Address. Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Technical University of Brno, Technicka 8, 616 00 Brno, Czech Republic
E-mail: diblik@dmat.fee.vutbr.cz
Abstract. This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form $$ \dot x(t)= -c(t)x(t-\tau (t)) \eqno{(^*)} $$ with positive function $c(t).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation $$ \dot y(t)=\beta (t)[y(t)-y(t-\tau (t))] $$ where the function $\beta (t)$ is positive. A result concerning the behaviour of solutions of Eq.~(*) in critical case is given and, moreover, an analogy with behaviour of solutions of the second order ordinary differential equation $$ x''(t)+a(t)x(t)=0 $$ for positive function $a(t)$ in critical case is considered.
AMS classification. 34K15, 34K25
Key words. Positive solution, oscillating solution, convergent solution, linear differential equation with delay, topological principle of Wazewski (Rybakowski's approach)