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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 43, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/43\hfil Interval oscillation criteria]
{Interval oscillation criteria for second-order forced delay
 differential equations under impulse effects}

\author[Qiaoluan Li, Wing-Sum Cheung  \hfil EJDE-2013/43\hfilneg]
{Qiaoluan Li, Wing-Sum Cheung}  % in alphabetical order

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

\address{Wing-Sum Cheung \newline
Department of Mathematics, The University of Hong Kong\\
Pokfulam Road, Hong Kong, China}
\email{wscheung@hkucc.hku.hk}

\thanks{Submitted July 11, 2012. Published February 8, 2013.}
\subjclass[2000]{34K11, 34C15}
\keywords{Oscillation; impulse; delay differential equation}

\begin{abstract}
 We establish some oscillation criteria for a forced
 second-order differential equation with impulses.
 These results extend some well-known results for forced
 second-order impulsive differential equations with delay.
\end{abstract}

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

\section{Introduction}

   In this article, we consider the second-order impulsive delay
differential equation
\begin{equation}
\begin{gathered}
(p(t)x'(t))'+q(t)x(t-\tau)+\sum_{i=1}^{n}q_i(t)\Phi_{\alpha_i}(x(t-\tau))=e(t),
\quad t\geq t_0,\;t\neq t_k,\\
x(t_k^+)=a_kx(t_k),\quad x'(t_k^+)=b_kx'(t_k),\quad k=1,2,\dots,
\end{gathered}\label{e1.1}
\end{equation}
where
\begin{gather*}
x(t_k^-) :=\lim_{t \to t_k^-}x(t),\quad
x(t_k^+) :=\lim_{t \to t_k^+}x(t), \\
x'(t_k^-):=\lim_{h\to 0^-} \frac{x(t_k+h)-x(t_k)}h,\quad
x'(t_k^+):=\lim_{h\to 0^+} \frac{x(t_k+h)-x(t_k)}h,
\end{gather*}
$\Phi_{\alpha}(s):=|s|^{\alpha-1}s$,
 $0\leq t_0<t_1<\dots <t_k<\dots$,
$\lim_{k\to\infty}t_k=\infty$,
the exponents of the nonlinearities satisfy
$$
\alpha_1 >\alpha_2>\dots>\alpha_m>1>\alpha_{m+1}>\dots>\alpha_n>0,
$$
the functions
$p, q, q_i, e $ are piecewise left continuous at each $t_k$, more precisely,
they belong to the set
\begin{align*}
PLC[t_0, \infty):=\big\{&h:[t_0,\infty)\to \mathbb R |,\; h
\text{ is continuous on each interval }(t_k, t_{k+1}),\\
& h(t_k^{\pm}) \text{ exists, and $h(t_k)=h(t_k^-)$ for all $k\in \mathbb{N}$}
\big\},
\end{align*}
and $p>0$ is a nondecreasing function.


 A function $x\in PLC[t_0,\infty)$ is
said to be a solution of \eqref{e1.1} if
 $x(t)$ satisfies \eqref{e1.1}, and
 $x(t)$ and $x'(t)$ are left continuous at every $t_k$, $k\in\mathbb{N}$.

In the past few decades, there has been a great deal of work on
the oscillatory behavior of the solutions of second order
differential equations, see \cite{a2,z1}
and the references cited therein.
Impulsive differential equations are an effective
tool for the simulation of processes and phenomena observed in
control theory, population dynamics, economics, etc. Research in
this direction was initiated by Gopalsamy and Zhang in \cite{g1}. Since
then there has been an increasing interest in finding the
oscillation criteria for such equations, see \cite{a1,b1,b2,l1,l2,o1,o2}
their references.

 Liu and Xu \cite{l1}, obtained several oscillation theorems for the
equation
\begin{equation} \label{e1.2}
\begin{gathered}
(r(t)x'(t))'+p(t)|x(t)|^{\alpha-1}x(t)=q(t),\quad t\geq t_0,\;t\neq t_k,\\
x(t_k^+)=a_kx(t_k),\quad x'(t_k^+)=b_kx'(t_k),\quad
x(t_0^+)=x_0,\quad x'(t_0^+)=x'_0,
\end{gathered}
\end{equation}
which is a special case of \eqref{e1.1}.

More recently, Guvenilir \cite{g2}
established interval criteria for the oscillation of second-order
functional differential equations with oscillatory potentials for
the equation
\begin{equation}
(k(t)x'(t))'+p(t)x(g(t))+q(t)|x(g(t))|^{\gamma-1}x(g(t))=e(t), \quad
t\geq 0.\label{e1.3}
\end{equation}
We note that when $g(t)=t-\tau$, this equation is included in \eqref{e1.1}.

