The theories of nonlinear partial functional differential equations are applied in many fields. In recent years the research of oscillation to impulsive partial differential systems caught more and more attention. In this paper, we study the oscillation properties of the solutions to impulsive delay hyperbolic equation

$ \begin{eqnarray} \frac{\partial}{\partial t}((r(t)\frac{\partial}{\partial t}u(x, t))&=&a(t)h(u(x, t))\triangle u(x, t) -\sum\limits_{i=1}^{n}b_{i}(t)h_{i}(u(x, \tau_{i}(t)))\triangle u(x, \tau_{i}(t))\nonumber\\ &&+\sum\limits_{j=1}^{m}q_{j}(x, t)\varphi_{j}(u(x, t)), \ t\neq t_{k}, (x, t)\in\Omega\equiv G\times (0, +\infty), \end{eqnarray} $

(1.1)

$ u(x, t_{k}^{+})-u(x, t_{k}^{-})=\alpha_{k}u(x, t_{k}), \ \ \ t=t_{k}, k=1, 2, \cdots, $

(1.2)

$ u_{t}(x, t_{k}^{+})-u_{t}(x, t_{k}^{-})=\beta_{k}u_{t}(x, t_{k}), \ \ \ t=t_{k}, k=1, 2, \cdots. $

(1.3)

The following is the boundary condition

$ \frac{\partial u(x, t)}{\partial n}+\gamma(x, t)u(x, t)=0, ~~ (x, t)\in \partial G\times[0, +\infty). $

(1.4)

where $G$ is a bounded domain of $R^{n}$ with the smooth boundary $\partial G$ and $n$ is the unit exterior normal vector to $\partial G$.

Following are the basic hypothesis

(H1) $r(t)\in C([0, +\infty);(0, +\infty))$, $a(t), b_{i}(t)\in PC([0, +\infty);[0, +\infty)), \ i=1, 2, \cdots, n$. $\gamma(x, t)\in C(R_{+}\times \partial G, R_{+})$. $q_{j}(x, t)\in C(\bar{\Omega};[0, +\infty)), \ j=1, 2, \cdots, m$, where PC denotes the class of functions which are piecewise continuous in $t$ with discontinuities of the first kind only at $t=t_{k}, k=1, 2, \cdots$.

(H2) $\tau_{i}(t)\in C([0, +\infty);R), \ \lim\limits_{t\rightarrow+\infty}\tau_{i}(t)=+\infty, \ i=1, 2, \cdots, n$.

(H3) $h(u), h_{i}(u)\in C(R, R)$, $uh(u)\geq0$, $uh'(u)\geq0$, $uh'_{i}(u)\geq0$, $i=1, 2, \cdots, n$; $\varphi_{j}(s)\in C(R, R)$, $\frac{\varphi_j(s)}{s}\geq C_{j}={\rm const.}>0$ for $s\neq0$. $\alpha_{k}, \beta_{k}={\rm const.}>-1$, $0<t_{1}<t_{2}<\cdots<t_{k}<\cdots$, $\lim\limits_{t\rightarrow+\infty}t_{k}=+\infty$, $k=1, 2, \cdots$.

We introduce the notations $U(t)=\displaystyle\int_{G}u(x, t)dx$ and $q_{j}(t)= \min\limits_{x\in \bar{G}} q_{j}(x, t)$.

Definition 1.1   The solution $u(x, t)$ of the problems (1.1)-(1.4) is said to be nonoscillatory in domain $\Omega$ if it is either eventually positive or eventually negative. Otherwise, it is called oscillatory.

Definition 1.2   We say that functions $H_{i}, \ i=1, 2, $ belong to a function class $H$, if $H_{i}\in C(D;[0, +\infty)), \ i=1, 2$, satisfy

1. $H_{i}(t, s)=0, \ i=1, 2\ {\rm for}\ t=s, $

2. $ H_{i}(t, s)>0, \ i=1, 2\ {\rm for}\ t>s, $

where $D=\{(t, s):0<s\leq t<+\infty\}$. Moreover, the partial derivatives $\partial H_{1}/\partial s$ and $\partial H_{2}/\partial s$ exist on $D$ such that

$ \frac{\partial H_{1}}{\partial s}(t, s)=h_{1}(t, s)H_{1}(t, s)\ {\rm and}\ \frac{\partial H_{2}}{\partial s}(t, s)=-h_{2}(t, s)H_{2}(t, s), $

where $h_{1}, h_{2}\in C_{loc}(D;\mathbb{R}).$

In recent years, there was much research activity concerning the oscillation theory of nonlinear hyperbolic equations with functional arguments by employing Riccati technique. Riccati techniques were used to obtain various oscillation results. Recently, Shoukaku and Yoshida [2] derived oscillation criteria by using oscillation criteria of Riccati inequality. In this work, we study the hyperbolic equation with impulsive.

