second-order nonlinear
 impulsive dynamic equations on time scales. Using Riccati
 transformation techniques, we obtain sufficient conditions for
 oscillation of all solutions.
 An example is given to show that the impulses play a dominant
 part in oscillations of dynamic equations on time scales.
\end{abstract}

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

 \section{Introduction}

This paper is concerned with the oscillations of second-order nonlinear
impulsive dynamic equations on time scales. We consider the  problem
\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{e1.1}
\end{equation}
where $\mathbb{T}$ is a time scale, unbounded-above,  with $0\in
\mathbb{T}$, $t_k\in \mathbb{T}$, $0\leq
t_0<t_1<t_2<\dots<t_k<\dots$, $\lim_{k\to \infty }t_k=\infty$.
\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 and left 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 always 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 $\frac{f(t, x)}{\varphi(x)}\geq p(t)$
 ($x\neq 0$), 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}
Throughout 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 space
\begin{align*}
AC^{i}=\big\{&y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R}
\text{is $i$-times $\Delta$-differentiable, whose}\\
&\text{$i$-th delta-derivative $y^{\Delta^{(i)}}$ is
absolutely continuous}\big\}.
\end{align*}
\begin{align*}
PC=\big\{&y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R}
 \text{ is right dense continuous at
$t_k, k=1, 2, \dots$ for which }\\
&\quad \text{$y(t^{-}_k), y(t^{+}_k), y^{\Delta}(t^{-}_k),
y^{\Delta}(t^{+}_k)$ exist and
$y(t^{-}_k)=y(t_k), y^{\Delta}(t^{-}_k)=y^{\Delta}(t_k)$}\big\}.
\end{align*}

\begin{definition} \label{def1} \rm
 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 condition $y(t^{+}_0)=y_0,
y^{\Delta}(t^{+}_0)=y^{\Delta}_0$.
\end{definition}

\begin{definition} \label{def2} \rm
 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.
\end{definition}

In recent years, the theory of dynamic equations on time scales,
which provides powerful new tools for exploring connections between
the traditionally separated fields, has been developing rapidly and
has received much attention. We refer the reader to the book by
Bohner and Peterson $[4]$ and to the papers cited therein. The time
scales calculus has a tremendous potential for applications in
mathematical models of real processes, for instance, in
biotechnology, chemical technology, economic, neural networks,
physics, social sciences and so on, see the monographs of Aulbach
and Hilger [2], Bohner and Perterson $[4]$ and the references
therein.

Very recently, impulsive dynamic equations on time scales have been
investigate by Agarwal et al.$[1]$, Belarbi et al.$[5]$, Benchohra et al.
$[6-9]$ and so forth.
Benchohra et al [9]. 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 guaranted by some simple conditions.
In $[6]$, M.Benchohra et al. discuss the existence of oscillatory and
nonoscillatory solutions for first order impulsive dynamic equations
on time scales using lower and upper solutions method.

The oscillations of impulsive differential equations have been
investigated  by many authors and they gained many classical
results. See  Chen and Feng $[10]$ and the papers cited therein.
Using the method of  Chen and  Feng $[10]$, the present paper is
devoted to study the oscillations of a kind of very extensive
second order impulsive nonlinear dynamic equations on time scales.
An example is given to show that though a dynamic equations 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.

In the following, we always assume the solutions of \eqref{e1.1}
exist in $\mathbb{J}_{\mathbb{T}}$. Our attention is restricted to
those  solution $y$ of \eqref{e1.1} which exist on half line
$\mathbb{J}_{\mathbb{T}}$ with $\sup\{|y(t)|: t\geq t_0\}\neq 0$ for
any $t_0\geq t_{y}$, where $t_{y}$ is dependent on the solution $y$
of $\eqref{e1.1}$. To the best of our knowledge, the question of the
oscillations for second order nonlinear impulsive dynamic equations
has not been yet considered. Hence, these results can be considered
as a contribution to this field.

\section{Main results }

In this section, we give some new oscillation criteria for
\eqref{e1.1}. In order to prove our main results, we need the
following auxiliary result.