In this article, some new sufficient conditions for the oscillation
of solutions of \eqref{e1.1} are presented, and  illustrated
by an example.  It should be noted that the derivation in
this work adopts new estimates which are not a routine
extension of the existing techniques used for the non-delay case.

As is customary, a solution of \eqref{e1.1} is said to be oscillatory
if it has arbitrarily large zeros; otherwise the solution is said
to be non-oscillatory.

\section{Main Results}

We will assume  the following three conditions throughout this article.
\begin{itemize}
 \item[(H1)] $\tau \geq 0$, $b_k,a_k> 0$,
$t_{k+1}-t_k>\tau$,  $k=1,2,\dots$;
$\alpha_1>\alpha_2>\dots>\alpha_m>1> \alpha_{m+1}>\dots>\alpha_n>0$,
$(n>m\geq 1)$.

\item[(H2)] $p, q, q_i, e \in PLC[t_0, \infty)$.

\item[(H3)] For any $T\geq 0$, there exist intervals
$[c_1,d_1]$ and  $[c_2,d_2] $ contained in $ [T,\infty)$ such
that $c_1<d_1\leq d_1+\tau\leq c_2<d_2$,
$c_j,d_j \not\in \{t_k\}$, $j=1,2$, $k=1,2,\dots$ and
\begin{gather*}
q(t)\geq 0, \quad q_i(t)\geq 0\quad\text{for }
t\in [c_1-\tau,d_1]\cup  [c_2-\tau,d_2], \quad i=1,2,\dots,n;\\
e(t)\leq 0 \quad \text{for }t\in [c_1-\tau,\,d_1];\\
e(t)\geq 0 \quad\text{for }t\in [c_2-\tau,\,d_2].
\end{gather*}
\end{itemize}
Denote
\begin{gather*}
I(s):=\max\{i: t_0<t_i<s\};\quad
\beta_j:=\max\{p(t): t\in [c_j, d_j]\},\quad j=1,2;\\
\Omega:=\{\omega\in C^1[c_j, d_j]: \omega(t)\not\equiv 0,\;
\omega(c_j)=\omega(d_j)=0,\; j=1,2\};\\
\begin{aligned}
\Gamma:=\big\{&G\in C^1[c_j, d_j]: G\geq 0, \;G\not\equiv 0,\;
G(c_j)=G(d_j)=0,\\
& G'(t)=2g(t)\sqrt{G(t)},\; j=1,2\big\}.
\end{aligned}
\end{gather*}
Before giving the main results, we  introduce the following Lemma.

 \begin{lemma}[\cite{s1}] \label{lem2.1}
For any n-tuple $(\alpha_1,\dots,\alpha_n)$
 satisfying $\alpha_1>\dots>\alpha_m>1 >\alpha_{m+1}>\dots>\alpha_n>0$,
there exists an n-tuple
 $(\eta_1,\dots, \eta_n)$ satisfying either (a)
\[
\sum_{i=1}^{n}\alpha_i\eta_i=1,\quad
\sum_{i=1}^{n}\eta_i<1,\quad 0<\eta_i<1,
\]
or   (b)
\[
\sum_{i=1}^{n}\alpha_i\eta_i=1,\quad \sum_{i=1}^{n}\eta_i=1,\quad 0<\eta_i<1.
\]
\end{lemma}

\begin{theorem} \label{thm2.1}
 Assume that conditions {\rm (H1)--(H3)}
hold, and that there exists $\omega\in \Omega$ such that
\begin{equation}
\begin{aligned}
&\int_{c_j}^{d_j}p(t)\omega'^{2}(t)dt -
\int_{c_j}^{t_{I(c_j)+1}}Q(t)Q_{I(c_j)}^{j}(t)\omega^2(t)dt\\
&-\sum_{k=I(c_j)+2}^{I(d_j)}\int_{t_{k-1}}^{t_k}Q(t)Q_k^j(t)\omega^2(t)dt
-\int_{t_{I(d_j)}}^{d_j}Q(t)Q_{I(d_j)}^j(t)\omega^2(t)dt\\
&\leq L(\omega, c_j,d_j),
\end{aligned}\label{e2.1}
\end{equation}
where $L(\omega, c_j, d_j):=0$ for $I(c_j)=I(d_j)$, and
\begin{align*}
&L(\omega, c_j,d_j)\\
&:=\beta_j
\big\{\omega^2(t_{I(c_j)+1})\frac{a_{I(c_j)+1}-b_{I(c_j)+1}}{a_{I(c_j)+1}
(t_{I(c_j)+1}-c_j)}+\sum_{k=I(c_j)+2}^{I(d_j)}\omega^2(t_k)
\frac{a_k-b_k}{a_k(t_k-t_{k-1})}\big\}
\end{align*}
for $I(c_j)<I(d_j)$, $j=1,2$,
\begin{gather*}
  Q(t) := q(t)+\eta_0^{-\eta_0}\prod_{i=1}^{n}\eta_i^{-\eta_i}q_i^{\eta_i}(t)
|e(t)|^{\eta_0},\\
Q_k^j(t) := \begin{cases}
\frac{t-t_k}{a_k\tau+b_k(t-t_k)}, & t\in (t_k,t_k+\tau),\\
\frac{t-t_k-\tau}{t-t_k}, & t\in[t_k+\tau,t_{k+1}],
\end{cases} \\
 k=I(c_j),I(c_j)+1,\dots,I(d_j),
\end{gather*}
and $\eta_1,\eta_2,\dots,\eta_n$ are positive constants
satisfying part (a) of Lemma \ref{lem2.1}.
Then all solutions of \eqref{e1.1} are oscillatory.
\end{theorem}

