\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 24, pp. 1--6.\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 2004 Texas State University - San Marcos.}
\vspace{9mm} }

\begin{document}

\title[\hfilneg EJDE-2004/24\hfil Oscillation of nonlinear equations]
{Oscillation of nonlinear impulsive hyperbolic
 equations with several delays}

\author[Anping Liu, Li Xiao, \& Ting Liu\hfil EJDE-2004/24\hfilneg]
{Anping Liu, Li Xiao, \& Ting Liu}

\address{Department of Mathematics and Physics\\
 China University of  Geosciences \\
 Wuhan, Hubei, 430074, China}
\email[Anping Liu]{wh\_apliu@263.net}

\date{}
\thanks{Submitted December 1, 2003. Published February 19, 2004.}
\thanks{Supported by grant 10071026 from the National
Natural Science Foundation of China, and  \hfill\break\indent
by grant 40373003 from
the Natural Science Foundation of China.}
\subjclass[2000]{35L10, 35R12, 35R10}
\keywords{Impulse, hyperbolic differential equation, oscillation, delay}


\begin{abstract}
 In this paper, we study oscillatory properties of solutions
 to nonlinear impulsive hyperbolic equations with several
 delays. Sufficient conditions for oscillations of
 the solutions are established
\end{abstract}

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

 \section{Introduction}

The theory of partial functional differential
 equations can be applied to fields,
 such as to biology, population growth, engineering, medicine,
 physics and chemistry. In the last few years,
 a few of papers have been published on
 oscillation theory of partial differential equations.
 The qualitative theory of this class of equations, however,
 is still in an initial stage of development.
 We may easily visualize
 situations in nature where abrupt changes such as
 shock and disasters may occur. These phenomena are short-time
 perturbations. Consequently, it
 is natural to assume, in modelling these problems, that these
 perturbations act instantaneously, that is, in the form of
 impulses. In 1991, the first paper \cite{e1} on this class of equations
 was published. But for instance,
 on oscillation theory of impulsive partial differential equations
 only a few of papers have been published. Recently,
 Bainov, Minchev, Fu and Luo \cite{b2,b3,f1,f2,l1} investigated
 the oscillation  of solutions of impulsive partial differential equations with
 or without deviating argument.
 But there is a scarcity in the study of oscillation theory of
 nonlinear impulsive hyperbolic equations
 with several delays.
 In this paper, we discuss the oscillatory properties
 of solutions for nonlinear impulsive hyperbolic
 equations with several delays \eqref{e1}, under the boundary condition
 \eqref{e4}.
 It should be noted that the equation we discuss here is nonlinear
 and that we could not find work for oscillations of this kind of problem.
\begin{gather}
\begin{aligned}
\frac{\partial^2 u}{\partial t^2}
&=a(t)h(u)\Delta u+ \sum_{i=1}^{m} a_i(t)h_{i}(u(t-\tau_i,x))\Delta u(t-\tau _i,x) \\
&\quad -q(t,x)f(u(t,x)) -\sum_{j=1}^{n} g_j(t,x)f_{j}(u(t-\sigma_j,x)),\\
&t\neq t_k,\quad  (t,x)\in \mathbb{R}_+\times \Omega =G,
\end{aligned}    \label{e1} \\
u(t_k^{+},x)-u(t_k^{-},x)=q_ku(t_k,x),  \label{e2}\\
u_{t}(t_k^{+},x)-u_{t}(t_k^{-},x)=b_ku_{t}(t_k,x),\quad
 k=1,2,\dots. \label{e3}
\end{gather}
with the boundary condition
\begin{equation} \label{e4}
\frac{\partial u}{\partial n}=0,\quad(t,x)\in \mathbb{R}_+ \times \partial \Omega ,
\end{equation}
and the initial condition $u(t,x)=\Phi(t,x)$,
$(t,x)\in[-\delta,0] \times\Omega$. Here
$\Omega\subset \mathbb{R}^N$ is a bounded domain with boundary
$\partial \Omega $  smooth enough and $n$ is a unit exterior normal vector
of $\partial \Omega$, $\delta=\max\{ \tau_{i},\sigma_{j}\}$,
$\Phi(t,x)\in C^2 ([-\delta,0]\times\Omega,\mathbb{R})$.