\begin{lemma} \label{lem1}
 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)]
$$
(t_1-t_0)+\frac{b^{*}_1}{a_1}(t_2-t_1)
+\frac{b^{*}_1 b^{*}_2}{a_1 a_2}(t_3-t_2)
+\dots+\frac{b^{*}_1 b^{*}_2\dots b^{*}_n}{a_1 a_2\dots a_n}(t_{n+1}-t_n)
+\dots=\infty,
$$
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{itemize}
\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
$$
y^{\Delta}(t^{+}_j)=h_j(y^{\Delta}(t_j))\leq b^{*}_j y^{\Delta}(t_j)<0.
$$
Let $y^{\Delta}(t^{+}_j)=-\alpha \ \ (\alpha>0)$. From \eqref{e1.1},
for $t\in(t_{j+i-1}, t_{j+i}]_{\mathbb{T}}, i=1, 2, \dots$, we
obtain
\begin{equation}
y^{\Delta \Delta}(t)=-f(t, y^{\sigma}(t))\leq
-p(t)\varphi(y^{\sigma}(t))\leq 0, \label{e2.1}
\end{equation}
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}(y^{\Delta}(t_{j+1}))
\leq b^{*}_{j+1}y^{\Delta}(t_{j+1})
\leq -b^{*}_{j+1}\alpha<0.
\end{gathered} \label{e2.2}
\end{equation}
It is easy to show that for any positive integer $n\geq 2$,
\begin{equation}
y^{\Delta}(t_{j+n})\leq -b^{*}_{j+n-1}b^{*}_{j+n-2}\dots b^{*}_{j+1}\alpha <0.
 \label{e2.3}
\end{equation}
Now, we claim that for any positive integer $n\geq 2$,
\begin{equation}
\begin{aligned}
y(t_{j+n})
&\leq a_{j+n-1}a_{j+n-2}\dots a_{j+1}\Big[y(t^{+}_j)-\alpha(t_{j+1}-t_j)
-\frac{b^{*}_{j+1}}{a_{j+1}}\alpha(t_{j+2}-t_{j+1})\\
&\quad -\dots -\frac{b^{*}_{j+n-1}b^{*}_{j+n-2}\dots b^{*}_{j+1}}{a_{j+n-1}
 a_{j+n-2} \dots a_{j+1}}\alpha(t_{j+n}-t_{j+n-1})\Big].
\end{aligned} \label{e2.4}
\end{equation}
Since $y^{\Delta}(t)$ is nonincreasing in $(t_j,
t_{j+1}]_{\mathbb{T}}$, hence
\begin{equation}
y^{\Delta}(t)\leq y^{\Delta}(t^{+}_j) \quad  t\in (t_j,
t_{j+1}]_{\mathbb{T}}. \label{e2.5}
\end{equation}
Integrating \eqref{e2.5} and using \eqref{e2.2}, we obtain
\begin{equation}
y(t_{j+1})\leq y(t^{+}_j)+y^{\Delta}(t^{+}_j)(t_{j+1}-t_j)
=y(t^{+}_j)-\alpha(t_{j+1}-t_j). \label{e2.6}
\end{equation}
Similarly to \eqref{e2.6} and using (H2), \eqref{e2.2} and \eqref{e2.6}, we get
\begin{align*}
y(t_{j+2})
&\leq y(t^{+}_{j+1})+y^{\Delta}(t^{+}_{j+1})(t_{j+2}-t_{j+1})\\
&=g_{j+1}(y(t_{j+1}))+h_{j+1}(y^{\Delta}(t_{j+1}))(t_{j+2}-t_{j+1})\\
&\leq a_{j+1}y(t_{j+1})+b^{*}_{j+1}y^{\Delta}(t_{j+1})(t_{j+2}-t_{j+1})\\
&\leq a_{j+1}\big[y(t^{+}_j)-\alpha(t_{j+1}-t_j)
-\frac{b^{*}_{j+1}}{a_{j+1}}\alpha(t_{j+2}-t_{j+1})\big].
\end{align*}
Then \eqref{e2.4} holds for $n=2$. Now we suppose that \eqref{e2.4}
holds for $n=m$, i.e.
\begin{equation}
\begin{aligned}
y(t_{j+m})
&\leq a_{j+1}a_{j+2}\dots a_{j+m-1}\Big[y(t^{+}_j)-\alpha(t_{j+1}-t_j)
-\frac{b^{*}_{j+1}}{a_{j+1}}\alpha(t_{j+2}-t_{j+1}) \\
&\quad -\dots-\frac{b^{*}_{j+1}b^{*}_{j+2}\dots b^{*}_{j+m-1}}
{a_{j+1}a_{j+2}\dots a_{j+m-1}}\alpha(t_{j+m}-t_{j+m-1})\Big],
\end{aligned} \label{e2.7}
\end{equation}
now we  prove that \eqref{e2.4} holds for $n=m+1$. Since
$y^{\Delta}(t)$ is nonincreasing in $(t_{j+m},
t_{j+m+1}]_{\mathbb{T}}$, we have
$$
y^{\Delta}(t)\leq y^{\Delta}(t^{+}_{j+m})  \quad t\in (t_{j+m},
t_{j+m+1}]_{\mathbb{T}}.
$$
Integrating  and using (H2), \eqref{e2.2}, \eqref{e2.3} and \eqref{e2.7},
we obtain
\begin{align*}
y(t_{j+m+1})
&\leq y(t^{+}_{j+m})+y^{\Delta}(t^{+}_{j+m})(t_{j+m+1}-t_{j+m})\\
&\leq a_{j+m}y(t_{j+m})+b^{*}_{j+m}y^{\Delta}(t_{j+m})(t_{j+m+1}-t_{j+m})\\
&\leq a_{j+1}a_{j+2}\dots a_{j+m}\Big[y(t^{+}_j)-\alpha(t_{j+1}-t_j)
-\frac{b^{*}_{j+1}}{a_{j+1}}\alpha(t_{j+2}-t_{j+1})\\
&\quad-\dots
 -\frac{b^{*}_{j+1}b^{*}_{j+2}\dots b^{*}_{j+m-1}}
{a_{j+1}a_{j+2}\dots a_{j+m-1}}\alpha(t_{j+m}-t_{j+m-1})\Big]\\
&\quad -b^{*}_{j+1}b^{*}_{j+2}\dots b^{*}_{j+m}\alpha(t_{j+m+1}-t_{j+m})\\
&=a_{j+1}a_{j+2}\dots a_{j+m}\Big[y(t^{+}_j)-\alpha(t_{j+1}-t_j)
 -\frac{b^{*}_{j+1}}{a_{j+1}}\alpha(t_{j+2}-t_{j+1}) \\
&\quad -\dots-\frac{b^{*}_{j+1}b^{*}_{j+2}\dots b^{*}_{j+m-1}}
{a_{j+1}a_{j+2}\dots a_{j+m-1}}\alpha(t_{j+m}-t_{j+m-1})\\
&\quad -\frac{b^{*}_{j+1}b^{*}_{j+2}\dots b^{*}_{j+m}}{a_{j+1}a_{j+2}
 \dots a_{j+m}}
\alpha(t_{j+m+1}-t_{j+m})\Big].
\end{align*}
Then \eqref{e2.4} holds for $n=m+1$. By induction, \eqref{e2.4}
holds for any positive integer $n\geq 2$. \eqref{e2.4} and (H3) is
contrary to $y(t)>0$. Therefore, $y^{\Delta}(t_k)\geq 0$ ($t_k\geq
t'_0$). From (H2), we get 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 nonincreasing in $(t_k, t_{k+1}]_{\mathbb{T}}$,
we know $y^{\Delta}(t)\geq y^{\Delta}(t_{k+1})\geq 0, t\in (t_k,
t_{k+1}]_{\mathbb{T}}$. The proof of Lemma \ref{lem1} is complete.
\end{proof}

