\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 80, pp. 1--8.\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-2004/80\hfil Oscillation of solutions]
{Oscillation of solutions for systems of hyperbolic equations of neutral type}

\author[Peiguang Wang, Yonghong Wu\hfil EJDE-2004/80\hfilneg]
{Peiguang Wang, Yonghong Wu} % in alphabetical order

\address{Peiguang Wang\hfill\break
College of Electronic and Information Engineering,
Hebei University, Baoding, 071002,  China}
\email{pgwang@mail.hbu.edu.cn}

\address{Yonghong Wu\hfill\break
Department of Mathematics and Statistics, 
Curtin University of Technology, GPO BOX U1987,
Perth, WA6845, Australia}
\email{yhwu@maths.curtin.edu.au}

\date{}
\thanks{Submitted December 19, 2003. Published June 7, 2004.}
\thanks{Supported by the  Natural Science 
 Foundation of Hebei Province of China (A2004000089)}
\subjclass[2000]{35B05, 35R10} 
\keywords{Oscillation, systems of hyperbolic equations, \hfill\break\indent
 boundary value problem, distributed deviating arguments}

\begin{abstract}
 In this paper, we obtain sufficient conditions
 for the oscillation of solutions to systems of
 hyperbolic differential equations of neutral type.
 We consider such systems subject to two kinds of
 boundary conditions.
\end{abstract}

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

\section{Introduction}

We consider the following system of hyperbolic differential
equations of neutral type
\begin{equation} \label{e1}
\begin{aligned}
& \frac{\partial^{2}}{\partial{t^2}}
\Big[u_{i}(x,t)-\sum_{j=1}^{m}c_{ij}u_{j}(x,t-\tau)\Big]\\
&=a_{i}(t)\Delta {u_{i}(x,t)}+b_{i}(t)\Delta {u_{i}(x,t-\rho)}
 -p_{i}(x,t)u_{i}(x,t)\\
 &\quad -\sum_{k=1}^{m}
\int_{a}^{b}q_{ik}(x,t,\xi)u_{k}[x,g(t,\xi)]d\mu(\xi),
(x,t)\in{\Omega\times\mathbb{R}_+\equiv{G},}
\end{aligned}
\end{equation}
subject to either of the following boundary conditions
\begin{gather}
\frac{\partial{u_{i}}}{\partial{n}}+\nu_{i}(x,t)u_{i}=0,
\quad (x,t)\in{\partial\Omega\times\mathbb{R}_+} \label{e2}\\
u_{i}(x,t)=0, \quad (x,t)\in{\partial\Omega\times\mathbb{R}_+}, \label{e3}
\end{gather}
where $\Omega$ is a bounded domain in $R^{n}$ with a piecewise
smooth boundary $\partial{\Omega}$, $R_+=[0,\infty)$, $\Delta$ is
the Laplacian operator in $\mathbb{R}^n$, $c_{ij}$, $\tau>0$ and
$\rho>0$ are constants, $n$ is the unit outward normal vector of
$\partial{\Omega}$.

There has been considerable interest in obtaining sufficient
conditions for oscillatory solutions of partial functional
differential equations, as this type of equations arise frequently
in many application fields (see for example the monograph
\cite{wu1}). Recently, several papers concerning systems of
hyperbolic functional differential equations have appeared in
literatures \cite{b1,li1,li2}. It is noted that previous work
focused only on the cases where the neutral coefficient number
lies between $-1$ and $0$, that is $-1\le c(t)\le 0$. To the best
of our knowledge, very little work has been done for other cases.