  Assume that the following conditions are fulfilled:
\begin{itemize}
\item[(H1)]  $a(t),a_i(t)\in PC(\mathbb{R}_+,\mathbb{R}_+)$,
$\tau _i=\mathrm{const.}>0$, $\sigma_j=\mathrm{const.}>0$, $i=1,2,\dots m$,
 $ j=1,2,\dots n$,
 $q(t,x),g_j(t,x)\in C(\mathbb{R}_+\times \overline {\Omega },
 (0,\infty))$;
  where $PC$ denote the class of functions which are piecewise
  continuous in $t$ with
   discontinuities of first kind only at
  $t=t_k$, $k=1,2,\dots$ and left continuous at $t=t_k$,
  $k=1,2,\dots$.

\item[(H2)]  $h'(u),h'_{i}(u),f(u),f_{j}(u)\in C(\mathbb{R},\mathbb{R})$;
   $f(u)/u\geq C=\mathrm{const.}>0$,
   $f_{j}(u)/u\geq C_{j}=\mathrm{const.}>0$,
for $u\neq0$; $q_k>-1$, $b_k>-1$, $b_k<q_k$,
   $0<t_1<t_2< \dots<t_k<\dots$, $\lim_{t\to \infty }t_k= \infty$.

\item[(H3)] $ u(t,x)$ and their derivatives $u_{t}(t,x)$ are
piecewise continuous in $t$ with
   discontinuities of first kind only at
  $t=t_k$, $k=1,2,\dots$ and left continuous at $t=t_k$,
  $u(t_k,x)=u(t_k^{-},x)$, $u_{t}(t_k,x)=u_{t}(t_k^{-},x)$,
  $k=1,2,\dots $.
\end{itemize}
%
  Let us construct the sequence $\{\bar{t}_k\}=\{t_k\}
  \cup\{t_{k\tau _i}\}\cup \{t_{k\sigma _j}\}$,
   where $t_{k\tau _i}=t_k+\tau _i,t_{k\sigma _j}=t_k+\sigma _j$ and
 $\bar{t}_k<\bar{t}_{k+1},i=1,2,\dots ,m,k=1,2,\dots$. \smallskip

\noindent\textbf{Definition.} %1.1.
By a solution of problem \eqref{e1}, \eqref{e4} with initial condition,
we mean that any function $u(t,x)$ for which the following
 conditions are valid:
\begin{enumerate}
\item If $-\delta \leq t \leq 0$, then $u(t,x)=\Phi(t,x)$.

\item If $0 \leq t \leq \bar{t}_1=t_1$, then $u(t,x)$
 coincides with the solution of the problem \eqref{e1}--\eqref{e4}
 with initial condition.

\item If $\bar{t}_k < t \leq \bar{t}_{k+1}$,
$\bar{t}_k\in\{t_k\} \setminus (\{t_{ki}\}\bigcup \{t_{kj}\})$,
then $u(t,x)$ coincides with the solution of the problem
\eqref{e1}--\eqref{e4}.

\item If $\bar{t}_k < t \leq \bar{t}_{k+1}$,
$\bar{t}_k\in \{t_{ki}\}\bigcup \{t_{kj}\}$, then $u(t,x)$
 coincides with the solution of the problem \eqref{e4}
   and the following equations
 \begin{gather*}
\frac{\partial^2u}{\partial t^2}=a(t)h(u(t^{+},x))
 \Delta u(t^{+},x)+
 \sum_{i=1}^{m} a_i(t)h_{i}(u((t-\tau_i)^{+},x))\Delta u((t-\tau _i)^{+},x)\\
  -q(t,x)f(u(t^{+},x))-\sum_{j=1}^{n} g_j(t,x)f_{j}(
 u((t-\sigma_j^{+}),x)) \quad  (t,x)\in \mathbb{R}_+\times \Omega =G
\\
u(\bar{t}_k^{+},x)=u(\bar{t}_k,x),\quad
u_{t}(\bar{t}_k^{+},x)=u_{t}(\bar{t}_k,x),\\
  \mbox{for } \bar{t}_k\in (\{t_{k\tau_{i}}\}\cup \{t_{k\sigma_{j}}\})
   \setminus \{t_k \},
\end{gather*}
or
\begin{gather*}
 u(\bar{t}_k^{+},x)=(1+q_{k_{i}})u(\bar{t}_k,x),\quad
   u_{t}(\bar{t}_k^{+},x)=(1+b_{k_{i}})u_{t}(\bar{t}_k,x)\\
\mbox{for }  \bar{t}_k\in (\{t_{k\tau_{i}}\}\cup \{t_{k\sigma_{j}}\})
 \cap \{t_k \}.
\end{gather*}
Where the number $k_{i}$ is determined by the equality $\bar{t}_k=t_{k_{i}}$.
\end{enumerate}

