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Razumikhin type stability theorems for impulsive functional differential equations. (English) Zbl 0933.34083

The uniform asymptotic stability is considered for impulsive functional-differential equations of the form \[ \dot x(t)=f(t,x_t),\;t\geq t_0,\quad x(t_k)= J_k\bigl(x(t_k^-) \bigr),\;k\in\mathbb{N}, \tag{1} \] where \(\mathbb{N}\) is the set of all positive integers, \(f:[t_0,\infty)\times PC\to\mathbb{R}^n\) and \(J_k(x):S(p) \to\mathbb{R}^n\), for each \(k\in\mathbb{N}\), \(PC=PC([-\tau,0], \mathbb{R}^n)= \{\varphi: [-\tau,0] \to\mathbb{R}^n\), \(\varphi(t)\) is continuous everywhere except a finite number of points \(\widetilde t\) at which \(\varphi(\widetilde t^+)\) and \(\varphi (\widetilde t^-)\) exist and \(\varphi (\widetilde t^+)=\varphi(\widetilde t^-)\}\), \(S(p)=\{x \in \mathbb{R}^n: |x|<p\}\), \(t_0\leq t_1< t_2<\cdots <t_k<t_{k+1} <\dots\) with \(t_k\to \infty\) as \(k\to\infty\). The uniform asymptotic stability of Lyapunov-Razumikhin type theorems is established.

MSC:

34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
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