\begin{remark} \label{rmk1} \rm
 In the case of $y(t)$ 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 $t_k \geq T$.
\end{remark}

\begin{theorem} \label{thm1}
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}
\begin{aligned}
&\int^{t_1}_{t_0}p(t)\Delta t+\frac{1}{b_1}\int^{t_2}_{t_1}p(t)\Delta t
+\frac{1}{b_1 b_2}\int^{t_3}_{t_2}p(t)\Delta t\\
&+\dots +\frac{1}{b_1 b_2\dots b_n}\int^{t_{n+1}}_{t_n}p(t)\Delta t
+\dots=\infty,
\end{aligned} \label{e2.8}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}  Suppose to the contrary that \eqref{e1.1} has a
nonoscillatory solution $y(t)$, without loss of generality, we may
assume that $y(t)$ is eventually positive solution of \eqref{e1.1},
i.e. $y(t)>0, t\geq t_0$ and $k_0=1$. From lemma \ref{lem1}, 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{e2.9}
\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}, we get 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'(y(t)+h\mu(t)y^{\Delta}(t))dh y^{\Delta}(t)\\
&\leq -p(t)-\frac{\varphi(y(t))}{\varphi(y^{\sigma}(t))}
(\frac{y^{\Delta}(t)}{\varphi(y(t))})^2
\int^1_0\varphi'(y(t)+h\mu(t)y^{\Delta}(t))dh\\
&\leq -p(t),
\end{aligned} \label{e2.10}
\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{e2.11}
\end{equation}
Integrating \eqref{e2.10}, we have
\begin{equation}
w(t_1)\leq w(t^{+}_0)-\int^{t_1}_{t_0}p(t)\Delta t.
 \label{e2.12}
\end{equation}
Using \eqref{e2.11} and \eqref{e2.12}, we obtain
$$
w(t^{+}_1)\leq b_1w(t_1)\leq b_1w(t^{+}_0)-b_1\int^{t_1}_{t_0}p(t)\Delta t.
% \eqref{e2.13}}
$$
Similarly, we get
\begin{equation}
\begin{aligned}
w(t^{+}_2)&\leq b_2w(t_2)\leq b_2\Big[w(t^{+}_1)-\int^{t_2}_{t_1}p(t)
\Delta t\Big] \\
&\leq b_1 b_2 w(t^{+}_0)-b_1 b_2\int^{t_1}_{t_0}p(t)\Delta t
-b_2\int^{t_2}_{t_1}p(t)\Delta t.
\end{aligned} \label{e2.14}
\end{equation}
By induction, for any positive integer $n$, we have
\begin{equation}
\begin{aligned}
w(t^{+}_n)
&\leq b_1b_2\dots b_n w(t^{+}_0)-b_1b_2\dots b_n\int^{t_1}_{t_0}p(t)\Delta t
-b_2\dots b_n\int^{t_2}_{t_1}p(t)\Delta t\\
&\quad -\dots -b_{n-1}b_n\int^{t_{n-1}}_{t_{n-2}}p(t)\Delta t
-b_n\int^{t_{n}}_{t_{n-1}}p(t)\Delta t\\
&=b_1b_2\dots b_n\Big[w(t^{+}_0)-\int^{t_1}_{t_0}p(t)\Delta t
-\frac{1}{b_1}\int^{t_2}_{t_1}p(t)\Delta t-\dots \\
&\quad -\frac{1}{b_1b_2\dots b_{n-2}}\int^{t_{n-1}}_{t_{n-2}}p(t)\Delta t
-\frac{1}{b_1b_2\dots b_{n-1}}\int^{t_{n}}_{t_{n-1}}p(t)\Delta t\Big].
\end{aligned}  \label{e2.15}
\end{equation}
Using \eqref{e2.8} and $b_k>0$, $k=1, 2, \dots$, we obtain
$w(t^{+}_n)<0$, $n\to \infty$, which contradicts to $w(t^{+}_n)\geq
0$.
\end{proof}