The aim of this paper is to study the oscillation problem for the
above system of hyperbolic differential equations of neutral type,
and give some oscillatory criteria for such systems. Throughout
the paper, we assume that the following conditions hold:
\begin{itemize}
\item[(H1)] $a_{i}(t)$, $b_{i}(t)\in{C(R_{+},R_{+})}$;

\item[(H2)] $p_{i}(x,t)\in{C(G,R_+)}$,
$q_{ik}(x,t,\xi)\in{C(G\times{[a,b]},R)}$;

\item[(H3)] $\nu_{i}(x,t)\in{C(\partial\Omega\times\mathbb{R}_+,R_{+})}$;

\item[(H4)] $g(t,\xi)\in{C(R_+\times{[a,b]},R)}$, $g(t,\xi)\le t$
for $\xi\in{[a,b]}$, and \\
 $\liminf_{t\to\infty,\xi\in{[a,b]}}\{g(t,\xi)\}=\infty$;

\item[(H5)] $\mu(\xi)\in{([a,b],R)}$ is nondecreasing, and the
integral in \eqref{e1} is a Stieltjes integral.
\end{itemize}

\noindent{\bf Definition.} % 1.}
A vector function
$u(x,t)=\{u_{1}(x,t),\dots ,u_{n}(x,t)\}^{T}$ is said to be a
 {\it solution} of \eqref{e1} and \eqref{e2} or \eqref{e1} and \eqref{e3} if it
satisfies equation \eqref{e1} in
$G\equiv{\Omega\times\mathbb{R}_+}$ and the associated boundary
condition on the boundary of $G$. \smallskip

\noindent {\bf Definition.} % 2\quad
 A vector solution $u(x,t)$ of the boundary value problem
 is said to be {\it oscillatory} in the domain $G$ if at least one
of its nontrivial components is oscillatory. Otherwise, the vector
solution $u(x,t)$ is said to be {\it nonoscillatory}. \smallskip

For the following neutral differential inequality
\begin{equation} \label{e4}
\frac{d^2}{dt^2}[y(t)-\lambda(t)y(t-\tau)]
+p(t)y(t)+\int_{a}^{b}q(t,\xi)y[g(t,\xi)]d\mu(\xi)\le 0,
\quad t\ge{0},
\end{equation}
where $\lambda(t)\in{C'(R_+, R_+)}$, $p(t)\in{C(R_+,R_+)}$,
$q(t,\xi)\in{C(R_+\times{[a,b]},R_+)}$, we assume that the following
conditions hold.
\begin{itemize}
\item[(A1)] There exists a function $h(t,\xi)\in{C(R_+\times{[a,b]},R_+)}$,
such that $h(h(t,\xi),\xi)=g(t,\xi)$, and $h(t,\xi)$ is
nondecreasing with respect to $t$ and $\xi$, and also $t\ge
h(t,\xi)\ge g(t,\xi)$;

\item[(A2)]  $\liminf_{t\to\infty}\int_{g(t,b)}^{t}
 \int_{a}^{b}q(s,\xi)d\mu(\xi)ds>1/e$;

\item[(A3)]  $\liminf_{t\to\infty}
\int_{h(t,b)}^{t}\int_{a}^{b}q(s,\xi)d\mu(\xi)ds>0$.
\end{itemize}
We remark here that the existence of the function $h(t,\xi)$ has
been proved in  \cite{r1}. The following two Lemmas are derived
from the known literatures and are useful to the proof of the main
results of this paper.

\begin{lemma}[\cite{r1}] \label{lm1}
Under assumptions (A1)--(A3),
the first-order differential inequality
\begin{equation} \label{e5}
x'(t)+\int_{a}^{b}q(t,\xi)x[g(t,\xi)]d\mu(\xi)\le 0
\end{equation}
has no eventually positive solution.
\end{lemma}


\begin{lemma}[\cite{w1}] \label{lm2}
Assume that $\lambda(t)\le{1}$, and (A1) holds. If for
some $0\le\varepsilon_{0}\le 1$ and $\xi_{0}\in{[a,b]}$,
\begin{gather}
\liminf_{t\to\infty}\int_{h(t,b)}^{t}\int_{a}^{b}
\varepsilon_{0}q(s,\xi)g(s,\xi)d\mu(\xi)ds>0, \label{e6}\\
\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\int_{a}^{b}
\varepsilon_{0}q(s,\xi)g(s,\xi)d\mu(\xi)ds
>\frac{1}{e}\exp\Big[
-\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\varepsilon_{0}sp(s)ds\Big],
\label{e7}
\end{gather}
then inequality \eqref{e4} has no eventually unbounded positive
solution.
\end{lemma}

