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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 40, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/40\hfil Interval criteria for oscillation]
{Interval criteria for oscillation of second-order impulsive
 differential equation with mixed nonlinearities}

\author[V. Muthulakshmi, E. Thandapani \hfil EJDE-2011/40\hfilneg]
{Velu Muthulakshmi, Ethiraju Thandapani}  % in alphabetical order

\address{Velu Muthulakshmi \newline
 Department of Mathematics, Periyar University,
Salem-636 011, India}
\email{vmuthupu@gmail.com}

\address{Ethiraju Thandapani \newline
 Ramanujan Institute For Advanced Study in Mathematics \\
 University of Madras \newline Chennai-600005, India}
\email{ethandapani@yahoo.co.in}

\thanks{Submitted October 6, 2010. Published March 9, 2011.}
\subjclass[2000]{34C10, 34A37}
\keywords{Oscillation; second order; impulse; damping term;
\hfill\break\indent mixed nonlinearities}

\begin{abstract}
 We establish sufficient conditions for the oscillation of
 all solutions to the second-order impulsive differential equation
 \begin{gather*}
 \big(r(t)x'(t)\big)'+p(t)x'(t)+q(t)x(t)+\sum_{i=1}^n
 q_i(t)|x(t)|^{\alpha_i}\operatorname{sgn} x(t)=e(t),\quad  t\neq \tau_k,\\
 x({\tau_k}^+)=a_k x({\tau_k}),\quad
 x'({\tau_k}^+)=b_k x'({\tau_k}).
 \end{gather*}
 The results obtained in this paper extend  some of the existing results
 and are illustrated by examples.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Consider the second-order impulsive differential equation,
with mixed nonlinearities,
\begin{equation} \label{e1}
\begin{gathered}
\big(r(t)x'(t)\big)'+p(t)x'(t)+q(t)x(t)
+\sum_{i=1}^n q_i(t)|x(t)|^{\alpha_i}\operatorname{sgn}  x(t)=e(t),
\quad t\neq \tau_k,\\
x({\tau_k}^+)=a_k x({\tau_k}),\quad
 x'({\tau_k}^+)=b_k x'({\tau_k}),
\end{gathered}
\end{equation}
where  $t \geq t_0$, $k \in \mathbb{N}$,
${\tau_k}$ is the impulse moments sequence with
 \begin{gather*}
 0\leq t_0=\tau_0< \tau_1<\dots <\tau_k<\dots ,
 \lim_{k\to \infty} \tau_k=\infty,\\
x(\tau_k)=x(\tau_k ^-)=\lim_{t\to \tau_k^{-0}}x(t), \quad
x({\tau_k}^+)= \lim_{t\to \tau_k^{+0}}x(t),\\
x'(\tau_k)= x'({\tau_k}^-)= \lim_{h\to 0^-}
\frac{x(\tau_k+h)-x(\tau_k)}{h}, \quad
 x'({\tau_k}^+)= \lim_{h\to 0^+} \frac{x(\tau_k+h)-x({\tau_k}^+)}{h}.
\end{gather*}
Throughout this paper, assume that the following conditions hold
without further mention:
\begin{itemize}
\item[C1)]  $r\in C^1([t_0,\infty),(0,\infty))$,
 $p,q,q_i,e \in C([t_0,\infty),\mathbb{R})$, $i=1,2\dots ,n$;

\item[(C2)] $\alpha_1>\dots >\alpha_m>1>\alpha_{m+1}>\dots >\alpha_n>0$
are constants;

\item[(C3)] $b_k \geq a_k >0$, $k \in \mathbb{N}$ are constants.
\end{itemize}

Let $J\subset \mathbb{R}$ be an interval and define
\begin{align*}
PC(J,\mathbb{R})
=\{&x:J\to \mathbb{R}:x(t) \text{ is piecewise-left-continuous}\\
&\text{and has discontinuity of first kind at }  \tau_k's\}.
\end{align*}

 By a \emph{solution} of  \eqref{e1}, we mean a
function $ x \in PC([t_0,\infty),\mathbb{R})$ with a property
$(rx')'\in PC([t_0,\infty),\mathbb{R})$ such that \eqref{e1}
is satisfied for all $ t \geq t_0$.
 A nontrivial solution is called \emph{oscillatory} if it
has arbitrarily large zeros;
otherwise, it is called \emph{nonoscillatory}.
An equation is called oscillatory if all its solutions are
oscillatory.


 In recent years the oscillation theory of impulsive differential
equations emerging as an important area of research, since such
equations have applications in control theory, physics, biology,
population dynamics, economics, etc. For further applications
and questions concerning existence and uniqueness of solutions
of impulsive differential equation, see for example \cite{l1} and
the references cited therein.

 In \cite{j1,l3,s1}, the authors established several
oscillation criteria for second-order impulsive differential
equations which are particular cases of \eqref{e1}.
Compared to second order ordinary differential
equations \cite{k1,l2,p1,s2,s3,t1,x1},
the oscillatory behavior of impulsive
second order differential equations received less attention
even though such equations have many applications.
Motivated by this observation, in this paper, we establish some
new oscillation criteria for all solutions of  \eqref{e1}.
Our results extend those obtained in \cite{t1} for equation without
impulses. Finally some examples are given to illustrate the results.

\section{Main results}

We begin with the following notation.
Let $k(s)= \max\{i:t_0<\tau_i<s\}$ and for $c_j<d_j$,
let $r_j= \max\{r(t):t\in [c_j,d_j]\}$, $j=1,2$.
For two constants $c,d \notin \{\tau_k\}$ with $c<d$
and a function $\phi \in C([c,d],\mathbb{R})$, we define an
operator $\Omega:C([c,d],\mathbb{R})\to \mathbb{R}$ by
\[
\Omega_c^d[\phi]=  \begin{cases}
 0,  & \text{for } k(c)=k(d),\\
 \phi(\tau_{k(c)+1})\theta(c)
+\sum_{i=k(c)+2}^{k(d)}\phi(\tau_i)\varepsilon(\tau_i),
&\text{for } k(c)<k(d),
\end{cases}
\]
where
\[
\theta(c)=\frac{b_{k(c)+1}-a_{k(c)+1}}{a_{k(c)+1}(\tau_{k(c)+1}-c)}, \quad
\varepsilon(\tau_i)=\frac{b_i-a_i}{a_i(\tau_i-\tau_{i-1})}.
\]
 Following Kong \cite{k1} and Philos \cite{l3},
we introduce a class of functions:
Let $D=\{(t,s):t_0\leq s \leq t\}$,
$H_1,H_2 \in C^1(D,\mathbb{R})$. A pair of functions
$(H_1,H_2)$ is said to belong to a function class $\mathcal{H}$,
if $H_1(t,t)=H_2(t,t)=0$,
$ H_1(t,s)>0$, $H_2(t,s)>0$ for $t>s$ and there exist
$h_1,h_2 \in L_{\rm loc}(D,\mathbb{R})$ such that
\begin{equation} \label{e2}
\frac{\partial H_1(t,s)}{\partial t}=h_1(t,s)H_1(t,s),\quad
\frac{\partial H_2(t,s)}{\partial s}=-h_2(t,s)H_2(t,s).
\end{equation}

To prove our main results we need the following lemma
due to Sun and Wong \cite{s3}.