\begin{theorem} \label{thm2}
 Assume that {\rm (H1)--(H3)} hold and
$\varphi(ab)\geq \varphi(a)\varphi(b)$ for any $ab>0$. If
\begin{equation}
\begin{aligned}
&\int^{t_1}_{t_0}p(t)\Delta t+\frac{\varphi(a^{*}_1)}{b_1}
\int^{t_2}_{t_1}p(t)\Delta t
+\frac{\varphi(a^{*}_1)\varphi(a^{*}_2)}{b_1 b_2}\int^{t_3}_{t_2}p(t)\Delta t\\
&+\dots+\frac{\varphi(a^{*}_1)\varphi(a^{*}_2)\dots
\varphi(a^{*}_n)}{b_1 b_2\dots b_n}
\int^{t_{n+1}}_{t_n}p(t)\Delta t +\dots=\infty,
\end{aligned} \label{e2.16}
\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{lem1} yields
$y^{\Delta}(t)\geq 0, t\geq t_0$,
define $w(t)$ as in \eqref{e2.9} and we get
$w(t_k)\geq 0$, $t\geq t_0$, $w(t^{+}_k)\geq 0$,
$k=1, 2, \dots$ and \eqref{e2.10} 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{e2.17}
\end{equation}
As in  the proof of \eqref{e2.15}, by induction,
for any positive integer $n$,
\begin{align*}
&w(t^{+}_n) \\
&\leq \frac{b_1b_2\dots b_n}{\varphi(a^{*}_1)\varphi(a^{*}_2)\dots \varphi(a^{*}_n)}
\Big[w(t^{+}_0)-\int^{t_1}_{t_0}p(t)\Delta t
-\frac{\varphi(a^{*}_1)}{b_1}\int^{t_2}_{t_1}p(t)\Delta t -\dots\\
&\quad -\frac{\varphi(a^{*}_1)\varphi(a^{*}_2)\dots \varphi(a^{*}_{n-2})}
{b_1b_2\dots b_{n-2}}\int^{t_{n-1}}_{t_{n-2}}p(t)\Delta t
-\frac{\varphi(a^{*}_1)\varphi(a^{*}_2)\dots \varphi(a^{*}_{n-1})}
{b_1b_2\dots b_{n-1}}\int^{t_{n}}_{t_{n-1}}p(t)\Delta t\Big].
\end{align*}
Let $n\to \infty$ and use \eqref{e2.16}, we obtain the desired contradiction.
\end{proof}

In the following , we will use the hypothesis
\begin{itemize}
\item[(H4)] $\int^{\pm\infty}_{\pm\epsilon}\frac{\Delta u}{\varphi(u)}<\infty$,
for any $\epsilon>0$,
where $\int^{\pm\infty}_{\pm\epsilon}\frac{\Delta u}{\varphi(u)}<\infty$
denotes
$\int^{\infty}_{\epsilon}\frac{\Delta u}{\varphi(u)}<\infty$ and
$\int^{-\infty}_{-\epsilon}\frac{\Delta u}{\varphi(u)}<\infty$.
\end{itemize}

