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Zbl 0933.34083
Shen, Jianhua; Yan, Jurang
Razumikhin type stability theorems for impulsive functional differential equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 33, No.5, 519-531 (1998). ISSN 0362-546X

The uniform asymptotic stability is considered for impulsive functional-differential equations of the form $$\dot x(t)=f(t,x_t),\ t\ge t_0,\quad x(t_k)= J_k\bigl(x(t_k^-) \bigr),\ k\in\bbfN, \tag 1$$ where $\bbfN$ is the set of all positive integers, $f:[t_0,\infty)\times PC\to\bbfR^n$ and $J_k(x):S(p) \to\bbfR^n$, for each $k\in\bbfN$, $PC=PC([-\tau,0], \bbfR^n)= \{\varphi: [-\tau,0] \to\bbfR^n$, $\varphi(t)$ is continuous everywhere except a finite number of points $\widetilde t$ at which $\varphi(\widetilde t^+)$ and $\varphi (\widetilde t^-)$ exist and $\varphi (\widetilde t^+)=\varphi(\widetilde t^-)\}$, $S(p)=\{x \in \bbfR^n: |x|<p\}$, $t_0\le t_1< t_2<\cdots <t_k<t_{k+1} <\dots$ with $t_k\to \infty$ as $k\to\infty$. The uniform asymptotic stability of Lyapunov-Razumikhin type theorems is established.
[I.Foltyńska (Poznań)]
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K45 Equations with impulses

Keywords: uniform asymptotic stability; impulsive functional-differential equations

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