## 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 $\begin{split} x'(t)&=f(t,x(\cdot)),\;\;t\neq t_k,\\ \left.\Delta\right|_{t=t_k}&=x(t_k)-x(t_k^-)=I_k(t_k,x(t_k^-)), \;\;k=1,2,\ldots, \end{split}$ $$0\leq t_0<t_1<\cdots$$, $$x'$$ 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 $$\sigma\geq t_0$$ $$\exists\delta>0$$ such that $$\|\phi\|<\delta$$ implies $$\alpha(\|x(t,\sigma,\phi)\|)<\varepsilon e^{-\lambda(t-\sigma)}$$ for $$t\geq\sigma$$ and some $$\lambda>0$$ and a strictly increasing $$\alpha: \mathbb{R}_+\to \mathbb{R}_+$$. 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.

### MSC:

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

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