\section{Main Results}

We introduce the following notation:
\begin{equation} \label{e8}
\begin{gathered}
p_{i}(t)=\min_{x\in{\overline{\Omega}}}\{p_{i}(x,t)\},
\quad P(t)=\min_{1\le i\le n}p_{i}(t); \\
Q_{ii}(t,\xi)=\min_{x\in{\overline{\Omega}}}\{q_{ii}(x,t,\xi)\},
\quad Q^{*}_{ik}(t,\xi)=\max_{x\in{\overline{\Omega}}}\{q_{ik}(x,t,\xi)\}.
\end{gathered}
\end{equation}

\begin{theorem} \label{thm1}
Suppose that $0\le C\le 1$ and (A1) holds. If for some constants
$0\le\varepsilon_{0}\le 1$ and $\xi_{0}\in{[a,b]}$,
\begin{equation} \label{e9}
\liminf_{t\to\infty}\int_{h(t,b)}^{t}\int_{a}^{b}
\varepsilon_{0}Q(s,\xi)g(s,\xi)d\mu(\xi)ds>0,
\end{equation}
\begin{equation} \quad\quad
\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\int_{a}^{b}
\varepsilon_{0}Q(s,\xi)g(s,\xi)d\mu(\xi)ds
>\frac{1}{e}\exp\Big[
-\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\varepsilon_{0}sP(s)ds \Big],
\label{e10}
\end{equation}
then each unbounded solution $u(x,t)$ of \eqref{e1}--\eqref{e2} is
oscillatory in the domain $G$, where
\[
 C=\max_{1\le i\le n}\Big\{c_{ii}
 +\sum_{j=1,j\neq{i}}^{m}|c_{ji}|\Big\},
\quad Q(t,\xi)=\min_{i\le i\le n}
\Big\{Q_{ii}(t,\xi)-\sum_{j=1,j\neq{i}}^{m}Q^{*}_{ji}(t,\xi)\Big\}.
\]
\end{theorem}

\begin{proof}  Suppose to the contrary that there is a nonoscillatory solution
$u(x,t)=(u_{1}(x,t),\dots ,u_{n}(x,t))^{T}$ of the problem
\eqref{e1} and \eqref{e2}, and that $|u_{i}(x,t)|>0$ for
 $t\ge 0$, $i=1,2,\dots ,n$. From (H4), there exists a
 $t_{1}\ge 0$, for $j=1,2,\dots ,m$, $k=1,2,\dots ,m$, such that
\[
|u_{j}(x,t-\tau)|>0, \quad |u_{i}(x,t-\rho)|>0, \quad
 |u_{k}[x,g(t,\xi)]|>0, \quad t\ge t_{1},
\xi\in{[a,b]}.
\]
Let
\begin{equation} \label{e11}
\begin{gathered}
\delta_{i}=\mathop{\rm sgn}{u_{i}(x,t)}, \quad
  \delta_{k}=\mathop{\rm sgn}{u_{k}[x,g(t,\xi)]}; \\
 Y_{i}(x,t)=\delta_{i}u_{i}(x,t),\quad
  Y_{k}[x,g(t,\xi)]=\delta_{k}u_{k}[x,g(t,\xi)],
\end{gathered}
\end{equation}
then $Y_{i}(x,t)>0$, $Y_{j}(x,t-\tau)>0$, $Y_{i}(x,t-\rho)>0$ and
$Y_{k}[x,g(t,\xi)]>0$ for $t\ge t_{1}$, $\xi\in{[a,b]}$,
$i=1,2,\dots ,n$, $j=1,2,\dots ,m$ and $k=1,2,\dots ,m$.

