Behaviour of Solutions of Linear Differential Equations with Delay

Josef Diblik

Address. Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Technical University of Brno, Technicka 8, 616 00 Brno, Czech Republic


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)