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Zbl 0801.34076
Malygina, V.V.
Some criteria for stability of equations with retarded argument.
(English. Russian original)
[J] Differ. Equations 28, No.10, 1398-1405 (1992); translation from Differ. Uravn. 28, No.10, 1716-1723 (1992). ISSN 0012-2661

The author considers the delay differential equation $(*)$ $\dot x(t)+ \int\sp t\sb \tau [d\sb s R(t,s)] x(s)=0$, $t\ge \tau$ $(\tau\in \bbfR\sb +)$, where $R: \{(t,s)\in \bbfR\sb +\times \bbfR\sb +$: $t\ge s\}\to L(B)$ is such that $R(\cdot,s)$ is Bochner-integrable on any finite interval of $\bbfR\sb +$ and $R(t,\cdot)$ is a function of bounded variation on any compact interval $[0,T]$. Here $L(B)$ denotes the set of the bounded linear operators on a Banach space $B$. In the paper, there are some nice results concerning the stability, exponential stability and asymptotic stability of the zero solution of equation $(*)$. These results are based on some simple but interesting exponential estimates of the fundamental solution of $(*)$. Some special examples are also discussed.
MSC 2000:
*34K20 Stability theory of functional-differential equations
34K30 Functional-differential equations in abstract spaces

Keywords: delay differential equation; exponential stability; asymptotic stability; exponential estimates

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