Integrating equation \eqref{e1} with respect to $x$ over the
domain $\Omega$, for $t\ge t_{1}$, we obtain
\begin{equation} \label{e12}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[\int_{\Omega}u_{i}(x,t)dx
 -\sum_{j=1}^{m}\int_{\Omega}c_{ij}u_{j}(x,t-\tau)dx\Big] \\
&+\int_{\Omega}p_{i}(x,t)u_{i}(x,t)dx
 +\sum_{k=1}^{m}\int_{\Omega} \int_{a}^{b}q_{ik}(x,t,\xi)
 u_{k}[x,g(t,\xi)]d\mu(\xi)dx \\
& =a_{i}(t)\int_{\Omega}\Delta {u_{i}(x,t)}dx
+b_{i}(t)\int_{\Omega}\Delta {u_{i}(x,t-\rho)}dx.
\end{aligned}
\end{equation}
Furthermore, it follows from \eqref{e11} that
\begin{equation} \label{e13}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[\int_{\Omega}Y_{i}(x,t)dx
-\sum_{j=1}^{m}\int_{\Omega}c_{ij}
\frac{\delta_{i}}{\delta_{j}}Y_{j}(x,t-\tau)dx\Big] \\
&+\int_{\Omega}p_{i}(x,t)Y_{i}(x,t)dx
+\sum_{k=1}^{m}\int_{\Omega}\frac{\delta_{i}}{\delta_{k}}
\int_{a}^{b}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]d\mu(\xi)dx \\
&=a_{i}(t)\int_{\Omega}\Delta {Y_{i}(x,t)}dx
+b_{i}(t)\int_{\Omega}\Delta {Y_{i}(x,t-\rho)}dx\,.
\end{aligned}
\end{equation}
It is clear that
\begin{equation} \label{e14}
\int_{\Omega}\int_{a}^{b}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]d\mu(\xi)dx
=\int_{a}^{b}\int_{\Omega}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]dxd\mu(\xi).
\end{equation}
 From the Green's formula and the boundary condition, we have
\begin{gather}
\int_{\Omega}\Delta {Y_{i}(x,t)}dx
=\int_{\partial\Omega}\frac{\partial{Y_{i}(x,t)}}{\partial{n}}d\omega
=-\int_{\partial\Omega}\nu_{i}(x,t)Y_{i}(x,t)d\omega\le 0, \label{e15}\\
\int_{\Omega}\Delta {Y_{i}(x,t-\rho)}dx
=-\int_{\partial\Omega}\nu_{i}(x,t-\rho)Y_{i}(x,t-\rho)d\omega
 \le 0, \label{e16}
 \end{gather}
where $d\omega$ is the surface integral element on
$\partial\Omega$.
Moreover, it follows from \eqref{e8} that
\begin{equation} \label{e17}
\int_{\Omega}p_{i}(x,t)Y_{i}(x,t)dx
 \ge p_{i}(t)\int_{\Omega}Y_{i}(x,t)dx.
\end{equation}
Combining \eqref{e14}--\eqref{e17} and noting \eqref{e8}, we
obtain
\begin{equation} \label{e18}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[\int_{\Omega}Y_{i}(x,t)dx
-\sum_{j=1}^{m}\int_{\Omega}c_{ij}
\frac{\delta_{i}}{\delta_{j}}Y_{j}(x,t-\tau)dx\Big] \\
&+p_{i}(t)\int_{\Omega}Y_{i}(x,t)dx+\int_{a}^{b}Q_{ii}(t,\xi)
 \Big[\int_{\Omega}Y_{i}[x,g(t,\xi)]dx\Big]d\mu(\xi) \\
&-\sum_{k=1,k\neq{i}}^{m}\int_{a}^{b}Q^{*}_{ik}(t,\xi)
\Big[\int_{\Omega}Y_{k}[x,g(t,\xi)]dx\Big]d\mu(\xi)
 \le 0.
\end{aligned}
\end{equation}
Let
\begin{equation} \label{e19}
V_{i}(t)=\int_{\Omega}Y_{i}(x,t)dx,\qquad i=1,2\dots ,n,
\end{equation}
then $V_{i}(t)>0$, and it follows from \eqref{e18} that
\begin{equation} \label{e20}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[V_{i}(t)-\sum_{j=1}^{m}c_{ij}
