具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

该文首先研究具有脉冲的线性Dirichlet边值问题 $\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) $ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, 该文首先研究具有脉冲的线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+a(t)x(t)=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m) $$ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, $0<\tau_{1}<\tau_{2}\cdots<\tau_{m}<T$为脉冲时刻. 其次利用上面的线性边值问题仅有零解这个性质和Leray-Schauder度理论, 研究具有脉冲的非线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+f(t,x(t))=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \ \Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \ x(0)=x(T)=0 \end{array} \right. (k=1,2\cdots,m) $$ 解的存在性和唯一性, 其中 $f\in C(0,T]\times R,R)$, $I_{k},M_{k}\in C(R, R),k=1,2,\cdots,m$. 该文主要定理的一个推论将经典的Lyaponov不等式比较完美地推广到脉冲系统.

Generalization of Lyapunov Inequality for Dirichlet BVPs with Impulses and its Applications

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.