\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 \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} 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 $$\label{e4} \frac{d^2}{dt^2}[y(t)-\lambda(t)y(t-\tau)] +p(t)y(t)+\int_{a}^{b}q(t,\xi)y[g(t,\xi)]d\mu(\xi)\le 0, \quad t\ge{0},$$ 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 $$\label{e5} x'(t)+\int_{a}^{b}q(t,\xi)x[g(t,\xi)]d\mu(\xi)\le 0$$ 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: $$\label{e8} \begin{gathered} p_{i}(t)=\min_{x\in{\overline{\Omega}}}\{p_{i}(x,t)\}, \quad P(t)=\min_{1\le i\le n}p_{i}(t); \\ Q_{ii}(t,\xi)=\min_{x\in{\overline{\Omega}}}\{q_{ii}(x,t,\xi)\}, \quad Q^{*}_{ik}(t,\xi)=\max_{x\in{\overline{\Omega}}}\{q_{ik}(x,t,\xi)\}. \end{gathered}$$ \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 $\xi_{0}\in{[a,b]}$, $$\label{e9} \liminf_{t\to\infty}\int_{h(t,b)}^{t}\int_{a}^{b} \varepsilon_{0}Q(s,\xi)g(s,\xi)d\mu(\xi)ds>0,$$ $$\quad\quad \liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\int_{a}^{b} \varepsilon_{0}Q(s,\xi)g(s,\xi)d\mu(\xi)ds >\frac{1}{e}\exp\Big[ -\liminf_{t\to\infty}\int_{g(t,\xi_{0})}^{t}\varepsilon_{0}sP(s)ds \Big], \label{e10}$$ 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 $$\label{e11} \begin{gathered} \delta_{i}=\mathop{\rm sgn}{u_{i}(x,t)}, \quad \delta_{k}=\mathop{\rm sgn}{u_{k}[x,g(t,\xi)]}; \\ Y_{i}(x,t)=\delta_{i}u_{i}(x,t),\quad Y_{k}[x,g(t,\xi)]=\delta_{k}u_{k}[x,g(t,\xi)], \end{gathered}$$ 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 \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} Furthermore, it follows from \eqref{e11} that \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} It is clear that $$\label{e14} \int_{\Omega}\int_{a}^{b}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]d\mu(\xi)dx =\int_{a}^{b}\int_{\Omega}q_{ik}(x,t,\xi)Y_{k}[x,g(t,\xi)]dxd\mu(\xi).$$ 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 $$\label{e17} \int_{\Omega}p_{i}(x,t)Y_{i}(x,t)dx \ge p_{i}(t)\int_{\Omega}Y_{i}(x,t)dx.$$ Combining \eqref{e14}--\eqref{e17} and noting \eqref{e8}, we obtain \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} Let $$\label{e19} V_{i}(t)=\int_{\Omega}Y_{i}(x,t)dx,\qquad i=1,2\dots ,n,$$ then $V_{i}(t)>0$, and it follows from \eqref{e18} that \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} Furthermore, let $V(t)=\sum_{i=1}^{m}V_{i}(t)$, then $V(t)>0$, and it follows from \eqref{e8} that \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} 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 $$\label{e22} \quad \frac{d^2}{dt^2}[V(t)-CV(t-\tau)]+P(t)V(t) +\int_{a}^{b}Q(t,\xi)V[g(t,\xi)]d\mu(\xi)\le 0.$$ 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 $$\label{e23} \begin{gathered} \Delta {u}+\alpha u=0, \quad (x,t)\in \Omega\times\mathbb{R}_+ \\ u=0, \quad (x,t)\in \partial\Omega\times\mathbb{R}_+, \end{gathered}$$ 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 $$\label{e24} A(t)=\min_{1\le i\le n}\{a_{i}(t)\}, \quad B(t)=\min_{1\le i\le n}\{b_{i}(t)\}$$ \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 $\xi_{0}\in{[a,b]}$, \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} 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 \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} Furthermore, we obtain \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} 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 $$U_{i}(t)=\int_{\Omega}Y_{i}(x,t)\varphi(x)dx,\qquad i=1,2\dots ,n,$$ then the remainder of the proof is the same as that for Theorem \ref{thm1} and thus is omitted here. 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