\begin{lemma} \label{lem1}
 Let $\{\alpha_i\}$, $i=1,2,\dots ,n, $ be the n-tuple satisfying
$\alpha_1>\dots >\alpha_m>1>\alpha_{m+1}>\dots >\alpha_n>0$.
Then there exist an $n$-tuple $(\eta_1,\eta_2,\dots ,\eta_n)$
satisfying
\begin{equation} \label{e3}
\sum_{i=1}^n \alpha_i\eta_i=1,
\end{equation}
which also satisfies either
\begin{equation} \label{e4}
\sum_{i=1}^n \eta_i<1,\quad 0<\eta_i<1,
\end{equation}
or
\begin{equation} \label{e5}
\sum_{i=1}^n \eta_i=1,\quad 0<\eta_i<1.
\end{equation}
\end{lemma}

\begin{remark} \label{rmk1} \rm
 For a given set of exponents $\{\alpha_i\}$ satisfying
$\alpha_1>\dots >\alpha_m>1>\alpha_{m+1}>\dots >\alpha_n>0$,
Lemma 1 ensures the existence of an $n$-tuple
$(\eta_1,\eta_2,\dots ,\eta_n)$ such that either \eqref{e3}
and \eqref{e4} hold or \eqref{e3} and \eqref{e5} hold.
 When $n=2$ and $\alpha_1>1>\alpha_2>0$, in the first case, we have
\[
\eta_1=\frac{1-\alpha_2(1-\eta_0)}{\alpha_1-\alpha_2}, \quad
\eta_2=\frac{\alpha_1(1-\eta_0)-1}{\alpha_1-\alpha_2},
\]
where $\eta_0$ can be any positive number satisfying
$0<\eta_0<(\alpha_1-1)/\alpha_1$.
This will ensure that $0<\eta_1,\eta_2<1$ and
conditions \eqref{e3} and \eqref{e4} are satisfied.
In the second case, we simply solve \eqref{e3} and \eqref{e5}
and obtain
\[
\eta_1=\frac{1-\alpha_2}{\alpha_1-\alpha_2}, \quad
\eta_2=\frac{\alpha_1-1}{\alpha_1-\alpha_2}.
\]
\end{remark}

\begin{theorem} \label{thm1}
 Assume that for any $T>0$ , there exist
$c_j,d_j,\delta_j \notin \{\tau_k\}$, $j=1,2$ such that
$c_1<\delta_1<d_1\leq c_2<\delta_2<d_2$, and
\begin{equation} \label{e6}
 \begin{gathered}
q(t),q_i(t) \geq 0, \quad t \in [c_1,d_1]\cup [c_2,d_2],\;
 i=1,2\dots ,n;\\
e(t)\leq 0, \quad t \in [c_1,d_1]; \\
e(t)\geq 0, t \in [c_2,d_2]
\end{gathered}
\end{equation}
and if there exists $(H_1,H_2)\in \mathcal{H}$ such that
\begin{equation} \label{e7}
\begin{split}
&\frac{1}{H_1(\delta_j,c_j)}\int_{c_j}^{\delta_j}H_1(t,c_j)
\Big[Q(t)-\frac{1}{4}r(t) \Big(h_1(t,c_j)
-\frac{p(t)}{r(t)}\Big)^2\Big]dt \\
&+\frac{1}{H_2(d_j,\delta_j)}\int_{\delta_j}^{d_j}H_2(d_j,t)
\Big[Q(t)-\frac{1}{4}r(t) \Big(h_2(d_j,t)+\frac{p(t)}{r(t)}
\Big)^2\Big]dt\\
&> \Lambda(H_1,H_2;c_j,d_j),
\end{split}
\end{equation}
where
\begin{equation} \label{e8}
\Lambda(H_1,H_2;c_j,d_j)=\frac{r_j}{H_1(\delta_j,c_j)}
\Omega_{c_j}^{\delta_j}[H_1(.,c_j)]
+\frac{r_j}{H_2(d_j,\delta_j)}\Omega_{\delta_j}^{d_j}[H_2(d_j,.)]
\end{equation}
and
\begin{equation} \label{e9}
Q(t)=q(t)+k_0|e(t)|^{\eta_0}\prod_{i=1}^n q_i^{\eta_i}(t),\quad
k_0=\prod_{i=0}^n \eta_i^{-\eta_i}, \ \eta_0=1-\sum_{i=1}^n \eta_i
\end{equation}
and $\eta_1,\eta_2,\dots ,\eta_n$ are positive constants
satisfying \eqref{e3} and \eqref{e4} in Lemma 1, then
\eqref{e1} is oscillatory.
\end{theorem}

\begin{proof}
Let $x(t)$ be a solution of \eqref{e1}.
Suppose $x(t)$ does not have any zero in $[c_1,d_1]\cup [c_2,d_2]$.
Without loss of generality, we may assume that
$x(t)> 0 $ for $t \in [c_1,d_1] $.
When $x(t)< 0 $ for $t \in [c_2,d_2] $, the proof follows the same
argument using the interval $[c_2,d_2] $
instead of $[c_1,d_1] $. Define
\begin{equation} \label{e10}
w(t)=-\frac{r(t)x'(t)}{x(t)}, \quad t \in [c_1,d_1].
\end{equation}
Then for $t \in [c_1,d_1]$ and $t\neq \tau_k$, we have
\begin{equation} \label{e11}
w'(t)=q(t)+\sum_{i=1}^n q_i(t)x^{\alpha_i-1}(t)
 -e(t)x^{-1}(t)-\frac{p(t)}{r(t)}w(t)+\frac{w^2(t)}{r(t)}.
\end{equation}
Recall the arithmetic-geometric mean inequality,
\begin{equation} \label{e12}
\sum_{i=0}^n\eta_i u_i\geq \prod_{i=0}^n u_i^{\eta_i}, \quad
u_i\geq 0
\end{equation}
where $\eta_i>0, i=0,1,2,\dots ,n$, are chosen according to given
$\alpha_1,\alpha_2,\dots ,\alpha_n$ as in
 Lemma 1 satisfying \eqref{e3} and \eqref{e4}.
Now identify $u_0=\eta_0^{-1}|e(t)|x^{-1}(t)$ and
 $u_i=\eta_i^{-1}q_i(t)x^{\alpha_i-1}(t)$, $i=1,2,\dots ,n$, in
\eqref{e12}. Then equation \eqref{e11} becomes
\begin{equation} \label{e13}
\begin{split}
w'(t)& \geq  q(t)+\eta_0^{-\eta_0}|e(t)|^{\eta_0}x^{-\eta_0}(t)
 \prod_{i=1}^n \eta_i^{-\eta_i} q_i^{\eta_i}(t)
 x^{\eta_i(\alpha_i-1)}(t)-\frac{p(t)}{r(t)}w(t)
 +\frac{w^2(t)}{r(t)} \\
& =  Q(t)-\frac{p(t)}{r(t)}w(t)+\frac{w^2(t)}{r(t)}, \quad
  t \in (c_1,d_1), \; t\neq \tau_k.
\end{split}
\end{equation}
For $t = \tau_k$, $k=1,2,\dots $,
from  \eqref{e10}, we have
\begin{equation} \label{e14}
w({\tau_k}^+)=\frac{b_k}{a_k}w(\tau_k).
\end{equation}
 Notice that whether there are or not impulsive moments
in $[c_1,\delta_1]$ and $[\delta_1,d_1]$,
 we must consider the following 4 cases, namely,
$k(c_1)<k(\delta_1)<k(d_1)$;
$k(c_1)=k(\delta_1)<k(d_1)$;
$k(c_1)<k(\delta_1)=k(d_1)$ and
$k(c_1)=k(\delta_1)=k(d_1)$.