We introduce the notation:
$\Gamma_k=\{(t,x): t\in(t_k,t_{k+1}),x\in\Omega \}$,
$\Gamma=\bigcup^{\infty}_{k=0}\Gamma_k$,
$\bar{\Gamma}_k=\{(t,x):  t\in(t_k,t_{k+1}),x\in\bar{\Omega} \}$,
$\bar{\Gamma}=\bigcup^{\infty}_{k=0}\bar{\Gamma}_k$,
$v(t)=\int_\Omega u(t,x)dx$ and $p(t)=min q(t,x)$,
$p_{j}(t)=min g_{j}(t,x),x\in \bar{\Omega}$. \smallskip

\noindent\textbf{Definition.}% 1.2.
 The solution $u \in C^2(\Gamma) \cap C^{1}(\bar {\Gamma})$
 of problem \eqref{e1}, \eqref{e4} is called nonoscillatory
 in the domain $G$ if it is either eventually positive or
 eventually negative.
 Otherwise, it is called oscillatory.

\section{Oscillation properties of the problem \eqref{e1}, \eqref{e4}}

 For the main theorem of this paper, we need following lemmas.

\begin{lemma} \label{lm2.1}
Let $u \in C^2(\Gamma) \cap C^{1}(\bar{\Gamma})$
 be a positive solution of \eqref{e1}, \eqref{e4} in $G$,
 then function $v(t)$ satisfies the impulsive differential
 inequality
\begin{gather}
 v''(t)+Cp(t)v(t)+\sum_{j=1}^{n} C_{j}p_j(t)v(t-\sigma_j)\leq0,\;,
 t\neq t_k, \label{e5} \\
v(t^{+}_k)=(1+q_k)v(t_k)\;\;k=1,2,\dots. \label{e6}\\
 v'(t^{+}_k)=(1+b_k)v'(t_k)\;\;k=1,2,\dots. \label{e7}
\end{gather}
\end{lemma}

\begin{proof}
Let $u(t,x)$ be a positive  solution of the problem \eqref{e1}, \eqref{e4} in
$G$.  Without loss of generality, we may assume that $u(t,x)>0$,
 $u(t-\tau _i,x)>0$,
 $i=1,2,\dots ,m,u(t-\sigma _j,x)>0$,
 $j=1,2,\dots ,n$, for any $(t,x)\in [t_0,\infty )\times \Omega$.

For $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$,
 integrating \eqref{e1} with respect to $x$ over $\Omega $ yields
\begin{equation} \label{e8}
\begin{aligned}
\frac {d^2}{dt^2}[\int_\Omega udx]
&= a(t)\int_\Omega h(u)\Delta udx
 + \sum_{i=1}^m a_i(t)\int_\Omega h_{i}(u(t-\tau_{i},x))
 \Delta u(t-\tau _i, x)dx \\
&\quad - \int_\Omega q(t,x)f(u(t,x))dx
 -  \sum_{j=1}^{n} \int_\Omega g_j(t,x)f_{j}(u(t-\sigma_j,x)).
\end{aligned}
\end{equation}
 By Green's formula and the boundary condition, we have
\begin{gather*}
\int_\Omega h(u)\Delta udx=\int_{\partial \Omega }h(u)\frac
 {\partial u}{\partial n}ds-\int_\Omega h'(u)|gradu|^2dx
\leq - \int_\Omega h'(u)|gradu|^2dx\leq 0,\\
\int_\Omega h_{i}(u(t-\tau _i,x))\Delta u(t-\tau _i,x)dx\leq 0.
\end{gather*}
 From condition (H2), we can easily obtain
\begin{gather*}
\int_\Omega q(t,x)f(u(t,x))dx\geq Cp(t) \int_\Omega u(t,x)dx,\\
\int_\Omega q_{j}(t,x)f_{j}(u(t-\sigma_{j},x))dx\geq
C_{j}p_{j}(t) \int_\Omega u(t-\sigma_{j},x)dx.
\end{gather*}
It follows that from above that
\begin{equation} \label{e9}
v''+Cp(t)v(t)+\sum_{j=1}^{n} C_{j}p_j(t)v(t-\sigma_j)
 \leq 0,\quad (t\geq t_0,\;t\neq t_k).
\end{equation}
Where $v(t)>0$.
For $t>t_0$, $t=t_k$, $k=1,2,\dots$, we have
\begin{gather*}
\int_\Omega u(t_k^{+},x)dx- \int_\Omega u(t_k^{-},x)dx= q_k\int_\Omega
u(t_k,x)dx,\\
\int_\Omega u_{t}(t_k^{+},x)dx- \int_\Omega u_{t}(t_k^{-},x)dx =b_k\int_\Omega
 u_{t}(t_k,x)dx.
\end{gather*}
This implies
\begin{gather}
v(t_k^{+})=(1+q_k)v(t_k), \label{e10}\\
v'(t^{+}_k)=(1+b_k)v'(t_k)\quad k=1,2,\dots. \label{e11}
\end{gather}
Hence we obtain that $v(t)>0$ is a positive solution of
 differential inequality \eqref{e5}--\eqref{e7}. This completes
 the proof.
\end{proof}