\frac{\delta_{i}}{\delta_{j}}V_{j}(t-\tau)\Big]
+p_{i}(t)V_{i}(t)\\
&+\int_{a}^{b}Q_{ii}(t,\xi)V_{i}[g(t,\xi)]d\mu(\xi)
-\sum_{k=1,k\neq{i}}^{m}
\int_{a}^{b}Q^{*}_{ik}(t,\xi)V_{k}[g(t,\xi)]d\mu(\xi) \le 0\,.
\end{aligned}
\end{equation}
Furthermore, let $V(t)=\sum_{i=1}^{m}V_{i}(t)$, then $V(t)>0$, and
it follows from \eqref{e8} that
\begin{equation} \label{e21}
\begin{aligned}
&\frac{d^2}{dt^2}\Big\{\sum_{i=1}^{m}
\Big[V_{i}(t)dx-\sum_{j=1}^{m}c_{ij}
\frac{\delta_{i}}{\delta_{j}}V_{j}(t-\tau)\Big]\Big\}
+P(t)V(t) \\
&+\int_{a}^{b}\sum_{i=1}^{m}
 \Big\{Q_{ii}(t,\xi)V_{i}[g(t,\xi)]-\sum_{k=1,k\neq{i}}^{m}
Q^{*}_{ik}(t,\xi)V_{k}[g(t,\xi)]\Big\}d\mu(\xi)\le 0.
\end{aligned}
\end{equation}
Noting that
\begin{align*}
&\sum_{i=1}^{m}\Big[V_{i}(t)
-\sum_{j=1}^{m}c_{ij}\frac{\delta_{i}}{\delta_{j}}V_{j}(t-\tau)\Big]\\
&=\sum_{i=1}^{m}\Big[V_{i}(t)-c_{ii}V_{i}(t-\tau)
-\sum_{j=1,j\neq{i}}^{m}c_{ij}\frac{\delta_{j}}{\delta_{i}}
V_{j}(t-\tau)\Big] \\
&\ge\sum_{i=1}^{m}\Big[V_{i}(t)-c_{ii}V_{i}(t-\tau)
-\sum_{j=1,j\neq{i}}^{m}|c_{ij}|V_{j}(t-\tau)\Big] \\
&=\Big[V_{1}(t)-c_{11}V_{1}(t-\tau)
-\sum_{j=1,j\neq{1}}^{m}|c_{1j}|V_{j}(t-\tau)\Big] \\
&+\cdots+\Big[V_{n}(t)-c_{nn}V_{n}(t-\tau)
-\sum_{j=1,j\neq{n}}^{m}|c_{nj}|V_{j}(t-\tau)\Big] \\
&=\sum_{i=1}^{m}V_{i}(t)- \Big(c_{11}+
\sum_{j=1,j\neq{1}}^{m}|c_{j1}|\Big)V_{1}(t-\tau) \\
&\quad -\cdots-\Big(c_{nn}
+\sum_{j=1,j\neq{n}}^{m}|c_{jn}|\Big)V_{n}(t-\tau) \\
&\ge\sum_{i=1}^{m}V_{i}(t) -\max_{1\le i\le n}\Big\{c_{ii}
+\sum_{j=1,j\neq{i}}^{m}|c_{ji}|\Big\}\sum_{i=1}^{m}V_{i}(t-\tau)
\\
&=V(t)-CV(t-\tau), \,
\end{align*}
and
\begin{align*}
&\sum_{i=1}^{m}\Big\{ Q_{ii}(t,\xi)V_{i}[g(t,\xi)]
-\sum_{j=1,j\neq{i}}^{m}Q^{*}_{ij}(t,\xi)V_{j}[g(t,\xi)]\Big\} \\
& =\Big[Q_{11}(t,\xi)V_{1}[g(t,\xi)]
 -\sum_{j=1,j\neq{1}}^{m}Q^{*}_{1j}(t,\xi)V_{j}[g(t,\xi)]\Big] \\
&+\cdots+\Big[Q_{nn}(t,\xi)V_{n}[g(t,\xi)]
-\sum_{j=1,j\neq{n}}^{m}Q^{*}_{nj}(t,\xi)V_{j}[g(t,\xi)]\Big] \\
& = \Big[Q_{11}(t,\xi)
-\sum_{j=1,j\neq{1}}^{m}Q^{*}_{j1}(t,\xi)\Big]V_{1}[g(t,\xi)] \\
&\quad +\cdots+\Big[Q_{nn}(t,\xi)
-\sum_{j=1,j\neq{n}}^{m}Q^{*}_{jn}(t,\xi)\Big]V_{n}[g(t,\xi)] \\
&\ge\min_{1\le i\le n}\Big\{Q_{ii}(t,\xi)
-\sum_{j=1,j\neq{i}}^{m}Q^{*}_{ji}(t,\xi)\Big\}
\sum_{i=1}^{m}V_{i}[g(t,\xi)] \\
&=Q(t,\xi)V[g(t,\xi)], \,
\end{align*}
we have from \eqref{e21} that
\begin{equation} \label{e22}
\quad \frac{d^2}{dt^2}[V(t)-CV(t-\tau)]+P(t)V(t)
+\int_{a}^{b}Q(t,\xi)V[g(t,\xi)]d\mu(\xi)\le 0.
\end{equation}
 From Lemma \ref{lm2}, inequality \eqref{e22} has no eventually positive
solutions, which is in contradiction with $V(t)>0$. This completes
the proof of Theorem \ref{thm1}.
\end{proof}