\subsection*{Case 1}
 If $k(c_1)<k(\delta_1)<k(d_1)$, then there are impulsive moments
$\tau_{k(c_1)+1}$, $\tau_{k(c_1)+2}$,\dots , $\tau_{k(\delta_1)} $
in $[c_1,\delta_1]$ and
$\tau_{k(\delta_1)+1},\tau_{k(\delta_1)+2},\dots ,\tau_{k(d_1)} $ in
$[\delta_1,d_1]$ respectively. Multiplying both sides of
inequality \eqref{e13} by $H_1(t,c_1)$, then integrating
it from $c_1$ to $\delta_1$ and using \eqref{e14}, we have
 %\label{e15}
\begin{align}
&\int_{c_1}^{\delta_1}H_1(t,c_1)Q(t)dt \nonumber \\
& \leq  \int_{c_1}^{\delta_1}H_1(t,c_1)w'(t)dt
 -\int_{c_1}^{\delta_1}H_1(t,c_1)\frac{w^2(t)}{r(t)}dt
 +\int_{c_1}^{\delta_1}H_1(t,c_1)w(t)\frac{p(t)}{r(t)}dt  \nonumber\\
&= \Big(\int_{c_1}^{\tau_{k(c_1)+1}}
 +\int_{\tau_{k(c_1)+1}}^{\tau_{k(c_1)+2}}+\dots
 +\int_{k(\delta_1)}^{\delta_1}\Big)H_1(t,c_1)dw(t)\\
&\quad -\int_{c_1}^{\delta_1}H_1(t,c_1)
  \Big[\frac{w^2(t)}{r(t)}-w(t)\frac{p(t)}{r(t)}\Big]dt  \nonumber\\
&= \sum_{i=k(c_1)+1}^{k(\delta_1)}H_1(\tau_i,c_1)
 \frac{a_i-b_i}{a_i}w(\tau_i)+H_1(\delta_1,c_1)w(\delta_1)  \label{e15}\\
&\quad  -\int_{c_1}^{\delta_1}\frac{H_1(t,c_1)}{r(t)}
 \Big[w^2(t)+r(t)w(t)\Big(h_1(t,c_1)-\frac{p(t)}{r(t)}\Big)\Big]dt \nonumber \\
&\leq  \sum_{i=k(c_1)+1}^{k(\delta_1)}H_1(\tau_i,c_1)
 \frac{a_i-b_i}{a_i}w(\tau_i)+H_1(\delta_1,c_1)w(\delta_1)  \nonumber\\
&\quad -\int_{c_1}^{\delta_1}\frac{H_1(t,c_1)}{r(t)}
 \Big[w(t)+\frac{r(t)}{2}\Big(h_1(t,c_1)-\frac{p(t)}{r(t)}\Big)
 \Big]^2dt  \nonumber\\
&\quad +\int_{c_1}^{\delta_1}\frac{H_1(t,c_1)r(t)}{4}\Big(h_1(t,c_1)
 -\frac{p(t)}{r(t)}\Big)^2dt  \nonumber \\
&= \sum_{i=k(c_1)+1}^{k(\delta_1)}H_1(\tau_i,c_1)
 \frac{a_i-b_i}{a_i}w(\tau_i)+H_1(\delta_1,c_1)w(\delta_1) \nonumber \\
&\quad+\frac{1}{4}\int_{c_1}^{\delta_1}H_1(t,c_1)r(t)\Big(h_1(t,c_1)
 -\frac{p(t)}{r(t)}\Big)^2dt. \nonumber
\end{align}
On the other hand, multiplying both sides of inequality \eqref{e13}
 by $H_2(d_1,t)$, then  integrating it from $\delta_1$ to $d_1$,
 we have
\begin{equation} \label{e16}
\begin{split}
\int_{\delta_1}^{d_1}H_2(d_1,t)Q(t)dt
& \leq  \sum_{i=k(\delta_1)+1}^{k(d_1)}H_2(d_1,\tau_i)
 \frac{a_i-b_i}{a_i}w(\tau_i)-H_2(d_1,\delta_1)w(\delta_1)  \\
& \quad +\frac{1}{4}\int_{\delta_1}^{d_1}H_2(d_1,t)r(t)
\Big(h_2(d_1,t)+\frac{p(t)}{r(t)}\Big)^2dt.
\end{split}
\end{equation}
Dividing \eqref{e15} and \eqref{e16} by $H_1(\delta_1,c_1)$
and $H_2(d_1,\delta_1)$ respectively, then adding them, we obtain
\begin{equation} \label{e17}
\begin{split}
&\frac{1}{H_1(\delta_1,c_1)}\int_{c_1}^{\delta_1}H_1(t,c_1)
 \Big[Q(t)-\frac{1}{4}r(t) \Big(h_1(t,c_1)
 -\frac{p(t)}{r(t)}\Big)^2\Big]dt \\
&+\frac{1}{H_2(d_1,\delta_1)}\int_{\delta_1}^{d_1}H_2(d_1,t)
 \Big[Q(t)-\frac{1}{4}r(t) \Big(h_2(d_1,t)
 +\frac{p(t)}{r(t)}\Big)^2\Big]dt\\
&\leq \frac{1}{H_1(\delta_1,c_1)}\sum_{i=k(c_1)+1}^{k(\delta_1)}
 H_1(\tau_i,c_1)\frac{a_i-b_i}{a_i}w(\tau_i) \\
&\quad +\frac{1}{H_2(d_1,\delta_1)}\sum_{i=k(\delta_1)+1}^{k(d_1)}
H_2(d_1,\tau_i)\frac{a_i-b_i}{a_i}w(\tau_i).
\end{split}
\end{equation}
Now for $t \in (c_1,\tau_{k(c_1)+1}]$,
\[
(r(t)x'(t))'+p(t)x'(t)=e(t)-q(t)x(t)-\sum_{i=1}^n
q_i(t)x^{\alpha_i}(t)\leq 0
\]
which implies that
$x'(t)\exp\big(\int_{c_1}^t \frac{r'(s)+p(s)}{r(s)}ds\big)$
is non-increasing on $(c_1,\tau_{k(c_1)+1}]$.
So for any $t \in (c_1,\tau_{k(c_1)+1}]$,
we have
\begin{align*}
x(t)-x(c_1)
&=x'(\xi)(t-c_1) \\
&\geq  \frac{x'(t)\exp\big(\int_{c_1}^t
 \frac{r'(s)+p(s)}{r(s)}ds\big)}{\exp
\big(\int_{c_1}^\xi \frac{r'(s)+p(s)}{r(s)}ds\big)}(t-c_1)\\
&\geq  x'(t)(t-c_1).
\end{align*}
for some $\xi \in (c_1,t)$.
Since $x(c_1)>0$, we have
\[
-\frac{x'(t)}{x(t)}\geq -\frac{1}{t-c_1}.
\]
Letting $t\to \tau_{k(c_1)+1}^-$, it follows that
\begin{equation} \label{e18}
w(\tau_{k(c_1)+1})
\geq  -\frac{r(\tau_{k(c_1)+1})}{\tau_{k(c_1)+1}-c_1}
\geq -\frac{r_1}{\tau_{k(c_1)+1}-c_1}.
\end{equation}
Similarly we can prove that on $(\tau_{i-1},\tau_i)$,
\begin{equation} \label{e19}
w(\tau_i) \geq  -\frac{r_1}{\tau_i-\tau_{i-1}}\quad
 \text{for } i=k(c_1)+2,\dots ,k(\delta_1).
\end{equation}
Using \eqref{e18}, \eqref{e19} and (C3), we obtain
\begin{align*}
&\sum_{i=k(c_1)+1}^{k(\delta_1)}\frac{b_i-a_i}{a_i}w(\tau_i)
 H_1(\tau_i,c_1)\\
&= \frac{b_{k(c_1)+1}-a_{k(c_1)+1}}{a_{k(c_1)+1}}w(\tau_{k(c_1)+1})
 H_1(\tau_{k(c_1)+1},c_1)
 +\sum_{i=k(c_1)+2}^{k(\delta_1)}\frac{b_i-a_i}{a_i}
 w(\tau_i)H_1(\tau_i,c_1)\\
&\geq  -r_1\Big[H_1(\tau_{k(c_1)+1},c_1)\theta(c_1)
 +\sum_{i=k(c_1)+2}^{k(\delta_1)}H_1(\tau_i,c_1)
 \varepsilon(\tau_i)\Big]  \\
&= -r_1\Omega_{c_1}^{\delta_1}[H_1(.,c_1)].
\end{align*}
Thus, we have
\[
\sum_{i=k(c_1)+1}^{k(\delta_1)}\frac{a_i-b_i}{a_i}w(\tau_i)
H_1(\tau_i,c_1) \leq r_1\Omega_{c_1}^{\delta_1}[H_1(.,c_1)],
\]
and
\[
\sum_{i=k(\delta_1)+1}^{k(d_1)}\frac{a_i-b_i}{a_i}w(\tau_i)
H_2(d_1,\tau_i) \leq r_1\Omega_{\delta_1}^{d_1}[H_2(d_1,.)].
\]
Therefore,  \eqref{e17} becomes
\begin{equation} \label{e20}
\begin{split}
&\frac{1}{H_1(\delta_1,c_1)}\int_{c_1}^{\delta_1}H_1(t,c_1)\Big[Q(t)
-\frac{1}{4}r(t) \Big(h_1(t,c_1)-\frac{p(t)}{r(t)}\Big)^2\Big]dt\\
& +\frac{1}{H_2(d_1,\delta_1)}\int_{\delta_1}^{d_1}H_2(d_1,t)
\Big[Q(t)-\frac{1}{4}r(t) \Big(h_2(d_1,t)+\frac{p(t)}{r(t)}
\Big)^2\Big]dt   \\
&\leq \frac{r_1}{H_1(\delta_1,c_1)}\Omega_{c_1}^{\delta_1}[H_1(.,c_1)]
+ \frac{r_1}{H_2(d_1,\delta_1)}\Omega_{\delta_1}^{d_1}[H_2(d_1,.)]\\
&= \Lambda(H_1,H_2;c_1,d_1)
\end{split}
\end{equation}
which contradicts \eqref{e7}.