\noindent\textbf{Definition.} % 2.2.
The solution $v(t)$ of differential  inequality \eqref{e5}--\eqref{e7}
is called eventually positive
 (negative) if there exists a number $t^{*}$ such that
 $v(t)>0 (v(t)<0)$ for $ t \geq t^{*}$.


\begin{lemma}[{\cite[Theorem 1.4.1]{b4}}] \label{lm2.3}
Assume that \begin{itemize}
\item[(i)] $m(t)\in PC^{1}[\mathbb{R}^{+},\mathbb{R}]$
is left continuous at $t_k$  for $k=1,2,\dots$,

\item[(ii)] For $k=1,2,\dots$ and $t \geq t_0$,
\begin{gather*}
m'(t)\leq p(t)m(t)+q(t) \quad (t\neq t_k),\\
 m(t^{+}_k)\leq d_km(t_k)+e_k.
\end{gather*}
\end{itemize}
where $ p(t),q(t)\in C(\mathbb{R}^{+},R),d_k \geq 0 $ and $ e_k $
 are real constants,
 $ PC^{1}[\mathbb{R}^{+},\mathbb{R}]=\{x:\mathbb{R}^{+}\to \mathbb{R}; x(t)$
is continuous and continuously differentiable everywhere except some
 $t_k$ at which $x(t^{+}_k),x(t^{-}_k),x'(t^{+}_k) $
and $x'(t^{-}_k)$ exist and $x(t_k)=x(t^{-}_k),x'(t_k)= x'(t^{-}_k)\}$.
Then
\begin{align*}
 m(t)&\leq m(t_0)\prod_{t_0<t_k<t}d_k \exp(\int_{t_0}^t p(s)ds)
 +\int_{t_0}^t\prod_{s<t_k<t}d_k exp(\int_{s}^t
 p(r)dr)q(s)ds  \\
 &\quad +\sum_{t_0<t_k<t} \prod_{t_k<t_{j}<t}  d_{j}exp(\int_{t_k}^t p(s)ds)e_k.
 \end{align*}
\end{lemma}

 From the above lemma, we can obtain the following result;
 see also \cite{l1}.

\begin{lemma} \label{lm2.4}
Let $v(t)$ be eventually positive (negative) solution of
 differential inequality \eqref{e5}--\eqref{e7}. Assume that there exists
 $T \geq t_0$ such that
  $v(t)>0(v(t)<0)$ for $t \geq T$. If the following condition
  holds,
 \begin{equation} \label{e12}
\lim _{t \to + \infty } \int_{t_0}^t \prod_{t_0<t_k<s}
\frac{1+b_k}{1+q_k}ds=+\infty,
\end{equation}
then  $v'(t)\geq 0(v'(t) \leq 0)$ for $t \in [T,t_{l}]
 \bigcup(\bigcup^{+\infty} _{k=l}(t_k,t_{k+1}])$, where
 $l=\min\{k:t_k\geq T\}$.
\end{lemma}

\begin{theorem} \label{thm2.5}
If condition \eqref{e12} hold and, for some $j_0$,
\begin{equation} \label{e13}
\lim _{t \to + \infty } \int_{t_0}^t
 \prod_{t_0<t_k<s} \frac{1+q_k}{1+b_k} p_{j_0}(s)ds=+\infty,
\end{equation}
then each solution of \eqref{e1}--\eqref{e4} oscillates in $G$.
\end{theorem}

