Asymptotic estimation of the convergence of solutions of the equation $\dot x(t)=b(t)x(t-\tau(t))$

Josef Diblik, Denys Khusainov

Address. J. Diblik, Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Brno University of Technology, Technicka 8, 616 00 Brno, CZECH REPUBLIC

D. Khusainov, Department of Complex Systems Modelling, Faculty of Cybernetics, Kiev University, Vladimirskaja 64, Kiev 252033, UKRAINE

E-mail: diblik@dmat.fee.vutbr.cz

             denis@dh.cyb.univ.kiev.ua

Abstract. The main result of the present paper is obtaining new inequalities for solutions of scalar equation $\dot x(t)=b(t)x(t-\tau(t))$. Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution $x(t)$ reaches an $\varepsilon$ - neighbourhood of origin and remains in it.

AMSclassification. 34K20, 34K25

Keywords. Stability of trivial solution, estimation of convergence of nontrivial solutions