\subsection*{Case 2}
 If $k(c_1)=k(\delta_1)<k(d_1)$, there is no impulsive moment
in $[c_1,\delta_1]$.Then we have
\begin{equation} \label{e21}
\begin{split}
&\int_{c_1}^{\delta_1}H_1(t,c_1)Q(t)dt \\
&\leq  \int_{c_1}^{\delta_1}H_1(t,c_1)w'(t)dt
-\int_{c_1}^{\delta_1}H_1(t,c_1)\Big[\frac{w^2(t)}{r(t)}
-w(t)\frac{p(t)}{r(t)}\Big]dt  \\
&= H_1(\delta_1,c_1)w(\delta_1)
 -\int_{c_1}^{\delta_1}H_1(t,c_1)\Big[\frac{w^2(t)}{r(t)}
 +\Big(h_1(t,c_1)-\frac{p(t)}{r(t)}\Big)w(t)\Big]dt  \\
&\leq  H_1(\delta_1,c_1)w(\delta_1)
 +\frac{1}{4}\int_{c_1}^{\delta_1}H_1(t,c_1)r(t)\Big(h_1(t,c_1)
 -\frac{p(t)}{r(t)}\Big)^2dt.
\end{split}
\end{equation}
Thus using $\Omega_{c_1}^{\delta_1}[H_1(.,c_1)]=0$, we obtain
\begin{align*}
&\frac{1}{H_1(\delta_1,c_1)}\int_{c_1}^{\delta_1}H_1(t,c_1)
 \Big[Q(t)-\frac{1}{4}r(t) \Big(h_1(t,c_1)-\frac{p(t)}{r(t)}
 \Big)^2\Big]dt \\
&\quad +\frac{1}{H_2(d_1,\delta_1)}\int_{\delta_1}^{d_1}H_2(d_1,t)
 \Big[Q(t)-\frac{1}{4}r(t) \Big(h_2(d_1,t)
 +\frac{p(t)}{r(t)}\Big)^2\Big]dt \\
&\leq  \frac{1}{H_2(d_1,\delta_1)}
 \sum_{i=k(\delta_1)+1}^{k(d_1)}H_2(d_1,\tau_i)
 \frac{a_i-b_i}{a_i}w(\tau_i) \\
&\leq  \frac{r_1}{H_2(d_1,\delta_1)}\Omega_{\delta_1}^{d_1}[H_2(d_1,.)]\\
&\leq \Lambda(H_1,H_2;c_1,d_1),
\end{align*}
which is a contradiction.
By a similar argument, we can prove the other two cases.
Hence the proof is complete.
\end{proof}

\begin{remark} \label{rmk2} \rm
When $p(t)=0$, Theorem \ref{thm1} reduces to
\cite[Theorem 2.2]{l3} with $\rho(t)=1$.
\end{remark}

 The following theorem gives an interval oscillation criteria
for equation \eqref{e1} with $e(t)\equiv 0$.

\begin{theorem} \label{thm2}
Assume that for any $T>0$, there exist
$c_1,d_1,\delta_1 \notin \{\tau_k\}$ such that
$c_1<\delta_1<d_1$, and
$q(t),q_i(t) \geq 0$ for $t \in [c_1,d_1]$
and if there exists $(H_1,H_2)\in \mathcal{H}$ such that
\begin{equation} \label{e22}
\begin{split}
&\frac{1}{H_1(\delta_1,c_1)}\int_{c_1}^{\delta_1}H_1(t,c_1)
 \Big[\overline{Q}(t)-\frac{1}{4}r(t) \Big(h_1(t,c_1)
 -\frac{p(t)}{r(t)}\Big)^2\Big]dt  \\
&\quad +\frac{1}{H_2(d_1,\delta_1)}\int_{\delta_1}^{d_1}H_2(d_1,t)
\Big[\overline{Q}(t)-\frac{1}{4}r(t) \Big(h_2(d_1,t)
+\frac{p(t)}{r(t)}\Big)^2\Big]dt \\
&\geq \Lambda(H_1,H_2;c_1,d_1),
\end{split}
\end{equation}
where
\begin{equation} \label{e23}
\overline{Q}(t)=q(t)+k_1\prod_{i=1}^n q_i^{\eta_i}(t),\quad
k_1=\prod_{i=1}^n \eta_i^{-\eta_i},
\end{equation}
$\Lambda$ is defined as in Theorem \ref{thm1}
and $\eta_1,\eta_2,\dots ,\eta_n$ are positive constants
satisfying \eqref{e3} and \eqref{e5} in Lemma 1,
then  \eqref{e1} is oscillatory.
\end{theorem}

\begin{proof}
 The proof is immediate from Theorem \ref{thm1}, if we put
$e(t)\equiv 0, \eta_0=0$ and applying conditions \eqref{e3}
and \eqref{e5} of Lemma 1.
\end{proof}

 Next we introduce another function class:
a function $u$ belongs to the class $E_{c,d}$ if $u \in C^1[c,d]$,
$u(t)\not \equiv 0$ and $u(c)=u(d)=0$.

\begin{theorem} \label{thm3}
Assume that for any $T>0$, there exist $c_j,d_j \notin \{\tau_k\}$,
$j=1,2$ such that $c_1<d_1\leq c_2<d_2$, and
\eqref{e6} holds.
Moreover if there exists $u_j\in E_{c_j,d_j}$ such that
\begin{equation} \label{e24}
\int_{c_j}^{d_j}\Big[Q(t){u_j}^2(t)-\frac{1}{4}r(t)
\Big(2{u_j}'(t)-\frac{p(t)}{r(t)}u_j(t)\Big)^2\Big]dt
 > r_j\Omega_{c_j}^{d_j}[u_j^2],\quad j=1,2
\end{equation}
where $Q(t)$ is the same as in Theorem \ref{thm1}, then
\eqref{e1} is oscillatory.
\end{theorem}

\begin{proof}
 Proceed as in the proof of Theorem \ref{thm1} to get \eqref{e13} and
 \eqref{e14}.