Theorem 2.1   If for each $T\geq0$, there exist $(H_{1}, H_{2})\in H$ and $a, b, c\in \mathbb{R}$ such that $T\leq a<c<b$ and

$ \begin{eqnarray}\begin{aligned} &\frac{1}{H_{1}(c, a)}\int_{a}^{c}H_{1}(s, a)\prod\limits_{t_{1}\leq t_{k}< s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s) -\frac{1}{4}r(s)\lambda_{1}^{2}(s, a))\psi(s)ds\\ +&\frac{1}{H_{2}(b, c)}\int_{c}^{b}H_{2}(b, s)\prod\limits_{t_{1}\leq t_{k}< s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1} (C_{l}q_{l}(s) -\frac{1}{4}r(s)\lambda_{2}^{2}(b, s))\psi(s)ds>0, \end{aligned} \end{eqnarray} $

(2.1)

then every solution of the problems (1.1)-(1.4) oscillates in $\Omega$, where

$ \psi(t)\in C^{1}((T_{0}, +\infty);(0, +\infty)) $

for some $t_{1}>0$ and

$ \lambda_{1}(s, t)=\frac{\psi'(s)}{\psi(s)}+h_{1}(s, t), ~~\lambda_{2}(t, s)=\frac{\psi'(s)}{\psi(s)}-h_{2}(t, s). $

Proof  Suppose to the contrary that there is a nonoscillatory solution $u(x, t)$ of the problems (1.1)-(1.4). Without loss of generality we may assume that $u(x, t)>0$ in $G\times[t_{0}, +\infty)$ for some $t_{0}>0$ because the case where $u(x, t)<0$ can be treated similarly. Since (H2) holds, we see that $u(x, \tau_{i}(t))>0$ $(i=1, 2, \cdots n)$ in $G\times[t_{1}, +\infty)$ for some $t_{1}\geq t_{0}$.

(1) For $t\geq t_{1}, \ t\neq t_{k}, \ k=1, 2, \cdots$, integrating (1) with respect to $x$ over $G$, we obtain

$ \begin{eqnarray*} \frac{d}{dt}(r(t)\int_{G}\frac{\partial}{\partial t}u(x, t)dx)&= & a(t)\int_{G}h(u(x, t))\triangle u(x, t)dx-\sum\limits_{j=1}^{m}\int_{G}q_{j}(x, t)\varphi_{j}(u(x, t))\\ &&+\sum\limits_{i=1}^{n}b_{i}(t)\int_{G}h_{i}(u(x, \tau_{i}(t))) \triangle u(x, \tau_{i}(t))dx. \end{eqnarray*} $

By Green's formula and the boundary condition, we have

$ \begin{eqnarray*} &&\int_{G}h(u(x, t))\Delta u(x, t)dx =\int_{\partial G}h(u(x, t))\frac{\partial u(x, t)}{\partial n}ds-\int_{G}h'(u)|grad u|^{2}dx\\ &=&-\int_{G}\gamma(x, t)u(x, t)h(u(x, t))ds-\int_{G}h'(u)|grad u|^{2}dx\leq0, \\ &&\int_{G}h_{i}(u(x, \tau_{i}(t)))\Delta u(x, \tau_{i}(t))dx\leq0. \end{eqnarray*} $

