##
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

** 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)