If $k(c_1)<k(d_1)$, there are all impulsive moments in
$[c_1,d_1]$; $\tau_{k(c_1)+1}$, $\tau_{k(c_1)+2}$, \dots,
$\tau_{k(d_1)}$.
Multiplying both sides of \eqref{e13} by $u_1^2(t)$ and
integrating over $[c_1,d_1]$, then using  integration by parts,
we obtain
\begin{align*}
&\sum_{i=k(c_1)+1}^{k(d_1)}u_1^2(\tau_i)[w(\tau_i)-w(\tau_i^+)]\\
&\geq \int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)-\frac{1}{4}r(t)
 \Big(2u_1'(t)-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt
 + \Big(\int_{c_1}^{\tau_{k(c_1)+1}}\\
&\quad +\sum_{i=k(c_1)+1}^{k(d_1)}
 \int_{\tau_{i-1}}^{\tau_i}+\int_{\tau_{k(d_1)}}^{d_1}\Big)
\frac{1}{r(t)}\Big[u_1(t)w(t) +\frac{1}{2}r(t)\Big(2u_1'(t)
-\frac{p(t)}{r(t)}u_1(t)\Big)\Big]^2dt \\
&\geq \int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)-\frac{1}{4}r(t)\Big(2u_1'(t)
-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt.
\end{align*}
Thus, we have
\begin{equation} \label{e25}
\sum_{i=k(c_1)+1}^{k(d_1)}\frac{a_i-b_i}{a_i}w(\tau_i)u_1^2(\tau_i)
\geq \int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)-\frac{1}{4}r(t)\Big(2u_1'(t)
-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt.
\end{equation}
Proceeding as in the proof of Theorem \ref{thm1} and using \eqref{e18}
and \eqref{e19}, we obtain
\[
\int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)-\frac{1}{4}r(t)\Big(2u_1'(t)
-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt
\leq r_1\Omega_{c_1}^{d_1}[u_1^2]
\]
which contradicts our assumption \eqref{e24}.

If $k(c_1)=k(d_1)$ then $\Omega_{c_1}^{d_1}[u_1^2]=0$
and there is no impulsive moments in $[c_1,d_1]$.
Similar to the proof of \eqref{e25}, we obtain
\[
\int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)-\frac{1}{4}r(t)\Big(2u_1'(t)
-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt\leq 0.
\]
This is  a contradiction which completes the proof.
\end{proof}

\begin{remark} \label{rmk3}\rm
 When $p(t)=0$, Theorem \ref{thm3} reduces to \cite[Theorem 2.1]{l3}
 with $\rho(t)=1$.
\end{remark}

\begin{theorem} \label{thm4}
Assume that for any $T>0$, there exist $c_1,d_1 \notin \{\tau_k\}$
such that $c_1<d_1$, and
$q(t),q_i(t) \geq 0$ for $t \in [c_1,d_1]$
and if there exists $u\in E_{c_1,d_1}$ such that
\begin{equation} \label{e26}
\int_{c_1}^{d_1}\Big[\overline{Q}(t)u^2(t)
-\frac{1}{4}r(t) \Big(2u'(t)-\frac{p(t)}{r(t)}u(t)\Big)^2\Big]dt
 > r_1\Omega_{c_1}^{d_1}[u^2]
\end{equation}
where
$\overline{Q}(t)$ is the same as in Theorem \ref{thm2}, then
\eqref{e1} with $e(t)=0$ is oscillatory.
\end{theorem}

 The proof of the above theorem is immediate by putting
$e(t)= 0$ and $\eta_0=0$ in the proof of  Theorem \ref{thm3}.
 Next we discuss the oscillatory behavior of the equation
\begin{equation} \label{e27}
 \begin{gathered}
(r(t)x'(t))'+p(t)x'(t)+q(t)x(t)+q_1(t)x^{\alpha_1}(t)=0, \quad
 t\neq \tau_k,\\
x({\tau_k}^+)=a_k x({\tau_k}),\quad
x'({\tau_k}^+)=b_k x'({\tau_k}),
\end{gathered}
\end{equation}
where $\alpha_1$ is a ratio of odd positive integers.
Before stating our result, we prove another lemma.

\begin{lemma} \label{lem2}
Let $u,B$ and $C$ be positive real numbers and $l,m$ be
ratio of odd positive integers. Then
\begin{itemize}
\item[(i)] $l>m+1$, $0<m\leq 1$,
$u^{l-1}+ M_1B^\frac{l-1}{l-m-1}\geq Bu^m$,
\item[(ii)] $0<l+m<1$, $u^{l+m-1}+M_2C^\frac{1-l}{1-l-m}u^m \geq C$,
\end{itemize}
where
\[
M_1=\Big(\frac{m}{l-1}\Big)^\frac{m}{l-m-1}\Big(\frac{l-m-1}{l-1}\Big),
\quad
M_2=\Big(\frac{1-l-m}{1-l}\Big)\Big(\frac{m}{1-l}\Big)^\frac{m}{1-l-m}.
\]
\end{lemma}

The proof of the above lemma follows by using elementary
differential calculus, and hence it is omitted.

\begin{theorem} \label{thm5}
Assume that for any $T>0$ , there exist $c,d \notin \{\tau_k\}$
such that $c<d$, and $q(t)>0,q_1(t) \geq 0$ for $t \in [c,d]$
and if there exists $u\in E_{c,d}$ such that
\begin{equation} \label{e28}
\int_{c}^{d}\Big[Q_1(t)u^2(t)-\frac{1}{4}r(t) \Big(2u'(t)
-\frac{p(t)}{r(t)}u(t)\Big)^2\Big]dt
 > r_1\Omega_{c}^{d}[u^2]
\end{equation}
where $\alpha_1>\beta+1$, $0< \beta \leq 1$,
$Q_1(t)= q(t)-M_1q_1(t)(\rho(t))^{\frac{\alpha_1-1}{\alpha_1-\beta-1}}$
where
$M_1=\big(\frac{\beta}{\alpha_1-1}\big)
^{\frac{\beta}{\alpha_1-\beta-1}}
\big(\frac{\alpha_1-\beta-1}{\alpha_1-1}\big)$
and $\rho(t)$ is a positive continuous function, then
\eqref{e27}  is oscillatory.
\end{theorem}

