zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

$\begin{array}{cc}\hfill {x}^{\text{'}}\left(t\right)& =f\left(t,x\left(·\right)\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\ne {t}_{k},\hfill \\ \hfill {{\Delta }|}_{t={t}_{k}}& =x\left({t}_{k}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left({t}_{k},x\left({t}_{k}^{-}\right)\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}k=1,2,...,\hfill \end{array}$

$0\le {t}_{0}<{t}_{1}<\cdots$, ${x}^{\text{'}}$ denotes the right-hand derivative, $f$ is a continuous functional defined in the appropriate space, $f\left(t,0\right)={I}_{k}\left({t}_{k},0\right)=0$. It is supposed that the IVP has a unique solution $x\left(t,\sigma ,\varphi \right)$ which can be continued to $\infty$. The initial function $\varphi$ is piecewise continuous.

The zero solution is said to be weak exponentially stable if for any $\epsilon >0$ and $\sigma \ge {t}_{0}$ $\exists \delta >0$ such that $\parallel \varphi \parallel <\delta$ implies $\alpha \left(\parallel x\left(t,\sigma ,\varphi \right)\parallel \right)<\epsilon {e}^{-\lambda \left(t-\sigma \right)}$ for $t\ge \sigma$ and some $\lambda >0$ and a strictly increasing $\alpha :{ℝ}_{+}\to {ℝ}_{+}$. If $\alpha \left(s\right)=s$ 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