关键词: 脉冲;边值问题, Lyapunov不等式, Leray-Schauder度, 存在唯一性

Abstract:

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} 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.

Key words: Impulses,  Boundary value problem, Lyapunov inequality, Leray-Schauder degree,  Existence and uniqueness