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Keywords:
Positive solution; oscillating solution; convergent solution; linear differential equation with delay; topological principle of Ważewski (Rybakowski’s approach)
Positive solution; oscillating solution; convergent solution; linear differential equation with delay; topological principle of Ważewski (Rybakowski’s approach)
Summary:
This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form \[ \dot{x}(t)= -c(t)x(t-\tau (t)) \qquad \mathrm {{(^*)}}\] 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 \[ x^{\prime \prime }(t)+a(t)x(t)=0 \] for positive function $a(t)$ in critical case is considered.
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