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Explicit exponential stability conditions for linear differential equations with several delays. (English) Zbl 1118.34069

Summary: New explicit conditions of exponential stability are obtained for the nonautonomous linear equation

$\stackrel{˙}{x}\left(t\right)+\sum _{k=1}^{m}{a}_{k}\left(t\right)x\left({h}_{k}\left(t\right)\right)=0,$

where ${\sum }_{k=1}^{m}{a}_{k}\left(t\right)\ge 0$, ${h}_{k}\left(t\right)\le t$, by comparing this equation with a nonoscillatory exponentially stable equation of the form

$\stackrel{˙}{x}\left(t\right)+\sum _{k\in I}{a}_{k}\left(t\right)x\left({g}_{k}\left(t\right)\right)=0,$

where $I\subset \left\{1,\cdots ,m\right\}$, ${g}_{k}\left(t\right)\le t$. Every comparison result gives ${2}^{m}-1$ different stability conditions due to the a priori choice of a subset $I$.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K06 Linear functional-differential equations
##### Keywords:
delay equations; positive fundamental function