\begin{theorem} \label{thm3}
 Assume that {\rm (H1)--(H4)} hold and there exists a positive
integer $k_0$ such that $a^{*}_k\geq 1$ for $k\geq k_0$. If
\begin{equation}
\begin{aligned}
&\sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}\Big[\int^{t_{k+1}}_sp(t)\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t \\
&+\dots+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t
+\dots\Big]\Delta s=\infty,
\end{aligned} \label{e2.18}
\end{equation}
then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 As before, we may assume $y(t)>0, t\geq t_0$ be a
nonoscillatory solution of \eqref{e1.1} and $k_0=1$, Lemma \ref{lem1} shows that
$y^{\Delta}(t^{+}_k)\geq 0$, $k=1, 2, \dots$ and
$y^{\Delta}(t)\geq 0, t\geq t_0$.
Since $a^{*}_k\geq 1$, $k=1, 2, \dots$, we get
\begin{equation}
y(t^{+}_0)\leq y(t_1)\leq y(t^{+}_1)\leq y(t_2)\leq y(t^{+}_2)\leq \dots,
 \label{e2.19}
\end{equation}
its easy to see that $y(t)$ is nondecreasing in $[t_0, \infty)$,
hence \eqref{e1.1} yields
\begin{equation}
y^{\Delta \Delta}(t)=-f(t, y^{\sigma}(t))\leq
-p(t)\varphi(y^{\sigma}(t)), \quad t\neq t_k;
 \label{e2.20}
\end{equation}
hence, $y^{\Delta}(t_1)-y^{\Delta}(t^{+}_0)\leq
-\int^{t_1}_{t_0}p(t) \varphi(y^{\sigma}(t))\Delta t$. Using (H2),
we obtain
$$
y^{\Delta}(t^{+}_0)\geq
y^{\Delta}(t_1)+\int^{t_1}_{t_0}p(t)\varphi(y^{\sigma}(t)) \Delta t
\geq
\frac{y^{\Delta}(t^{+}_1)}{b_1}+\int^{t_1}_{t_0}p(t)\varphi(y^{\sigma}(t))\Delta
t.
$$
Similarly,
$$
y^{\Delta}(t^{+}_1) \geq
\frac{y^{\Delta}(t^{+}_2)}{b_2}+\int^{t_2}_{t_1}p(t)\varphi(y^{\sigma}(t))\Delta
t.
$$
Generally, for any positive integer $n$, we get
$$
y^{\Delta}(t^{+}_n)\geq y^{\Delta}(t_{n+1})
+\int^{t_{n+1}}_{t_n}p(t)\varphi(y^{\sigma}(t))\Delta t \geq
\frac{y^{\Delta}(t^{+}_{n+1})}{b_{n+1}}
+\int^{t_{n+1}}_{t_n}p(t)\varphi(y^{\sigma}(t))\Delta t.
% \eqref{e2.21}}
$$
 From this inequality and \eqref{e2.20}, noting that
$y^{\Delta}(t^{+}_k)\geq 0$, $k=1, 2, \dots$, we have for $s\in
(t_k, t_{k+1}]_{\mathbb{T}}$,
\begin{align*}
y^{\Delta}(s)
&\geq \int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t+y^{\Delta}(t_{k+1})\\
&\geq \int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{y^{\Delta}(t^{+}_{k+1})}{b_{k+1}}\\
&\geq \int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\Big[\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta
t
+\frac{y^{\Delta}(t^{+}_{k+2})}{b_{k+2}}\Big]\\
&\geq \int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta t\\
&\quad
+\frac{1}{b_{k+1}b_{k+2}}\int^{t_{k+3}}_{t_{k+2}}p(t)\varphi(y^{\sigma}(t))
\Delta t
+\frac{y^{\Delta}(t^{+}_{k+3})}{b_{k+1}b_{k+2}b_{k+3}}\\
&\geq \dots
 \geq \int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots\\
&\quad +\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{y^{\Delta}(t^{+}_{k+n+1})}{b_{k+1}b_{k+2}\dots b_{k+n+1}}.
\end{align*} % \eqref{e2.22}}
Noting that $b_k>0$ and $y^{\Delta}(t^{+}_k)\geq 0$, $k=1, 2, \dots$,
the above inequality yields
\begin{equation}
\begin{aligned}
y^{\Delta}(s) &\geq
\int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots\\
&\quad +\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\varphi(y^{\sigma}(t))\Delta t,
\end{aligned} \label{e2.23}
\end{equation}
holds for any positive integer $n$, then
\begin{equation}
\begin{aligned}
y^{\Delta}(s) &\geq
\int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots\\
&\quad +\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots.
\end{aligned} \label{e2.24}
\end{equation}
Using (H1) and the above inequality, we obtain that for $s\in (t_k,
t_{k+1}]_{\mathbb{T}}$,
\begin{align*}
\frac{y^{\Delta}(s)}{\varphi(y(s))} &\geq
\int^{t_{k+1}}_{s}p(t)\frac{\varphi(y^{\sigma}(t))}{\varphi(y(s))}\Delta
t +\frac{1}{b_{k+1}}
\int^{t_{k+2}}_{t_{k+1}}p(t)\frac{\varphi(y^{\sigma}(t))}{\varphi(y(s))}\Delta t+\dots\\
&\quad +\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\frac{\varphi(y^{\sigma}(t))}{\varphi(y(s))}\Delta t+\dots\\
&\geq \int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t\\
&\quad +\dots
+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots.
\end{align*}
Integrating it from $t_k$ to $t_{k+1}$ and using \eqref{e2.3}, we have
\begin{align*}
&\int^{t_{k+1}}_{t_k}\Big[\int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t+\dots\\
&+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots\Big]\Delta s\\
&\leq \int^{t_{k+1}}_{t_k}\frac{y^{\Delta}(s)}{\varphi(y(s))}\Delta s\\
&=\int^{y(t_{k+1})}_{y(t^{+}_k)}\frac{1}{\varphi(u)}\Delta u.
\end{align*} % \eqref{e2.25}