\begin{proof}
Let $u(t,x)$ be a nonoscillatory solution of \eqref{e1}, \eqref{e4}.
 Without loss of generality, we can assume that $u(t,x)>0$,
$u(t-\tau _i,x)>0$, $i=1,2,\dots ,m$,
 $u(t-\sigma _j,x)>0$, $j=1,2,\dots ,n$
 for any $(t,x)\in [t_0,\infty )\times \Omega$.
 From Lemma \ref{lm2.1}, we know that $v(t)$ is a positive solution
 of \eqref{e5}--\eqref{e7}. Thus from Lemma \ref{lm2.4}, we can find
that $v'(t)\geq 0$  for $t \geq t_0$.

  For $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$, define
\[
  w(t)=\frac{v'(t)}{v(t)},\quad t \geq t_0.
\]
Then we have $w(t)>0$, $t \geq t_0$, $v'(t)-w(t)v(t)=0$.
 We may assume that $v(t_0)=1$, thus in view of \eqref{e5}--\eqref{e7}
 we have that for $t \geq t_0$,
 \begin{gather}
 v(t)=\exp(\int_{t_0}^t w(s)ds) \label{e14} \\
 v'(t)=w(t)\exp(\int_{t_0}^t w(s)ds) \label{e15}\\
 v''(t)=w^2(t)\exp(\int_{t_0}^t w(s)ds)+w'(t)\exp(\int_{t_0}^t w(s)ds)
 \label{e16}
\end{gather}
We substitute \eqref{e14}--\eqref{e16} into \eqref{e5} and can obtain
\[
w^2(t)\exp(\int_{t_0}^t w(s)ds)+w'(t)\exp(\int_{t_0}^tw(s)ds)
+C_{j_0}p_{j_0}(t)exp(\int_{t_0}^ {t-\sigma_{j_0}}w(s)ds) \leq 0.
\]
Hence, we have
\[
  w^2(t)+w'(t)+C_{j_0}p_{j_0}(t)exp(-\int_{t-\sigma_{j_0}}
  ^{t}w(s)ds) \leq 0\,.
\]
From above and condition $b_k<q_k$, it is easy to see that the function $w(t)$ is
non-increasing for  $t \geq t_1 \geq \delta +t_0$. Thus $w(t) \leq w(t_1)$, for $t
\geq t_1$,
 which implies
 \[
w'(t)+C_{j_0}p_{j_0}(t)\exp(- \delta w(t_1)ds) \leq 0, \quad t \geq t_1.
 \]
 From \eqref{e6}, \eqref{e7} we get
\[
  w(t^{+}_k)=\frac{v'(t^{+}_k)}{v(t^{+}_k)}=\frac {1+b_k}{1+q_k}
  w(t_k),\quad k=1,2,\dots.
\]
 It follows that
\begin{gather}
w'(t)\leq -C_{j_0}p_{j_0}(t)exp(- \delta w(t_1)ds) \quad (t\neq t_k), \label{e17}\\
w(t^{+}_k)=\frac{1+b_k}{1+q_k}w(t_k) \quad (t= t_k). \label{e18}
\end{gather}
 Using Lemma \ref{lm2.3}, we obtain
\begin{align*}
 w(t)&\leq w(t_0)\prod_{t_0<t_k<t}
  \frac{1+b_k}{1+q_k}   +\int_{t_0}^t\prod_{s<t_k<t}\frac{1+b_k}
  {1+q_k}(-C_{j_0}p_{j_0}(s)\exp(- \delta w(t_1)))ds \\
 & =\prod_{t_0<t_k<t}   \frac{1+b_k}{1+q_k}
  \{w(t_0)-\int_{t_0}^t\prod_{t_0<t_k<s}  \frac{1+q_k}{1+b_k}C_{j_0}p_{j_0}(s)
  \exp(- \delta w(t_1))ds \}.
\end{align*}
  Since $w(t)>0$, the last inequality contradicts \eqref{e13}.
 Therefore, the proof is complete.
\end{proof}

It should be noted that obviously all solutions of
 problem \eqref{e1}--\eqref{e2} are oscillatory
 if there exists a subsequence $n_k$ of $n$ such that
 $q_{n_k}<-1$, for $k=1,2,\dots$.
So we  discuss only the case of $ q_k>-1$.


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\end{document}