\begin{proof}
Let $x(t)$ be a solution of \eqref{e27}. Suppose $x(t)$
does not have any zero in $[c,d]$. Without loss of generality,
we may assume that $x(t)> 0 $ for $t \in [c,d] $. Define
\[
w(t)=-\frac{r(t)x'(t)}{x(t)}, \quad t \in [c,d].
\]
Then for $t \in [c,d]$ and $t\neq \tau_k$, we have
\begin{equation} \label{e29}
w'(t)=q(t)+q_1(t)x^{\alpha_1-1}(t)-\frac{p(t)}{r(t)}w(t)
+\frac{w^2(t)}{r(t)},
\end{equation}
or
\begin{align*}
w'(t)
& \geq  q(t)+q_1(t)x^{\alpha_1-1}(t)-q_1(t)\rho(t)x^\beta(t)
 -\frac{p(t)}{r(t)}w(t)+\frac{w^2(t)}{r(t)} \\
&= q(t)+q_1(t)\Big(x^{\alpha_1-1}(t)-\rho(t)x^\beta(t)\Big)
 -\frac{p(t)}{r(t)}w(t)+\frac{w^2(t)}{r(t)}.
\end{align*}
Thus by Lemma 2(i), we have
\begin{equation} \label{e30}
w'(t)\geq Q_1(t)-\frac{p(t)}{r(t)}w(t)+\frac{w^2(t)}{r(t)}.
\end{equation}
Then following the proof of Theorem \ref{thm3}, we obtain a
contradiction to \eqref{e28}. Hence the proof is complete.
\end{proof}

\begin{theorem} \label{thm6}
 Assume that for any $T>0$ , there exist $c,d \notin \{\tau_k\}$
such that $c<d$, and
$q(t)>0,q_1(t) \geq 0$ for $t \in [c,d]$
and if there exists $u\in E_{c,d}$ such that
\begin{equation} \label{e31}
\int_{c}^{d}\Big[Q_2(t)u^2(t)-\frac{1}{4}r(t) \Big(2u'(t)
-\frac{p(t)}{r(t)}u(t)\Big)^2\Big]dt
 > r_1\Omega_{c}^{d}[u^2]
\end{equation}
where $0<\alpha_1+\alpha_2<1$,
$Q_2(t)= q(t)-M_2q_1(t)(\rho(t))^\frac{1-\alpha_1}{1-\alpha_1-\alpha_2}$ where \\
$M_2=\big(\frac{1-\alpha_1-\alpha_2}{1-\alpha_1}\big)
\big(\frac{\alpha_2}{1-\alpha_1}\big)
^{\frac{\alpha_2}{1-\alpha_1-\alpha_2}}$ and
$\rho(t)$ is a positive continuous function, then
\eqref{e27}  is oscillatory.
\end{theorem}

\begin{proof}
 Proceeding as in the proof of Theorem \ref{thm5}, we obtain  \eqref{e29} or
 \[
w'(t) \geq q(t)+q_1(t)\Big(x^{\alpha_1-1}(t)
-\rho(t)x^{-\alpha_2}(t)\Big)-\frac{p(t)}{r(t)}w(t)
+\frac{w^2(t)}{r(t)}.
\]
Now use Lemma 2(ii) and then proceed as in the proof of
Theorem \ref{thm3}.
Thus we obtain a contradiction to condition \eqref{e31}.
This completes the proof.
\end{proof}

\begin{remark} \label{rmk4}\rm
 When $q_1(t)\equiv 0$, then the results of Theorems
\ref{thm5} and \ref{thm6}
are the same and it seems to be new. However,
Theorem \ref{thm5} and Theorem \ref{thm6} are not applicable
when $q(t)\equiv 0$.  Therefore, it would be interesting
to obtain results similar to  Theorems \ref{thm5}
and \ref{thm6} which are applicable to the case $q(t)\equiv 0$
and $q_1(t) \not \equiv 0$.
\end{remark}

\section{Examples}

In this section, we give some examples to illustrate our results.

\begin{example} \label{exa1} \rm
Consider the impulsive differential equation
\begin{equation} \label{e32}
\begin{split}
 \begin{aligned}
 &x''(t)+ \sin t x'(t)+(l\cos t) x(t)+(l_1
 \sin t)|x(t)|^\frac{3}{2}\operatorname{sgn}  x(t)\\
&+(l_2 \cos t)|x(t)|^{1/2}\operatorname{sgn}  x(t)
=-\cos 2t, \quad t\neq \tau_k,
\end{aligned}\\
x({\tau_k}^+)=\frac{1}{2} x({\tau_k}),\quad
x'({\tau_k}^+)= \frac{3}{4} x'({\tau_k}),\quad
\tau_k= 2k\pi+\frac{\pi}{6},
\end{split}
\end{equation}
where $k\in \mathbb{N}$,
$t\geq t_0>0,\tau_{2n}=2n\pi+\frac{\pi}{6}$,
$\tau_{2n+1}=2n\pi+\frac{\pi}{3}$,
$n=0,1,2,\dots$, $l,l_1,l_2$ are positive constants.
Also note that $r_1=r_2=1$.
Now choose $\eta_0=\frac{1}{4}$, $ \eta_1=\frac{5}{8}$,
$\eta_2=\frac{1}{8}$ to get $k_0=4\frac{2^{3/4}}{5^{5/8}}$ and
$Q(t)=l\cos t+ 4\frac{2^{3/4}}{5^{5/8}}|
-\cos 2t|^{1/4}(l_1 \sin t)^{5/8}(l_2 \cos t)^{1/8}$.

 For any $T\geq 0$, we can choose $n$ large enough such that
$T<c_1=2n\pi<\delta_1=2n\pi+\frac{\pi}{8}<d_1
=2n\pi+\frac{\pi}{4}=c_2<\delta_2=2n\pi+\frac{3\pi}{8}< d_2
=2n\pi+\frac{\pi}{2}$, $n=0,1,2,\dots $.