Using \eqref{e2.19}, and (H4), the above inequality  yields
\begin{align*}
&\sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}\Big[\int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t+\dots\\
&+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t
+\dots\Big]\\
&\leq \sum^{\infty}_{k=0}\int^{y(t_{k+1})}_{y(t^{+}_k)}\frac{1}{\varphi(u)}
 \Delta u \\
&\leq \int^{\infty}_{y(t^{+}_0)}\frac{1}{\varphi(u)}\Delta u<\infty,
\end{align*} % \eqref{e2.26}
which contradicts  \eqref{e2.18}.
\end{proof}

\begin{theorem} \label{thm4}
 Suppose that {\rm (H1)--(H4)} hold and there exists a positive
integer $k_0$ such that $a^{*}_k\geq 1$ for $k\geq k_0$.
Assume, furthermore, that
$\varphi(ab)\geq \varphi(a)\varphi(b)$ for any $ab>0$ and
\begin{equation}
\begin{aligned}
&\sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}\Big[\int^{t_{k+1}}_sp(t)\Delta t
+\frac{\varphi(a^{*}_{k+1})}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t\\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})}{b_{k+1}b_{k+2}}
\int^{t_{k+3}}_{t_{k+2}}p(t)\Delta t+\dots \\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots
 \varphi(a^{*}_{k+n})} {b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t +\dots\Big]\Delta s=\infty.
\end{aligned}  \label{e2.27}
\end{equation}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
 As before, we may assume that $y(t)>0$, $t\geq t_0$, is a
nonoscillatory solution of \eqref{e1.1} and $k_0=1$. According to
the proof of Theorem \ref{thm3}, \eqref{e2.19} and \eqref{e2.24}
hold. Furthermore, from $(H1)$ and Lemma \ref{lem1}, we obtain
$\varphi(y)$ is nondecreasing and $y(t)$ is also nondecreasing in
$(t_k, t_{k+1}]_{\mathbb{T}}$, $k=0, 1, 2, \dots$. Therefore,
$\varphi(y(t))$ is nondecreasing in $(t_k, t_{k+1}]_{\mathbb{T}}$.
Hence,
$$
\varphi(y(t^{+}_{k+1}))\geq \varphi(a^{*}_{k+1}y(t_{k+1}))
\geq \varphi(a^{*}_{k+1})\varphi(y(t_{k+1})),
$$
and
$$
\varphi(y(t^{+}_{k+2}))\geq \varphi(a^{*}_{k+2}y(t_{k+2}))
\geq \varphi(a^{*}_{k+2})\varphi(y(t^{+}_{k+1}))
\geq \varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\varphi(y(t_{k+1})).
$$
By induction, it can be proved that for any positive integer $n$,
\begin{equation}
\varphi(y(t^{+}_{k+n}))\geq
\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots\varphi(a^{*}_{k+n})
\varphi(y(t_{k+1})).
 \label{e2.28}
\end{equation}
 From this inequality, \eqref{e2.24}, and using the fact that
$\varphi(y(t))$ is nondecreasing, we obtain, for $s\in (t_k,
t_{k+1}]_{\mathbb{T}}$,
\begin{align*}
y^{\Delta}(s) &\geq
\int^{t_{k+1}}_{s}p(t)\varphi(y^{\sigma}(t))\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots\\
&\quad +\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\varphi(y^{\sigma}(t))\Delta t+\dots\\
&\geq \varphi(y(s))\int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{\varphi(y(t^{+}_{k+1}))}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta
t
+\dots\\
&\quad +\frac{\varphi(y(t^{+}_{k+n}))}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots\\
&\geq \varphi(y(s))\int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{\varphi(a^{*}_{k+1})\varphi(y(t_{k+1}))}{b_{k+1}}
\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t+\dots\\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots\varphi(a^{*}_{k+n})
\varphi(y(t_{k+1}))}{b_{k+1}b_{k+2}\dots b_{k+n}}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots.
\end{align*}
Hence,
\begin{align*}
\frac{y^{\Delta}(s)}{\varphi(y(s))}
&\geq \int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{\varphi(a^{*}_{k+1})}{b_{k+1}}\frac{\varphi(y(t_{k+1}))}{\varphi(y(s))}
\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t+\dots\\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots\varphi(a^{*}_{k+n})}
{b_{k+1}b_{k+2}\dots b_{k+n}}\frac{\varphi(y(t_{k+1}))}{\varphi(y(s))}
\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots\\
&\geq \int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{\varphi(a^{*}_{k+1})}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t+\dots\\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots\varphi(a^{*}_{k+n})}
{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t+\dots.
\end{align*}
Integrating the above inequality  and using \eqref{e2.19}, \eqref{e2.8},
 we obtain
\begin{align*}
&\sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}\Big[\int^{t_{k+1}}_{s}p(t)\Delta t
+\frac{\varphi(a^{*}_{k+1})}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t
+\dots\\
&\quad +\frac{\varphi(a^{*}_{k+1})\varphi(a^{*}_{k+2})\dots\varphi(a^{*}_{k+n})}
{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t
+\dots\Big]\\
&\leq \sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}
\frac{y^{\Delta}(s)}{\varphi(y(s))}\\
&=\sum^{\infty}_{k=0}\int^{y(t_{k+1})}_{y(t^{+}_k)}\frac{1}{\varphi(u)}
 \Delta u \\
&\leq \int^{\infty}_{y(t^{+}_0)}\frac{1}{\varphi(u)}\Delta u<\infty,
\end{align*}
\end{proof}
which contradicts \eqref{e2.27}.

