## 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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### References:

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