 If we choose $H_1(t,s)=H_2(t,s)=(t-s)^2$ then
$h_1(t,s)=-h_2(t,s)=\frac{2}{t-s}$. Then by using the mathematical
software Mathematica 5.2, the left hand side of the
inequality \eqref{e7} with $j=1$ is
\begin{align*}
&\frac{1}{H_1(\delta_1,c_1)}\int_{c_1}^{\delta_1}H_1(t,c_1)\Big[Q(t)
-\frac{1}{4}r(t) \Big(h_1(t,c_1)-\frac{p(t)}{r(t)}\Big)^2\Big]dt \\
&+\frac{1}{H_2(d_1,\delta_1)}\int_{\delta_1}^{d_1}H_2(d_1,t)\Big[Q(t)
 -\frac{1}{4}r(t) \Big(h_2(d_1,t)+\frac{p(t)}{r(t)}\Big)^2\Big]dt\\
&= \frac{64}{\pi^2}\Big[l \int_{2n\pi}^{2n\pi+\frac{\pi}{8}}(\cos t)
(t-2n\pi)^2dt \\
&\quad -\frac{1}{4}\int_{2n\pi}^{2n\pi+\frac{\pi}{8}}(t-2n\pi)^2
 \Big(\frac{2}{t-2n\pi}-\sin t \Big)^2dt+
 l \int_{2n\pi+\frac{\pi}{8}}^{2n\pi+\frac{\pi}{4}}(\cos t)
(2n\pi+\frac{\pi}{4}-t)^2dt\\
&\quad + 4\frac{2^{3/4}}{5^{5/8}}{l_1}^{5/8} {l_2}^{1/8}
\int_{2n\pi+\frac{\pi}{8}}^{2n\pi+\frac{\pi}{4}}|
-\cos 2t|^{1/4}(\sin t)^{5/8}(\cos t)^{1/8}(2n\pi
+\frac{\pi}{4}-t)^2dt \\
&\quad -\frac{1}{4}\int_{2n\pi+\frac{\pi}{8}}^{2n\pi
 +\frac{\pi}{4}}(2n\pi+\frac{\pi}{4}-t)^2
 \Big(\frac{-2}{2n\pi+\frac{\pi}{4}-t}+\sin t \Big)^2dt\Big]\\
&\approx 0.240013l+0.306144{l_1}^{5/8} {l_2}^{1/8}-4.72546.
\end{align*}
Note that there is no impulsive moment in $(c_1,\delta_1)$ and
$\tau_{2n}\in (\delta_1,d_1)$. Also $k(\delta_1)=2n-1, k(d_1)=2n$.
 Hence the right side of the inequality \eqref{e7} with $j=1$ is
\begin{align*}
\Lambda(H_1,H_2;c_1,d_1)
&=\frac{r_1}{H_2(d_1,\delta_1)}\Omega_{\delta_1}^{d_1}[H_2(d_1,.)]\\
&= \frac{64}{\pi^2}H_2(d_1,\tau_{2n})\theta(\delta_1)\\
&= \frac{32}{3\pi}\Big(\frac{b_{2n}-a_{2n}}{a_{2n}}\Big)
=\frac{16}{3\pi}.
\end{align*}
Thus \eqref{e7} is satisfied with $j=1$ if
\[
0.240013l+0.306144{l_1}^{5/8} {l_2}^{1/8}>4.72546+\frac{16}{3\pi}
=6.42311.
\]
In a similar way, the left hand side of the inequality \eqref{e7}
with $j=2$ is
\begin{align*}
&\frac{1}{H_1(\delta_2,c_2)}\int_{c_2}^{\delta_2}H_1(t,c_2)\Big[Q(t)
-\frac{1}{4}r(t) \Big(h_1(t,c_2)-\frac{p(t)}{r(t)}\Big)^2\Big]dt\\
&+\frac{1}{H_2(d_2,\delta_2)}\int_{\delta_2}^{d_2}H_2(d_2,t)
\Big[Q(t)-\frac{1}{4}r(t) \Big(h_2(d_2,t)+\frac{p(t)}{r(t)}
\Big)^2\Big]dt \\
&\approx 0.0994167l+0.48437{l_1}^{5/8} {l_2}^{1/8}-4.23605.
\end{align*}
Note that $\tau_{2n+1}\in (c_2,\delta_2)$ and there is no
impulsive moment in $(\delta_2,d_2)$. Also
$k(c_2)=2n$, $k(\delta_2)=2n+1$. Hence the right side of
the inequality \eqref{e7} with $j=2$ is
\begin{align*}
\Lambda(H_1,H_2;c_2,d_2)
&=\frac{r_2}{H_1(\delta_2,c_2)}\Omega_{c_2}^{\delta_2}[H_1(.,c_2)]\\
&= \frac{64}{\pi^2}H_1(\tau_{2n+1},c_2)\theta(c_2)\\
&= \frac{16}{3\pi}\Big(\frac{b_{2n+1}-a_{2n+1}}{a_{2n+1}}\Big)
=\frac{8}{3\pi}.
\end{align*}
Thus \eqref{e7} is satisfied with $j=2$ if
\[
0.0994167l+0.48437{l_1}^{5/8} {l_2}^{1/8}>4.23605+\frac{8}{3\pi}
=5.08488.
\]
So, if we choose the constants $l,l_1,l_2$ large enough such that
\[
0.240013l+0.306144{l_1}^{5/8} {l_2}^{1/8}>6.42311,\\
0.0994167l+0.48437{l_1}^{5/8} {l_2}^{1/8}>5.08488,
\]
then by Theorem \ref{thm1}, equation \eqref{e32} is oscillatory.
In fact, for $l=20$, $l_1=30$, $l_2=40$, equation \eqref{e32}
is oscillatory.
\end{example}

\begin{example} \label{exa2} \rm
Consider the impulsive differential equation
\begin{equation} \label{e33}
 \begin{split}
\begin{aligned}
&\Big(\frac{1}{2+\sin 2t}x'(t)\Big)'+(2\cos 4t) x'(t)
 +(\gamma_0 \cos t) x(t)+(\gamma_1 \cos 2t)|x(t)|^{5/2}
\operatorname{sgn}  x(t) \\
&+(\gamma_2 \cos 2t)|x(t)|^{1/2}\operatorname{sgn}  x(t)
=\sin 2t,\quad t\neq 2k\pi-\frac{\pi}{8},
\end{aligned}\\
x({\tau_k}^+)=\frac{1}{3} x({\tau_k}), \quad
x'({\tau_k}^+)=\frac{2}{3} x'({\tau_k}),\quad
\tau_k= 2k\pi-\frac{\pi}{8},
\end{split}
\end{equation}
where $k\in \mathbb{N}$, $t\geq t_0>0$, $\gamma_i$, $i=0,1,2$
are positive constants.

Now choose $\eta_0=\frac{1}{2}$,
$\eta_1=3/8$,
$\eta_2=1/8$ to get $k_0=\frac{4}{3^{3/8}}$ and
$Q(t)=\gamma_0 \cos t+\frac{4}{3^{3/8}}|\sin 2t|^{1/2}
(\gamma_1 \cos 2t)^{3/8}(\gamma_2 \cos 2t)^{1/8}$.
For any $T\geq 0$, we can choose $n$ large enough such that
$T<c_1=2n\pi-\frac{\pi}{4}$, $d_1=c_2=2n\pi$,
$d_2=2n\pi+\frac{\pi}{4}$, $n=1,2,\dots $.
If we take $u_1(t)=\sin 4t$,
$u_2(t)=\sin 8t$ then by using the mathematical software
Mathematica 5.2, we obtain
\begin{align*}
&\int_{c_1}^{d_1}\Big[Q(t)u_1^2(t)
-\frac{1}{4}r(t)\Big(2u_1'(t)-\frac{p(t)}{r(t)}u_1(t)\Big)^2\Big]dt\\
&= \gamma_0\int_{2n\pi-\frac{\pi}{4}}^{2n\pi}\cos t \sin^2 4t\,dt\\
&\quad + \frac{4}{3^{3/8}}{\gamma_1}^{3/8}{\gamma_2}^{1/8}
\int_{2n\pi -\frac{\pi}{4}}^{2n\pi}|\sin 2t|^{1/2}(\cos 2t)^{3/8}
 (\cos 2t)^{1/8}\sin^2 4t\,dt \\
&\quad -\frac{1}{4}\int_{2n\pi-\frac{\pi}{4}}^{2n\pi}
 \frac{1}{2+\sin 2t}\Big(8\cos 4t-2\cos 4t(2+\sin 2t)\sin 4t\Big)^2dt \\
&\approx 0.359165 \gamma_0+ 0.673369 {\gamma_1}^{3/8}
{\gamma_2}^{1/8}-6.28071,
\end{align*}
and
\begin{align*}
&\int_{c_2}^{d_2}\Big[Q(t)u_2^2(t)-\frac{1}{4}r(t)\Big(2u_2'(t)
 -\frac{p(t)}{r(t)}u_2(t)\Big)^2\Big]dt\\
&= \gamma_0\int_{2n\pi}^{2n\pi+\frac{\pi}{4}}\cos t \sin^2 8t\,dt\\
&\quad + \frac{4}{3^{3/8}}{\gamma_1}^{3/8}{\gamma_2}^{1/8}
\int_{2n\pi}^{2n\pi+\frac{\pi}{4}}|\sin 2t|^{1/2}(\cos 2t)^{3/8}
 (\cos 2t)^{1/8}\sin^2 8t\,dt \\
&\quad -\frac{1}{4}\int_{2n\pi}^{2n\pi+\frac{\pi}{4}}\frac{1}{2+\sin 2t}
\Big(16\cos 8t-2\cos 4t(2+\sin 2t)\sin 8t\Big)^2dt\\
&\approx 0.35494 \gamma_0+ 0.598551 {\gamma_1}^{3/8}{\gamma_2}^{1/8}
-9.18481.
\end{align*}
Since $k(c_1)=n-1$, $k(d_1)=n$, $r_1=1$ and $k(c_2)=k(d_2)$,
we obtain
\[
r_1\Omega_{c_1}^{d_1}[u_1^2]=\frac{8(b_n-a_n)}{\pi a_n}
=\frac{8}{\pi}, \quad r_2\Omega_{c_2}^{d_2}[u_2^2]=0.
\]
So, if we choose the constants $\gamma_0$ or $\gamma_1,\gamma_2$
large enough such that
\begin{gather*}
0.359165 \gamma_0+ 0.673369 {\gamma_1}^{3/8}{\gamma_2}^{1/8}
>6.28071+ \frac{8}{\pi}=8.82719,\\
0.35494 \gamma_0+ 0.598551 {\gamma_1}^{3/8}{\gamma_2}^{1/8}>9.1848,
\end{gather*}
then by Theorem \ref{thm3}, equation \eqref{e33} is oscillatory.
In fact, for $\gamma_0=40,\gamma_1=20,\gamma_2=30$,
equation \eqref{e33} is oscillatory.
\end{example}