To investigate the boundary-value problem \eqref{e1}, \eqref{e3},
we consider the following Dirichlet problem
\begin{equation} \label{e23}
\begin{gathered}
\Delta {u}+\alpha u=0, \quad  (x,t)\in \Omega\times\mathbb{R}_+ \\
u=0, \quad (x,t)\in \partial\Omega\times\mathbb{R}_+,
\end{gathered}
\end{equation}
where $\alpha$ is a constant. It is well-known \cite{v1} that the
least eigenvalue $\alpha_0$ of problem (23) is positive and the
corresponding eigenfunction $\varphi(x)$ is positive for
$x\in{\Omega}$.

We further introduce the following notation
\begin{equation} \label{e24}
A(t)=\min_{1\le i\le n}\{a_{i}(t)\}, \quad
 B(t)=\min_{1\le i\le n}\{b_{i}(t)\}
\end{equation}

\begin{theorem} \label{thm2}
 Suppose that
 $0\le C\le 1$, (A1) and \eqref{e9} hold.
If for some constants $0\le\varepsilon_{0}\le 1$ and
$\xi_{0}\in{[a,b]}$,
\begin{equation} \label{e25}
\begin{aligned}
&\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\int_{a}^{b}
\varepsilon_{0}Q(s,\xi)g(s,\xi)d\mu(\xi)ds \\
&>\frac{1}{e}\exp\Big[
-\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}
\varepsilon_{0}s[\alpha_{0}A(s)+P(s)]ds\Big],
\end{aligned}
\end{equation}
then each unbounded solution $u(x,t)$ of the boundary-value
problem \eqref{e1} and \eqref{e3} is oscillatory in the domain
$G$.
\end{theorem}

\begin{proof} Suppose to the
contrary that there is a non-oscillatory solution
$u(x,t)=(u_{1}(x,t),\dots ,u_{n}(x,t))^{T}$ of the problem \eqref{e1} and
\eqref{e3}, and $|u_{i}(x,t)|>0$ for
 $t\ge 0$, $i=1,2,\dots ,n$. Proceeding as the proof of Theorem \ref{thm1}, there exists
a $t_1\ge 0$ such that $Y_{i}(x,t)>0$, $Y_{j}(x,t-\tau)>0$,
$Y_{i}(x,t-\rho)>0$ and $Y_{k}[x,g(t,\xi)]>0$ for $t\ge t_1$,
$\xi\in{[a,b]}$ and $j=1,2,\dots ,m$, $k=1,2,\dots ,m$.