\begin{proof}
 Suppose that $x(t)$ is a non-oscillatory solution of
 \eqref{e1.1}. By re-defining $t_0$ if necessary, without loss of generality,
we may assume that $x(t-\tau)>0$ for all $t\geq t_0>0$.
Define the Riccati transformation
\[
v(t):=\frac{p(t)x'(t)}{x(t)}.
\]
It follows from \eqref{e1.1} that $v(t)$ satisfies
\begin{equation}
v'(t)=-q(t)\frac{x(t-\tau)}{x(t)}-\sum_{i=1}^{n}q_i(t)|x(t-\tau)|^{\alpha_i-1}
\frac{x(t-\tau)}{x(t)}
+\frac{e(t)}{x(t)}-\frac{v^2(t)}{p(t)},\label{e2.2}
\end{equation}
for all $t\neq t_k,t\geq t_0$, and $v(t_k^+)=\frac{b_k}{a_k}v(t_k)$ for
all $k\in\mathbb{N}$.

From the assumptions, we can choose $c_1,d_1\geq t_0$ such that
$q(t)\geq 0$ and $q_i(t)\geq 0$ for $t\in [c_1-\tau, d_1]$,
$i=1,2,\dots,n$, and $e(t)\leq 0$ for $t\in [c_1-\tau, d_1]$.
By Lemma \ref{lem2.1}, there exist $\eta_i>0$, $i=1,\dots,n$, such that
$\sum_{i=1}^n \alpha_i\eta_i=1$ and $\sum_{i=1}^n \eta_i<1$.
Define $\eta_0:=1-\sum_{i=1}^n \eta_i$ and let
\begin{gather*}
u_0 :=\eta_0^{-1}\Big|\frac{e(t)x(t-\tau)}{x(t)}\Big|\,x^{-1}(t-\tau),\\
u_i :=\eta_i^{-1}q_i(t)\frac{x(t-\tau)}{x(t)}\,x^{\alpha_i-1}(t-\tau),\quad
i=1,2,\dots,n\,.
\end{gather*}
Then by the arithmetic-geometric mean inequality (see \cite{b3}), we
have
$$
\sum_{i=0}^{n}\eta_iu_i\geq \prod_{i=0}^{n}u_i^{\eta_i}
$$
and so
\begin{equation}
\begin{aligned}
v'(t)
&\leq -\eta_0^{-\eta_0}\prod_{i=1}^{n}\eta_i^{-\eta_i}
q_i^{\eta_i}(t)\frac{x^{\eta_i}(t-\tau)}{x^{\eta_i}(t)}
 x^{(\alpha_i-1)\eta_i}(t-\tau)|e(t)|^{\eta_0}\\
&\quad \times \frac{x^{\eta_0}(t-\tau)}{x^{\eta_0}(t)}x^{-\eta_0}(t-\tau)
 -q(t)\frac{x(t-\tau)}{x(t)}-\frac{v^2(t)}{p(t)},\quad
 t\neq t_k.
\end{aligned}\label{e2.3}
\end{equation}
Since
\begin{gather*}
\prod_{i=0}^{n}\frac{x^{\eta_i}(t-\tau)}{x^{\eta_i}(t)}
=\frac{x^{\eta_0+\eta_1+\dots+\eta_n}(t-\tau)}
{x^{\eta_0+\eta_1+\dots+\eta_n}(t)}=\frac{x(t-\tau)}{x(t)}\,,
\\
\prod_{i=1}^{n}x^{(\alpha_i-1)\eta_i}(t-\tau)x^{-\eta_0}(t-\tau)=1\,,
\end{gather*}
we obtain
\begin{equation}
\begin{aligned}
v'(t)
& \leq -q(t)\frac{x(t-\tau)}{x(t)}-\eta_0^{-\eta_0}
 \prod_{i=1}^{n}\eta_i^{-\eta_i}
  q_i^{\eta_i}(t)\frac{x(t-\tau)}{x(t)}|e(t)|^{\eta_0}
           -\frac{v^2(t)}{p(t)}\\
& = -Q(t)\frac{x(t-\tau)}{x(t)}-\frac{v^{2}(t)}{p(t)},\quad t\neq
      t_k.
\end{aligned} \label{e2.4}
\end{equation}
Multiply both sides of \eqref{e2.4} by $\omega^2(t)$, with $w$ as
prescribed in the hypothesis of the theorem. Then integrate from
 $c_1$ to $d_1$; using
integration by parts on the left side, we have
\begin{equation}
\begin{aligned}
& \sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\
&\leq  2\int_{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt
 - \int_{c_1}^{d_1}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt
 -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt\\
&=  2\int _{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt
 - \int_{c_1}^{t_{I(c_1)+1}}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt\\
& \quad -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}
  \omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt\\
&\quad -\int_{t_{I(d_1)}}^{d_1}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt
       -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt.
\end{aligned} \label{e2.5}
\end{equation}
To estimate $\frac{x(t-\tau)}{x(t)}$, we first consider the
situation where $I(c_1)<I(d_1)$. In this case, all the impulsive
moments in $[c_1,d_1]$ are
$t_{I(c_1)+1},\,\,t_{I(c_1)+2},\dots,\,t_{I(d_1)}.$


