Convergence tests for one scalar differential equation with vanishing delay

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


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.