For condition (H3) we can easily obtain

$ \int_{G}q_{j}(x, t)\varphi_{j}(u(x, t))dx\geq C_{j}q_{j}(t)\int_{G}u(x, t)dx, $

then $U(t)>0$, and it follows that

$ (r(t)U'(t))'+\sum\limits_{i=1}^{m}C_{i}(t)q_{i}(t)U(t)\leq0. $

For some $l\in\{1, 2, \cdots, m\}$, we can get

$ (r(t)U'(t))'+C_{l}q_{l}(t)U(t)\leq0, \ \ t\geq t_{1}, t\neq t_{k}. $

(2) For $t=t_{k}$, $k=1, 2, \cdots$. From (1.2)-(1.3), we have that

$ \begin{eqnarray*} &&\int_{G}u(x, t_{k}^{+})dx-\int_{G}u(x, t_{k}^{-})dx=\alpha_{k}\int_{G}u(x, t_{k}), \\ &&\int_{G}u_{t}(x, t_{k}^{+})dx-\int_{G}u_{t}(x, t_{k}^{-})dx=\beta_{k}\int_{G}u_{t}(x, t_{k}), \end{eqnarray*} $

that is

$ U(t_{k}^{+}) =(1+\alpha_{k})U(t_{k}), \ U'(t_{k}^{+}) =(1+\beta_{k})U'(t_{k}). $

Thus we obtain that the functions $U(t)$ is a eventually positive solution of the impulsive differential inequality

$ \begin{eqnarray}\begin{aligned} &(r(t)y'(t))'+ C_{l}q_{l}(t)y(t)\leq 0, \\ &y(t_{k}^{+}) =(1+\alpha_{k})y(t_{k}), \\ &y'(t_{k}^{+}) =(1+\beta_{k})y'(t_{k}).\end{aligned} \end{eqnarray} $

(2.2)

Set $w(t)=\frac{r(t)U'(t)}{U(t)}$ for $t\geq t_{1}$. From (2.2), we obtain that

$ w'(t)+\frac{1}{r(t)}w^{2}(t)\leq-C_{l}q_{l}(t), ~~w(t_{k}^{+})=\frac{1+\beta_{k}}{1+\alpha_{k}}w(t_{k}). $

Define $v(t)=(\prod\limits_{t_{1}\leq t_{k}<t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})w(t)$. In fact, $w(t)$ is continuous on each interval $(t_{k}, t_{k+1}]$, and in view of $w(t_{k}^{+})=(\frac{1+\beta_{k}}{1+\alpha_{k}})w(t_{k})$, it follows that for $t\geq t_{1}$,

$ v(t_{k}^{+})=\prod\limits_{t_{1}\leq t_{j}\leq t_{k}}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}w(t_{k}^{+})=\prod\limits_{t_{1}\leq t_{j}< t_{k}}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}w(t_{k})=v(t_{k}), $

and for all $t\geq t_{1}$,

$ v(t_{k}^{-})=\prod\limits_{t_{1}\leq t_{j}\leq t_{k-1}}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}w(t_{k}^{-})=\prod\limits_{t_{1}\leq t_{j}< t_{k}}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}w(t_{k})=v(t_{k}), $

which implies that $v(t)$ is continuous on $[t_{1}, +\infty)$,

$ \begin{eqnarray*} & &v'(t)+(\prod\limits_{t_{1}\leq t_{k}< t}\frac{1+\beta_{k}}{1+\alpha_{k}})\frac{1}{r(t)}v^{2}(t)+(\prod\limits_{t_{1}\leq t_{k}<t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})C_{l}q_{l}(t)\\ &=&(\prod\limits_{t_{1}\leq t_{k}< t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})w'(t)+\frac{1}{r(t)}(\prod\limits_{t_{1}\leq t_{k}< t}\frac{1+\beta_{k}}{1+\alpha_{k}})(\prod\limits_{t_{1}\leq t_{k}< t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})^{2}w^{2}(t)\\ &&+(\prod\limits_{t_{1}\leq t_{k}<t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})C_{l}q_{l}(t)\\ &=&\prod\limits_{t_{1}\leq t_{k}< t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}[w'(t)+\frac{1}{r(t)}w^{2}(t)+C_{l}q_{l}(t)]\leq0. \end{eqnarray*} $

That is to say

$ v'(t)+(\prod\limits_{t_{1}\leq t_{k}\leq t}\frac{1+\beta_{k}}{1+\alpha_{k}})\frac{1}{r(t)}v^{2}(t)\leq-(\prod\limits_{t_{1}\leq t_{k}<t}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})C_{l}q_{l}(t). $

(2.3)

Multiplying (2.3) by $\psi(s)$, we obtain

$ (\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})\psi(s)C_{l}q_{l}(s)\leq-\psi(s)v'(s)-(\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}})\frac{\psi(s)}{r(s)}v^{2}(s). $

(2.4)