Case (1). $t\in (t_k,t_{k+1}]\subset[c_1,d_1]$.

(i) If $t\in [t_k+\tau,t_{k+1}]$, then $t-\tau\in
[t_k,t_{k+1}-\tau]$. Since $t_{k+1}-t_k>\tau$, there are no
impulsive moments in $(t-\tau,t)$. As in the proof of
\cite[Lemma 2.4]{a1}, we have
$$
x(t)>x(t)-x(t_k^+)=x'(\xi)(t-t_k),\quad \xi\in (t_k,\,\,t).
$$
Since the function $p(t)x'(t)$ is nonincreasing,
$$
x(t)>x'(\xi)(t-t_k)>\frac{p(t)x'(t)}{p(\xi)}(t-t_k).
$$
From the fact that $p(t)$ is nondecreasing, we have
$$
\frac{p(t)x'(t)}{x(t)}<\frac{p(\xi)}{t-t_k}<\frac{p(t)}{t-t_k}.
$$
We obtain $\frac{x'(t)}{x(t)}<\frac{1}{t-t_k}$. Upon integrating
from $t-\tau$ to $t$, we obtain
$\frac{x(t-\tau)}{x(t)}>\frac{t-t_k-\tau}{t-t_k}$.

(ii) If $t\in (t_k, t_k+\tau),$ then
$t-\tau\in(t_k-\tau,t_k)$, and there is an impulsive moment
$t_k$ in $(t-\tau, t)$. Similar to (i), we obtain
$\frac{x'(s)}{x(s)}< \frac{1}{s-t_k+\tau}$ for
$s\in (t_k-\tau, t_k]$. Upon integrating from $t-\tau$ to $t_k$, we
obtain $\frac{x(t-\tau)}{x(t_k)}>\frac{t-t_k}{\tau}$. Since
$x(t)-x(t_k^+)<x'(t_k^+)(t-t_k)$, we have
$$
\frac{x(t)}{x(t_k^+)}<1+\frac{x'(t_k^+)}{x(t_k^+)}(t-t_k)
=1+\frac{b_kx'(t_k)}{a_kx(t_k)}(t-t_k)\,.
$$
Using $\frac{x'(t_k)}{x(t_k)}<\frac{1}{\tau}$ and
$x(t_k^+)=a_kx(t_k)$, this implies
$$
\frac{x(t_k)}{x(t)}>\frac{\tau}{a_k\tau+b_k(t-t_k)}\,.
$$
Therefore,
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-t_k}{a_k\tau+b_k(t-t_k)}.
$$


Case (2). $t\in [c_1,t_{I(c_1)+1}]$.   We consider three sub-cases.

(i) If $t_{I(c_1)}>c_1-\tau$, $t\in[t_{I(c_1)}+\tau,t_{I(c_1)+1}]$,
then there are no impulsive
moments in $(t-\tau, t)$. Making a similar analysis of case
1(i), we obtain $\frac{x(t-\tau)}{x(t)}>\frac{t-\tau-t_{I(c_1)}}{
t-t_{I(c_1)}}$.

