On a linear differential equation with a proportional delay

This paper deals with the delay differential equation$$ {\dot x} (t) = c (t) [x (t) - px (\lambda t)] \,, \quad 0 < \lambda < 1 \,, \quad p \ne 0 \,, \quad t > 0 \,. $$We impose some growth conditions on c , under which we are able to give a precise description of the asymptotic properties of all solutions of this equation. Although we naturally have to distinguish the cases c eventually positive and c eventually negative, we show a certain resemblance between the asymptotic formulae corresponding to both cases. Moreover, using the transformation approach we generalize these results to the equation with a general form of a delay. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)