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Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems. (English) Zbl 1179.34079
The authors consider the impulsive FDE of the form $$ \split x'(t)&=f(t,x(\cdot)),\ \ t\ne t_k,\\ \left.\Delta\right|_{t=t_k}&=x(t_k)-x(t_k^-)=I_k(t_k,x(t_k^-)), \ \ k=1,2,\ldots, \endsplit $$ $0\le t_0<t_1<\cdots$, $x'$ denotes the right-hand derivative, $f$ is a continuous functional defined in the appropriate space, $f(t,0)=I_k(t_k,0)=0$. It is supposed that the IVP has a unique solution $x(t,\sigma,\phi)$ which can be continued to $\infty$. The initial function $\phi$ is piecewise continuous. The zero solution is said to be weak exponentially stable if for any $\varepsilon>0$ and $\sigma\ge t_0$ $\exists\delta>0$ such that $\|\phi\|<\delta$ implies $\alpha(\|x(t,\sigma,\phi)\|)<\varepsilon e^{-\lambda(t-\sigma)}$ for $t\ge\sigma$ and some $\lambda>0$ and a strictly increasing $\alpha: \bbfR_+\to \bbfR_+$. If $\alpha(s)=s$ we obtain the exponential stability. Two theorems on the weak exponential stability are proved. The paper ends with two illustrative examples. There are many misprints.

34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
Full Text: DOI
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