From Theorems \ref{thm1}--\ref{thm4}, we have the following corollaries.

\begin{corollary} \label{coro1}
 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
\begin{align*}
&\int^{t_1}_{t_0}p(t)\Delta t+\frac{1}{b_1}\int^{t_2}_{t_1}p(t)\Delta t
+\frac{1}{b_1b_2}\int^{t_3}_{t_2}p(t)\Delta t+\dots
+\frac{1}{b_1b_2\dots b_n}\int^{t_{n+1}}_{t_n}p(t)\Delta t\\
&\geq \int^{t_1}_{t_0}p(t)\Delta t+\int^{t_2}_{t_1}p(t)\Delta t
+\int^{t_3}_{t_2}p(t)\Delta t+\dots
+\int^{t_{n+1}}_{t_n}p(t)\Delta t\\
&=\int^{t_{n+1}}_{t_0}p(t)\Delta t.
\end{align*} %\eqref{e2.29}
Let $n\to \infty$, from $\int^{\infty}p(t)\Delta t=\infty$,
the above inequality yields
\eqref{e2.8}. By Theorem \ref{thm1}, we conclude that \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary} \label{coro2}
 Assume that {\rm (H1)--(H4)} 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
$\int^{\infty}\int^{\infty}_{s}p(t)\Delta t\Delta s=\infty$,
then \eqref{e1.1} is oscillatory.
\end{corollary}

The proof of the above result is similar to the proof of Corollary
\ref{coro1}
and using Theorem \ref{thm3}.

\begin{corollary} \label{coro3}
 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 (\frac{t_{k+1}}{t_k})^{\alpha},\quad
\text{for } k\geq k_0.
 \label{e2.30}
\end{equation}
If
\begin{equation}
\int^{\infty}t^{\alpha}p(t)\Delta t=\infty.  \label{e2.31}
\end{equation}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof} As before, let $k_0=1$. Then  \eqref{e2.30} yields
\begin{align*}
&\int^{t_1}_{t_0}p(t)\Delta t+\frac{1}{b_1}\int^{t_2}_{t_1}p(t)\Delta t
+\frac{1}{b_1b_2}\int^{t_3}_{t_2}p(t)\Delta t
\dots+\frac{1}{b_1b_2\dots b_n}\int^{t_{n+1}}_{t_n}p(t)\Delta t\\
&\geq \frac{1}{t^{\alpha}_1}\left[\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_{n+1}}_{t_n}t^{\alpha}_{n+1}p(t)\Delta t\right]\\
&\geq \frac{1}{t^{\alpha}_1}\left[\int^{t_2}_{t_1}t^{\alpha}p(t)\Delta t
+\int^{t_3}_{t_2}t^{\alpha}p(t)\Delta t+\dots
+\int^{t_{n+1}}_{t_n}t^{\alpha}p(t)\Delta t\right]\\
&=\frac{1}{t^{\alpha}_1}\int^{t_{n+1}}_{t_1}t^{\alpha}p(t)\Delta t.
\end{align*} % \eqref{e2.32}
Let $n\to \infty$. Then using \eqref{e2.31}, the above inequality
yields \eqref{e2.8}. By Theorem \ref{thm1}, we obtain that \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary} \label{coro4}
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
$$
\frac{\varphi(a^{*}_k)}{b_k}\geq (\frac{t_{k+1}}{t_k})^{\alpha}, \quad
\text{for } k\geq k_0.
$$
If $\int^{\infty}t^{\alpha}p(t)\Delta t=\infty$,
then \eqref{e1.1} is oscillatory.
\end{corollary}

The above corollary follows from Theorem \ref{thm2}, and  its proof is similar
to that of Corollary \ref{coro3}.

