具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

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

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Abstract:

In this paper, first the authors obtain the nonexistence of nontrivial solutions for the linear Dirichlet boundary value problem with impulses

{\begin{cases} x^{″} (t) + a (t) x (t) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = c_{k} x (τ_{k}), \\ Δ x^{'} (τ_{k}) = d_{k} x (τ_{k}), \\ x (0) = x (T) = 0, \end{cases} (k = 1, 2 \dots, m)

$\left\{ \begin{array}{ll} x''(t)+a(t)x(t)=0, t\neq \tau_{k}, \\ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \\ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m)$ where

a : [0, T] \to R

$a:[0,T]\rightarrow R$ ,

c_{k}

$c_{k}$ and

d_{k}

$d_{k}$ are
constants,

k = 1, 2, \dots, m

$k=1,2,\cdots,m$ ,

Δ x (τ_{k}) = x (τ_{k}^{+}) - x (τ_{k}^{-})

$\Delta x(\tau_{k})=x(\tau_{k}^{+})-x(\tau_{k}^{-})$ ,

Δ x^{'} (τ_{k}) = x^{'} (τ_{k}^{+}) - x^{'} (τ_{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

{\begin{cases} x^{″} (t) + f (t, x (t)) = 0, t \neq τ_{k}, \\ Δ x (τ_{k}) = I_{k} (x (τ_{k})), \\ Δ x^{'} (τ_{k}) = M_{k} (x (τ_{k})), \\ x (0) = x (T) = 0, \end{cases} (k = 1, 2 \dots, m)

$\left\{ \begin{array}{ll} x''(t)+f(t,x(t))=0, t\neq \tau_{k}, \\ \Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \\ \Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \\ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m)$ where

f \in C ([0, T] \times R, R)

$f\in C([0,T]\times R, R)$ ,

I_{k}, M_{k} \in C (R, R)

$I_{k},M_{k}\in C(R,R)$ ,

k = 1, 2, \dots, m

$k=1,2,\cdots,m$ .
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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