Multiplying (2.4) by $H_{2}(t, s)$ and integrating over $[c, t]$ for $t\in[c, b)$, we have

$ \begin{eqnarray*} & &\int_{c}^{t}(\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})H_{2}(t, s)\psi(s)C_{l}q_{l}(s)ds\\ &\leq &-\int_{c}^{t}H_{2}(t, s)\psi(s)v'(s)ds-\int_{c}^{t}H_{2}(t, s)(\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}})\frac{\psi(s)}{r(s)}v^{2}(s)ds \\ &=&H_{2}(t, c)v(c)\psi(c)-\int_{c}^{t}H_{2}(t, s)(\sqrt{\frac{\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}}}{r(s)}}v(s)-\frac{1}{2}\lambda_{2}(t, s)\sqrt{\frac{r(s)}{\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}}}})^{2}\psi(s)ds\\ &&+ \frac{1}{4}\int_{c}^{t}\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})H_{2}(t, s)r(s)\psi(s)\lambda^{2}_{2}(t, s)ds\\ &\leq&H_{2}(t, c)v(c)\psi(c)+\frac{1}{4}\int_{c}^{t}\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})H_{2}(t, s)r(s)\psi(s)\lambda^{2}_{2}(t, s)ds, \end{eqnarray*} $

and so

$ \frac{1}{H_{2}(t, c)}\int_{c}^{t}H_{2}(t, s)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda^{2}_{2}(t, s))\psi(s)ds\leq v(c)\psi(c). $

Let $t\rightarrow b^{-}$ in the above, we obtain

$ \frac{1}{H_{2}(b, c)}\int_{c}^{b}H_{2}(b, s)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda^{2}_{2}(b, s))\psi(s)ds\leq v(c)\psi(c). $

(2.5)

On the other hand, multiplying (2.4) by $H_{1}(s, t)$ and integrating over $[t, c]$ for $t\in(a, c]$, we obtain

$ \begin{eqnarray*} & &\int_{t}^{c}(\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1})H_{1}(s, t)\psi(s)C_{l}q_{l}(s)ds \\ & \leq & -\int_{t}^{c}H_{1}(s, t)\psi(s)v'(s)ds-\int_{t}^{c}H_{1}(s, t)(\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}})\frac{\psi(s)}{r(s)}v^{2}(s)ds \\ &=&-H_{1}(c, t)v(c)\psi(c)-\int_{t}^{c}H_{1}(s, t)(\sqrt{\frac{\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}}}{r(s)}}v(s)\\&&-\frac{1}{2}\lambda_{1}(s, t)\sqrt{\frac{r(s)}{\prod\limits_{t_{1}\leq t_{k}<s}\frac{1+\beta_{k}}{1+\alpha_{k}}}})^{2}\psi(s)ds\\ &&+ \frac{1}{4}\int_{t}^{c}\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})H_{1}(s, t)r(s)\psi(s)\lambda^{2}_{1}(s, t)ds\\ &\leq&-H_{1}(c, t)v(c)\psi(c)+\frac{1}{4}\int_{t}^{c}\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})H_{1}(s, t)r(s)\psi(s)\lambda^{2}_{1}(s, t)ds, \end{eqnarray*} $

and so

$ \frac{1}{H_{1}(c, t)}\int_{t}^{c}H_{1}(s, t)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda^{2}_{1}(s, t))\psi(s)ds\leq -v(c)\psi(c). $

Let $t\rightarrow a^{+}$ in the above, we get

$ \frac{1}{H_{1}(c, a)}\int_{a}^{c}H_{1}(s, a)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda^{2}_{1}(s, a))\psi(s)ds\leq -v(c)\psi(c). $

(2.6)

Adding (2.5) and (2.6), we easily obtain the following

$ \begin{eqnarray*} & &\frac{1}{H_{1}(c, a)}\int_{a}^{c}H_{1}(s, a)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda_{1}^{2}(s, a))\psi(s)ds\\ &+&\frac{1}{H_{2}(b, c)}\int_{c}^{b}H_{2}(b, s)\prod\limits_{t_{1}\leq t_{k}<s}(\frac{1+\beta_{k}}{1+\alpha_{k}})^{-1}(C_{l}q_{l}(s)-\frac{1}{4}r(s)\lambda_{2}^{2}(b, s))\psi(s)ds\leq0, \end{eqnarray*} $

which contradicts condition (2.1).