(ii) If $t_{I(c_1)}>c_1-\tau$, $t\in [c_1,t_{I(c_1)}+\tau)$,
then $t-\tau\in [c_1-\tau,\,t_{I(c_1)})$ and there is an impulsive
moment $t_{I(c_1)}$ in $(t-\tau,\, t)$. Similar to case 1(ii), we
have
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-t_{I(c_1)}}
{a_{I(c_1)}\tau+b_{I(c_1)}(t-t_{I(c_1)})}\,.
$$

(iii) If $t_{I(c_1)}<c_1-\tau$, then there are no impulsive
moments in $(t-\tau, t)$. So
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-\tau-t_{I(c_1)}}{t-t_{I(c_1)}}\,.
$$


Case (3). $t\in (t_{I(d_1)},\,d_1]$. There
are three sub-cases to consider:

(i) If $t_{I(d_1)}+\tau<d_1,t\in [t_{I(d_1)}+\tau,d_1]$, then
there are no impulsive moments in $(t-\tau,t)$. Similar to case
2(i), we have
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-\tau-t_{I(d_1)}}{t-t_{I(d_1)}}\,.
$$

(ii) If $t_{I(d_1)}+\tau<d_1,t\in [t_{I(d_1)},t_{I(d_1)}+\tau)$,
then there is an impulsive moment $t_{I(d_1)}$. Similar to case 2(ii),
we obtain
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-t_{I(d_1)}}{
a_{I(d_1)}\tau+b_{I(d_1)}(t-t_{I(d_1)})}\,.
$$

(iii) If $t_{I(d_1)}+\tau\geq d_1$, then there is an impulsive moment
$t_{I(d_1)}$ in $(t-\tau,\,t)$. Similar to case 3(ii), we obtain
$$
\frac{x(t-\tau)}{x(t)}>\frac{t-t_{I(d_1)}}{a_{I(d_1)}\tau+b_{I(d_1)}(t-t_{I(d_1)})}\,.
$$
Combining all these cases, we have
$$
\frac{x(t-\tau)}{x(t)}
>\begin{cases}
Q_{I(c_1)}^1(t), & \text{for }  t\in [c_1,t_{I(c_1)+1}],\\
Q_k^1(t),        & \text{for }  t\in (t_k,t_{k+1}],\; k=I(c_1)+1,\dots,I(d_1)-1,\\
Q_{I(d_1)}^1(t), & \text{for }  t\in (t_{I(d_1)},d_1].
\end{cases}
$$
Hence by \eqref{e2.5}, we have
\begin{align*}
  & \sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\
&\leq   2\int _{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt
 - \int_{c_1}^{t_{I(c_1)+1}}\omega^2(t)Q(t)Q_{I(c_1)}^1(t)dt\\
&\quad  -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}
 \omega^2(t)Q(t)Q_{k}^1(t)dt
  -\int_{t_{I(d_1)}}^{d_1}\omega^2(t)Q(t)Q_{I(d_1)}^1(t)dt\\
&\quad -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt\\
&=  -\int_{c_1}^{t_{I(c_1)+1}}\frac{1}{p(t)}[p(t)\omega'(t)-v(t)\omega(t)]^2dt\\
&\quad  -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}\frac{1}{p(t)}
 [p(t)\omega'(t)-v(t)\omega(t)]^2dt\\
&\quad -\int_{t_I(d_1)}^{d_1}\frac{1}{p(t)}[p(t)\omega'(t)-v(t)\omega(t)]^2dt\\
&\quad +\int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)]
 dt\\
&\quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}[p(t)\omega'^2(t)
  -Q(t)Q_{k}^1(t)\omega^2(t)]dt\\
&\quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t)-Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,.
\end{align*}
Hence we have
\begin{equation}
\begin{aligned}
&\sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\
&< \int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)]dt\\
& \quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}[p(t)\omega'^2(t)
 -Q(t)Q_k^1(t)\omega^2(t)]dt\\
& \quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t)
 -Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,,
\end{aligned}\label{e2.6}
\end{equation}
for if not, we must have $p(t)\omega'(t)=v(t)\omega(t)$ or
$x(t)\omega'(t)=x'(t)\omega(t)$ on $[c_1,d_1]$.  Upon integrating,
$x(t)$ will be a multiple of $\omega(t)$, which contradicts the
facts that $\omega$ vanishes at $c_1$ and $d_1$ while $x(t)$ does
not.

