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Convergence tests for one scalar differential equation with vanishing delay

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*Jaromir Bastinec, Josef Diblik and Zdenek Smarda*

**Address.** Dept. of Math., Faculty of Electrical Engineering, Brno
University of Technology, Technicka 8, 616 00 Brno,

Czech Republic
**E-mail:** bastinec@dmat.fee.vutbr.cz

diblik@dmat.fee.vutbr.cz

smarda@dmat.fee.vutbr.cz

**Abstract.** In the present paper the differential equation $$ \dot
y(t)=\alpha(t)[y(t)-y(t-\tau(t))] $$ with positive coefficient $\alpha$
and with positive bounded delay $\tau$ (which can have the property $\tau(+\infty)=0$)
is considered. Explicit tests for convergence of all its solutions (for
$t\to +\infty$) are proved.

**AMSclassification.** 34K15, 34K25

**Keywords.** Asymptotic convergence, point test, integral test.