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On exponential stability of linear differential equations with several delays. (English) Zbl 1112.34055

Studied in this paper is the exponential stability of the following nonautonomous delayed linear equation

$\stackrel{˙}{x}\left(t\right)+\sum _{k=1}^{m}{a}_{k}\left(t\right)x\left({h}_{k}\left(t\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

with ${\sum }_{k=1}^{m}{a}_{k}\left(t\right)\ge 0$, ${h}_{k}\left(t\right)\le t$. Applying the comparison method based on a Bohl-Perron-type theorem, the authors obtain some new stability conditions on exponential stability of ($*$). These conditions are in “iterative” and “limit” forms. The results are compared with some existing ones by several examples. The study is a continuation of the paper of the authors [J. Math. Anal. Appl. 314, No. 2, 391–411 (2006; Zbl 1101.34057)], where ordinary differential equations are applied as comparison equations, while in this paper under review delay differential equations with positive coefficients and a positive fundamental function are used for comparison.

MSC:
 34K20 Stability theory of functional-differential equations 34K06 Linear functional-differential equations