On the other hand, since $\big(p(t)x'(t)\big)'<0$ for all
 $t\in (c_1,\,t_{I(c_1)+1}]$, $p(t)x'(t)$
is nonincreasing in $(c_1,\,t_{I(c_1)+1}]$. Thus
$$
x(t)>x(t)-x(c_1)=x'(\xi)(t-c_1)\geq\frac{p(t)x'(t)}{p(\xi)}(t-c_1),\quad
\text{for some } \xi\in (c_1,\,t)\ ,
$$
and hence $\frac{p(t)x'(t)}{x(t)}<\frac{p(\xi)}{t-c_1}$.
Letting $t\to t_{I(c_1)+1}^-$,
we have
\begin{equation}
v(t_{I(c_1)+1})\leq\frac{\beta_1}{t_{I(c_1)+1}-c_1}\,.\label{e2.7}
\end{equation}
Making a similar analysis on $(t_{k-1},t_{k}]$, $k=I(c_1)+2,\,\dots, I(d_1)$, it
is not difficult to see that
\begin{equation}
v(t_k)\leq \frac{\beta_1}{t_k-t_{k-1}}\,.\label{e2.8}
\end{equation}
Here we must point out that \eqref{e2.7} and \eqref{e2.8} play a key role in our
method for estimating $v(t_j)$, which is different from the usual
techniques for the case without impulses. From \eqref{e2.7} and \eqref{e2.8},
and noting that $a_k\le b_k$, we have
\begin{align*}
&\sum_{k=I(c_1)+1}^{I(d_1)}\frac{a_k-b_k}{a_k}\omega^2(t_k)v(t_k)\\
&\geq  \beta_1\Big[\sum_{k=I(c_1)+2}^{I(d_1)}\frac{a_k-b_k}{a_k(t_k-t_{k-1})}
 \omega^2(t_k)+\frac{a_{I(c_1)+1}-
        b_{I(c_1)+1}}{a_{I(c_1)+1}(t_{I(c_1)+1}-c_1)}\omega^2(t_{I(c_1)+1})\Big]\\
&= L(\omega, c_1, d_1)\,.
\end{align*}
Since
$$
\sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]=
\sum_{k=I(c_1)+1}^{I(d_1)}\frac{a_k-b_k}{a_k}\omega^2(t_k)v(t_k)\,,
$$
by \eqref{e2.6}, we have
\begin{align*}
L(\omega, c_1, d_1)
& <  \int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)]dt\\
& \quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}
 [p(t)\omega'^2(t)-Q(t)Q_k^1(t)\omega^2(t)]dt\\
& \quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t)
 -Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,,
\end{align*}
which contradicts \eqref{e2.1}.


If $I(c_1)=I(d_1)$, then $L(\omega, c_1, d_1)=0$, and there are
no impulsive moments in $[c_1,d_1]$. Similar to the proof of
\eqref{e2.6}, we obtain
\begin{equation}
\int_{c_1}^{d_1}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}(t)\omega^{2}(t)]dt>0\,.
\label{e2.9}
\end{equation}
This again contradicts our assumption. Finally, if $ x(t)$ is
eventually negative, we can consider $[c_2,d_2]$ and reach a
similar contradiction. The proof of Theorem \ref{thm2.1} is complete.
\end{proof}


\begin{theorem} \label{thm2.2}
 Assume conditions {\rm (H1)--(H3)} hold,
$a_k\leq b_k$ and there exists a $G\in \Gamma$ such that
\begin{equation}
\begin{aligned}
&\int_{c_j}^{d_j}p(t)g^2(t)dt
 -\int_{c_j}^{t_{I(c_j)+1}}Q(t)G(t)Q_{I(c_j)}^j(t)dt\\
&-\sum_{k=I(c_j)+1}^{I(d_j)-1} \int_{t_k}^{t_{k+1}}Q(t)G(t)Q_k^j(t)dt
 -\int_{t_I(d_j)}^{d_j}Q(t)G(t)Q_{I(d_j)}^j(t)dt\\
&\leq R(G,c_j,d_j)\,,
\end{aligned}\label{e2.10}
\end{equation}
where $R(G,c_j,d_j):=0$ for $I(c_j)=I(d_j)$, $ j=1,2$, and
\begin{align*}
&R(G,c_j,d_j)\\
&:=\frac{a_{I(c_j)+1}-b_{I(c_j)+1}}{a_{I(c_j)+1}(t_{I(c_j)+1}-c_1)}
G(t_{I(c_j)+1})\beta_j + \sum_{k=I(c_j)+2}^{I(d_j)}
\frac{a_k-b_k}{a_k}\frac{\beta_j}{t_k-t_{k-1}}G(t_k)
\end{align*}
for $I(c_j)<I(d_j)$, then all solutions of \eqref{e1.1} are oscillatory.
\end{theorem}

