具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

Abstract

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

CLC Number:

34B37

WENG Ai-Zhi, SUN Ji-Tao. Generalization of Lyapunov Inequality for |Dirichlet BVPs with Impulses and its Applications[J].Acta mathematica scientia,Series A, 2011, 31(1): 82-91.

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

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