In this paper, first the authors obtain the nonexistence of nontrivial solutions for the linear Dirichlet boundary value problem with impulses
$$\left\{
\begin{array}{ll}
x''(t)+a(t)x(t)=0, t\neq \tau_{k}, \\
\Delta x(\tau_{k})=c_{k}x(\tau_{k}),\\
\Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \\
x(0)=x(T)=0,
\end{array}
\right. (k=1,2\cdots,m)
$$ where $a:[0,T]\rightarrow R$, $c_{k}$ and $d_{k}$ are
constants, $k=1,2,\cdots,m$, $\Delta
x(\tau_{k})=x(\tau_{k}^{+})-x(\tau_{k}^{-})$, $\Delta
x'(\tau_{k})=x'(\tau_{k}^{+})-x'(\tau_{k}^{-})$,
$0<\tau_{1}<\tau_{2}<\cdots<\tau_{m}<T$.
Secondly, by applying Leray-Schauder degree, the authors obtain the existence and uniqueness of solutions for the nonlinear Dirichlet boundary value problem with impulses
$$\left\{
\begin{array}{ll}
x''(t)+f(t,x(t))=0, t\neq \tau_{k}, \\
\Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \\
\Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \\
x(0)=x(T)=0,
\end{array}
\right. (k=1,2\cdots,m)
$$ where $f\in C([0,T]\times R, R)$,
$I_{k},M_{k}\in C(R,R)$,
$k=1,2,\cdots,m$.
As a corollary of the results, the Lyapunov inequality is extended to impulsive systems.
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