\begin{proof}
Similar to the proof of Theorem \ref{thm2.1}, suppose
$x(t-\tau)>0$ for $t\geq t_0$.
 If $I(c_1)<I(d_1)$, multiplying $G(t)$ throughout
\eqref{e2.4} and integrating over $[c_1,d_1]$, we obtain
\begin{align*}
&\sum_{k=I(c_1)+1}^{I(d_1)}G(t_k)\frac{a_k-b_k}{a_k}v(t_k)\\
&\leq  -\int_{c_1}^{d_1}Q(t)G(t)\frac{x(t-\tau)}{x(t)}dt
         -\int_{c_1}^{d_1}\frac{v^2(t)G(t)}{p(t)}dt
         +2\int_{c_1}^{d_1}v(t)g(t)\sqrt{G(t)}dt\\
&< -\int_{c_1}^{d_1}\Big(\sqrt{\frac{G(t)}{p(t)}}v(t)-\sqrt{p(t)}g(t)\Big)^2dt
 +\int_{c_1}^{d_1}p(t)g^2(t)dt\\
& \quad -\int_{c_1}^{t_{I(c_1)+1}}Q(t)G(t)Q_{I(c_1)}^1(t)dt
    -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}Q(t)G(t)Q_k^1(t)dt\\
& \quad -\int_{t_{I(d_1)}}^{d_1}Q(t)G(t)Q_{I(d_1)}^1(t)dt\\
&\le \int_{c_1}^{d_1}p(t)g^2(t)dt-\int_{c_1}^{t_{I(c_1)+1}}Q(t)G(t)Q_{I(c_1)}^1(t)
 dt\\
&\quad -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}Q(t)G(t)Q_k^1(t)dt
  -\int_{t_{I(d_1)}}^{d_1}Q(t)G(t)Q_{I(d_1)}^1(t)dt\,.
\end{align*}
On the other hand, from the proof of Theorem \ref{thm2.1}, we have
\begin{equation}
v(t_{I(c_1)+1})\leq \frac{\beta_1}{t_{I(c_1)+1}-c_1},\quad
v(t_k)\leq \frac{\beta_1}{t_k-t_{k-1}}, \label{e2.11}
\end{equation}
for $k=I(c_1)+2,\dots,I(d_1)$.
So
\begin{align*}
&\sum_{k=I(c_1)+1}^{I(d_1)}\frac{a_k-b_k}{a_k}G(t_k)v(t_k)\\
&\geq  \frac{a_{I(c_1)+1}-b_{I(c_1)+1}}{a_{I(c_1)+1}(t_{I(c_1)+1}-c_1)}G(t_{I(c_1)
 +1})\beta_1
 +\sum_{k=I(c_1)+2}^{I(d_1)}\frac{a_k-b_k}{a_k}\frac{\beta_1}{t_k-t_{k-1}}G(t_k)\\
&=  R(G, c_1, d_1)\,.
\end{align*}
This contradicts \eqref{e2.10}. If $I(c_1)=I(d_1)$, the proof is
similar to that of Theorem \ref{thm2.1}, and so it is omitted here.
The proof of Theorem \ref{thm2.2} is complete.
\end{proof}


Next, let $D=\{(t,s): t_0\leq s\leq t\}$. A
function $H\in C(D, \mathbb R)$ is said to belong to the class $\mathfrak{H}$
if
\begin{itemize}
\item[(A1)] $H(t,t)=0$, $H(t, s)>0$ for $t>s$; and
\item[(A2)] $H$ has partial derivatives $\frac{\partial H}{\partial t}$ and
$\frac{\partial H}{\partial s}$ on $D$ such that
$$
\frac{\partial H}{\partial t}=2h_1(t,s)\sqrt{H(t,s)},\quad
\frac{\partial H}{\partial s}=-2h_2(t,s)\sqrt{H(t,s)}.
$$
\end{itemize}
Similar to \cite[Theorem 2.3]{l1}, we have the
following Theorem.


\begin{theorem} \label{thm2.3}  Assume the conditions {\rm (H1)--(H3)}
hold. Suppose that there are $\delta_j\in (c_j, d_j)$, $j=1,2$,
and $H\in \mathfrak{H}$ such that
\begin{equation}
\begin{aligned}
  & \frac{1}{H(d_j,\delta_j)}\Big[\int_{\delta_j}^{d_j}Q(s)Q_j(s)H(d_j,s)ds
        -\int_{\delta_j}^{d_j}p(s)h_2^2(d_j, s)ds\Big]\\
 &+\frac{1}{H(\delta_j,c_j)}\Big[\int_{c_j}^{\delta_j}Q(s)Q_j(s)H(s,c_j)ds
        -\int_{c_j}^{\delta_j}p(s)h_1^2(s, c_j)ds\Big]\\
&>  P(H, c_j,d_j)\,,
\end{aligned}\label{e2.12}
\end{equation}
where $P(H, c_j, d_j):=0$ for $I(c_j)=I(d_j)$, and
\begin{equation}
\begin{aligned}
P(H, c_j, d_j)
&:=\frac{\beta_j}{H(d_j,\delta_j)}\Big(H(d_j,t_{I(\delta_j)+1})
  \frac{b_{I(\delta_j)+1} -a_{I(\delta_j)+1}}{a_{I(\delta_j)+1}(t_{I(\delta_j)+1}
  -\delta_j)}\\
&\quad + \sum_{i=I(\delta_j)+2}^{I(d_j)}
        H(d_j, t_i)\frac{b_i-a_i}{a_i(t_i-t_{i-1})}\Big)\\
& \quad +\frac{\beta_j}{H(\delta_j,c_j)}
         \Big(H(t_{I(c_j)+1},c_j)\frac{b_{I(c_j)+1}
 -a_{I(c_j)+1}}{a_{I(c_j)+1}(t_{I(c_j)+1}-c_j)} \\
&\quad + \sum_{i=I(c_j)+2}^{I(\delta_j)}H(t_i, c_j)
         \frac{b_i-a_i}{a_i(t_i-t_{i-1})}\Big)
\end{aligned}\label{e2.13}
\end{equation}
for $I(c_j)<I(d_j)$, $ j=1,2$. Then all  solutions of \eqref{e1.1}
are oscillatory.
\end{theorem}