\begin{example} \label{exa3} \rm
Consider the impulsive differential equation
\begin{equation} \label{e34}
 \begin{gathered}
 x''(t)+(2\sin t) x'(t)+(l \sin t) x(t)+(l_1 \cos t)x^5(t) =0,
\quad t\neq 2k\pi+\frac{\pi}{4},\\
x({\tau_k}^+)=4 x({\tau_k}),\quad x'({\tau_k}^+)= 5x'({\tau_k}),
\quad \tau_k= 2k\pi+\frac{\pi}{4},
\end{gathered}
\end{equation}
where $k\in \mathbb{N}$, $t\geq t_0>0$, $l,l_1$ are positive constants.
For any $T\geq 0$, we can choose $n$ large enough such that
$T<c=2n\pi+\frac{\pi}{6}$,
$d=2n\pi+\frac{\pi}{2}$, $n=1,2,\dots $.
 Then $q(t)=l \sin t> 0$, $q_1(t)=l_1 \cos t> 0$ on $[c,d]$.
If we take $u(t)=\sin 6t$,  $ \beta=1, \rho(t)=4$,
we have $M_1=(\frac{1}{4})^{1/3}(\frac{3}{4})$ and
$Q_1(t)=l\sin t-4^{4/3}M_1(l_1\cos t)$.
Thus by using the mathematical software Mathematica 5.2, we have
\begin{align*}
&\int_c^d\Big[Q_1(t)u^2(t)-\frac{1}{4}r(t)\Big(2u'(t)
 -\frac{p(t)}{r(t)}u(t)\Big)^2\Big]dt\\
&= l\int_{2n\pi+\frac{\pi}{6}}^{2n\pi+\frac{\pi}{2}}
 \sin t \sin^2 6t\,dt -4^{4/3}\Big(\frac{1}{4}\Big)^{1/3}
 \Big(\frac{3}{4}\Big)l_1\int_{2n\pi+\frac{\pi}{6}}^{2n\pi
 +\frac{\pi}{2}}\cos t \sin^2 6t\,dt\\
&\quad -\frac{1}{4}\int_{2n\pi+\frac{\pi}{6}}^{2n\pi
 +\frac{\pi}{2}}\Big(12\cos 6t-2\sin t \sin 6t\Big)^2dt \\
&\approx 0.436041l- 0.755245l_1-19.4744.
\end{align*}
Since $k(c)=n-1, k(d)=n,r_1=1$ and $k(c)<k(d)$, we obtain
\[
r_1\Omega_{c}^{d}[u^2]=\frac{12}{\pi}
\Big(\frac{b_n-a_n}{a_n}\Big)=\frac{3}{\pi}.
\]
So, if we choose the constants $l,l_1$ such that
\[
0.436041l- 0.755245l_1>19.4744+\frac{3}{\pi}=20.4293
\]
then by Theorem \ref{thm5}, equation \eqref{e34} is oscillatory.
In fact, for $l=50,l_1=0.01$, equation \eqref{e34} is oscillatory.
\end{example}

\begin{example} \label{exa4} \rm
 Consider the impulsive differential equation
\begin{equation} \label{e35}
 \begin{gathered}
 x''(t)-(\sin 2t) x'(t)+ke^{t/2} x(t)+k_1 e^{t/4}x^3(t) =0,\quad
 t\neq 2k\pi, \\
x({\tau_k}^+)=\frac{1}{4} x({\tau_k}),\quad
x'({\tau_k}^+)=\frac{1}{2} x'({\tau_k}),\quad
\tau_k= 2k\pi,
\end{gathered}
\end{equation}
where $k\in \mathbb{N}$, $t\geq t_0>0$, $k,k_1$ are positive constants.
Note that $\alpha_1=\frac{1}{3}$, $p(t)=-\sin 2t$.
For any $T\geq 0$, we can choose $n$ large enough such that
$T<c=2n\pi-\frac{\pi}{4}$,
$d=2n\pi+\frac{\pi}{4}$, $n=1,2,\dots $.
Then $q(t)=ke^{t/2}> 0$, $q_1(t)=k_1 e^{t/4}> 0$ on $[c,d]$.
If we take $u(t)=\cos 2t$, $\alpha_2=\frac{1}{3}$, $\rho(t)=3$,
we have $M_2=\frac{1}{4}$ and
 $Q_2(t)=ke^{t/2}-\frac{9}{4}k_1 e^{t/4}$.
Thus by using the mathematical software Mathematica 5.2, we have
\begin{align*}
&\int_c^d\Big[Q_2(t)u^2(t)-\frac{1}{4}r(t)\Big(2u'(t)
 -\frac{p(t)}{r(t)}u(t)\Big)^2\Big]dt\\
&= k\int_{2n\pi-\frac{\pi}{4}}^{2n\pi+\frac{\pi}{4}}
 e^{t/2} \cos^2 2t\,dt
-\frac{9}{4}k_1\int_{2n\pi-\frac{\pi}{4}}^{2n\pi+\frac{\pi}{4}}e^{t/4}
\cos^2 2t\,dt\\
&\quad -\frac{1}{4}\int_{2n\pi-\frac{\pi}{4}}^{2n\pi
+\frac{\pi}{4}}\Big(-4 \sin 2t+(\sin 2t)(\cos 2t) \Big)^2dt \\
&\approx 18.3585k-8.52225k_1-2.52401.
\end{align*}
Since $k(c)=n-1$, $k(d)=n$, $r_1=1$ and $k(c)<k(d)$, we obtain
\[
r_1\Omega_{c}^{d}[u^2]=\frac{4}{\pi}.
\]
So, if we choose the constants $k,k_1$ such that
\[
18.3585k-8.52225k_1>2.52401+\frac{4}{\pi}=3.79725
\]
then by Theorem \ref{thm6}, equation \eqref{e35} is oscillatory.
In fact, for $k=k_1=1$, equation \eqref{e35} is oscillatory.
\end{example}

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\end{document}