\begin{corollary} \label{coro5}
Suppose that {\rm (H1)--(H4)} 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 t^{\alpha}_{k+1},\quad
for \quad k\geq k_0. \label{e2.33}
\end{equation}
If
\begin{equation}
\sum^{\infty}_{k=0}(t_{k+1}-t_k)\int^{\infty}_{t_{k+1}}t^{\alpha}p(t)\Delta t
=\infty.  \label{e2.34}
\end{equation}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
 As before, we assume $k_0=1$, $t_1\geq 1$.
From \eqref{e2.33}, we get
\begin{align*}
\frac{1}{b_{k+1}}&\geq t^{\alpha}_{k+2},
\frac{1}{b_{k+1}b_{k+2}} \\
&\geq t^{\alpha}_{k+2}t^{\alpha}_{k+3}, \dots,
\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}} \\
&\geq t^{\alpha}_{k+2}t^{\alpha}_{k+3}\dots t^{\alpha}_{k+n+1},\dots.
\end{align*}
Similar to the proof of Corollary \ref{coro3}, we have
\begin{align*}
&\int^{t_{k+1}}_{s}p(t)\Delta t+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t
+\frac{1}{b_{k+1}b_{k+2}}\int^{t_{k+3}}_{t_{k+2}}p(t)\Delta t+\dots\\
&+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t\\
&\geq \int^{t_{k+n+1}}_{t_{k+1}}t^{\alpha}p(t)\Delta t.
\end{align*}
Let $n\to \infty$, we have
\begin{align*}
&\int^{t_{k+1}}_{s}p(t)\Delta t+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t
+\frac{1}{b_{k+1}b_{k+2}}\int^{t_{k+3}}_{t_{k+2}}p(t)\Delta t+\dots\\
&+\frac{1}{b_{k+1}b_{k+2}\dots b_{k+n}}\int^{t_{k+n+1}}_{t_{k+n}}p(t)\Delta t
+\dots\\
&\geq \int^{\infty}_{t_{k+1}}t^{\alpha}p(t)\Delta t.
\end{align*} % \eqref{e2.35}
Using \eqref{e2.34} and the above inequality,  we get
\begin{align*}
&\sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}
\Big[\int^{t_{k+1}}_sp(t)\Delta t
+\frac{1}{b_{k+1}}\int^{t_{k+2}}_{t_{k+1}}p(t)\Delta t
+\frac{1}{b_{k+1}b_{k+2}}\int^{t_{k+3}}_{t_{k+2}}p(t)\Delta t+\dots\Big]
 \Delta s\\
&\geq \sum^{\infty}_{k=0}\int^{t_{k+1}}_{t_k}
\int^{\infty}_{t_{k+1}}t^{\alpha}p(t)\Delta t\Delta s \\
&=\sum^{\infty}_{k=0}(t_{k+1}-t_k)\int^{\infty}_{t_{k+1}}t^{\alpha}p(t)\Delta t
=\infty.
\end{align*}
By Theorem \ref{thm3}, we obtain that \eqref{e1.1} is oscillatory.
\end{proof}

\begin{corollary} \label{coro6}
Suppose that {\rm (H1)--(H4)} hold and there exists a positive
integer $k_0$ and a constant $\alpha>0$ such that
$$
a^{*}_k\geq 1, \quad \frac{\varphi(a^{*}_k)}{b_k}\geq t^{\alpha}_{k+1},\quad
\text{for } k\geq k_0.
$$
Suppose that $\varphi(ab)\geq \varphi(a)\varphi(b)$ for any $ab>0$ and
$$
\sum^{\infty}_{k=0}(t_{k+1}-t_k)\int^{\infty}_{t_{k+1}}p(t)\Delta t=\infty.
$$
Then \eqref{e1.1} is oscillatory.
\end{corollary}

The proof of the above corollary is similar to that of Corollary \ref{coro5},
so we omit it.

\section{Example}

 Consider the 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 $\mu(t)\leq Kt$, and $K$ is a positive constant.
Since $a_k=a^{*}_k=\frac{k+1}{k}$, $b_k=b^{*}_k=1$,
$p(t)=\frac{1}{t\sigma^2(t)}$, $t_k=k$
and $\varphi(y)=y^{\gamma}$. it is easy to see that (H1)--(H3) hold. Let
$k_0=1$, $\alpha=3$, hence
\begin{gather*}
\frac{\varphi(a^{*}_k)}{b_k}=(\frac{k+1}{k})^{\gamma}
=(\frac{t_{k+1}}{t_k})^{\gamma}
\geq (\frac{t_{k+1}}{t_k})^{3},
\\
\int^{\infty}t^{\alpha}p(t)\Delta t=\int^{\infty}t^3
\frac{1}{t\sigma^2(t)}\Delta t
=\int^{\infty}(\frac{t}{\sigma(t)})^{2}\Delta t.
\end{gather*}
Since $\mu(t)\leq Kt$, we get
$$
\frac{t}{\sigma(t)}=\frac{t}{t+\mu(t)}\geq \frac{1}{1+K},
$$
hence
$$
\int^{\infty}(\frac{t}{\sigma(t)})^{2}\Delta t \geq
\frac{1}{(1+K)^2}\int^{\infty}\Delta t=\infty.
$$
By Corollary \ref{coro3}, we obtain that \eqref{e4.1} is oscillatory.
But by [3] we know that the
dynamic equation $y^{\Delta \Delta}(t)+\frac{1}{t\sigma^2(t)}y^{\gamma}(\sigma(t))=0$
is nonoscillatory.

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

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\end{document}