\begin{example} \label{examp2.1}
Consider the  impulsive differential equation
\begin{equation}
\begin{gathered}
\begin{aligned}
&x''(t)+m\cos(t/2)x(t-\frac{\pi}{8})+8\cos(t/2)
 |x(t-\frac{\pi}{8})|^{\frac{3}{2}}x(t-\frac{\pi}{8})\\
&+\cos^{3}\frac{t}{2}|x(t-\frac{\pi}{8})|^{-\frac{1}{2}}x(t-\frac{\pi}{8})
=\sin\frac{t}{2}, \quad t\neq 2k\pi\pm \frac{\pi}{4},
\end{aligned}\\
x(t_k^+)=a_kx(t_k),\quad x'(t_k^+)=a_kx'(t_k),
\quad  t_k=2k\pi\pm \frac{\pi}{4}.
\end{gathered}\label{e2.14}
\end{equation}
\end{example}

In this equation,
$\tau=\pi/8$, $t_{k+1}-t_k\geq \pi/2>\pi/8$,
$\alpha_1=5/2$, $\alpha_2=1/2$, and $m$ is a positive constant.
For any $T>0$, we can choose $k $ large enough such
that $T<c_1=4k\pi-\frac{\pi}{2}<d_1=4k\pi$ and
$c_2=4k\pi+\frac{\pi}{8}<d_2=4k\pi+\frac{\pi}{2}$ satisfy
(H3), then there is an impulsive moment $t_k=4k\pi-\frac{\pi}{4}$
in $[c_1, d_1]$ and an impulsive moment
$t_{k+1}=4k\pi+\frac{\pi}{4}$ in $[c_2, d_2]$.
 Let $\omega(t)=\sin(8t)\in \Omega_\omega(c_j, d_j)$, $j=1,2$, we have
\begin{equation}
\int_{c_1}^{d_1}(\omega'(t))^2dt=32\int_{c_1}^{d_1}(\cos16t+1)dt
=16\pi,\label{e2.15}
\end{equation}
$t_{I(c_1)}=4k\pi-\frac{7}{4}\pi$, $t_{I(d_1)}=4k\pi-\frac{\pi}{4}$.
Choose $\eta_0=\eta_1=\eta_2=1/3$. Then
\begin{equation}
\begin{aligned}
Q(t)&= m\cos(t/2)+\big[(\frac{1}{3})^{-1/3}\big]^3
\big(8\cos(t/2)\big)^{1/3}\cos(t/2)\
\big|\sin\frac{t}{2}\big|^{1/3}\\
&= \cos(\frac{t}{2})(m-3\sin^{1/3}t)\\
&\geq   m\cos(t/2).
\end{aligned}\label{e2.16}
\end{equation}
Hence
\begin{equation}
\begin{aligned}
&\int_{4k\pi-\frac{\pi}{2}}^{4k\pi-
\frac{\pi}{4}}Q(t)\frac{t-\frac{\pi}{8}-t_{I(c_1)}}{t-t_{I(c_1)}}\sin^2(8t)dt\\
&+\int_{4k\pi-\frac{\pi}{4}}^{4k\pi-\frac{\pi}{8}}Q(t)\frac{t-t_{I(d_1)}}
{a_{I(d_1)}(t+\frac{\pi}{8}-t_{I(d_1)})}\sin^2(8t)dt\\
&+\int_{4k\pi-\frac{\pi}{8}}^{4k\pi}Q(t)\frac{t-\frac{\pi}{8}-t_{I(d_1)}}
 {t-t_{I(d_1)}}\sin^2(8t)dt\\
&>\frac{9}{10}m\int_{4k\pi-\frac{\pi}{2}}^{4k\pi-
\frac{\pi}{4}}\cos(t/2)\sin^2(8t)dt>16\pi
\end{aligned} \label{e2.17}
\end{equation}
for $m$ large enough. On the other hand, note that $a_k=b_k>0$, so that
$L(\omega, c_j, d_j)=0$. It follows from Theorem \ref{thm2.1} that all
the solutions of \eqref{e2.14} are oscillatory.


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

W.-S. Cheung was partially supported by grant HKU7016/07P
from the Research Grants  Council of the Hong Kong SAR, China.
Q. Li was partially supported by grants 11071054 from
the NNSF of China, A2011205012 from the Natural Science Foundation of Hebei
Province, and L2009Z02 from the  Main Foundation of Hebei Normal University.

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\end{document}