Multiplying both sides of equation \eqref{e1} by the eigenfunction
$\varphi(x)$ and then integrating the equation with respect to $x$
over the domain $\Omega$, for $t\ge t_{1}$, we obtain
\begin{equation} \label{e26}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[\int_{\Omega}u_{i}(x,t)\varphi(x)dx
-\sum_{j=1}^{m}\int_{\Omega}c_{ij}u_{j}(x,t-\tau)\varphi(x)dx\Big] \\
&+\int_{\Omega}p_{i}(x,t)u_{i}(x,t)\varphi(x)dx
+\sum_{k=1}^{m}\int_{\Omega}
\int_{a}^{b}q_{ik}(x,t,\xi)u_{k}[x,g(t,\xi)]\varphi(x)d\mu(\xi)dx \\
&=a_{i}(t)\int_{\Omega}\Delta {u_{i}(x,t)}\varphi(x)dx
+b_{i}(t)\int_{\Omega}\Delta {u_{i}(x,t-\rho)}\varphi(x)dx.
\end{aligned}
\end{equation}
Furthermore, we obtain
\begin{equation} \label{e27}
\begin{aligned}
&\frac{d^2}{dt^2}\Big[\int_{\Omega}Y_{i}(x,t)\varphi(x)dx
-\sum_{j=1}^{m}\int_{\Omega}
c_{ij}\frac{\delta_{i}}{\delta_{j}}Y_{j}(x,t-\tau)\varphi(x)dx\Big]\\
& +\int_{\Omega}p_{i}(x,t)Y_{i}(x,t)\varphi(x)dx
+\sum_{k=1}^{m}\int_{\Omega}\frac{\delta_{i}}{\delta_{k}}
\int_{a}^{b}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]\varphi(x)d\mu(\xi)dx \\
& =a_{i}(t)\int_{\Omega}\Delta {Y_{i}(x,t)}\varphi(x)dx
+b_{i}(t)\int_{\Omega}\Delta {Y_{i}(x,t-\rho)}\varphi(x)dx.
\end{aligned}
\end{equation}
 From the Green's formula and the boundary condition \eqref{e3}, we have
\begin{gather}
\int_{\Omega}\Delta {Y_{i}(x,t)}\varphi(x)dx
=\int_{\Omega}Y_{i}(x,t)\Delta {\varphi(x)}dx
=-\alpha_{0}\int_{\Omega}Y_{i}(x,t)\varphi(x)dx, \label{e28}\\
\int_{\Omega}\Delta {Y_{i}(x,t-\tau)}\varphi(x)dx
=-\alpha_{0}\int_{\Omega}Y_{i}(x,t-\tau)\varphi(x)dx. \label{e29}
\end{gather}
Combining \eqref{e27}--\eqref{e29}, we obtain
\begin{align*}
& \frac{d^2}{dt^2}\Big[\int_{\Omega}Y_{i}(x,t)\varphi(x)dx
-\sum_{j=1}^{m}\int_{\Omega}c_{ij}
\frac{\delta_{i}}{\delta_{j}}Y_{j}(x,t-\tau)\varphi(x)dx\Big] \\
& +p_{i}(t)\int_{\Omega}Y_{i}(x,t)\varphi(x)dx
+\int_{a}^{b}Q_{ii}(t,\xi)\int_{\Omega}Y_{i}[x,g(t,\xi)]\varphi(x)dxd\mu(\xi)
\\
&-\sum_{k=1,k\neq{i}}^{m}\int_{a}^{b}Q^{*}_{ik}(t,\xi)
\int_{\Omega}Y_{k}[x,g(t,\xi)]\varphi(x)dxd\mu(\xi) \\
&=-\alpha_{0}a_{i}(t)\int_{\Omega}Y_{i}(x,t)\varphi(x)dx
 -\alpha_{0}b_{i}(t)\int_{\Omega}Y_{i}(x,t-\tau)\varphi(x)dx\,.
\end{align*}
Let
\begin{equation}
U_{i}(t)=\int_{\Omega}Y_{i}(x,t)\varphi(x)dx,\qquad i=1,2\dots ,n,
\end{equation}
then the remainder of the proof is the same as that for Theorem \ref{thm1}
and thus is omitted here. This completes the proof.
\end{proof}

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\end{document}