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On a linear differential equation with a proportional delay. (English) Zbl 1128.34051

The paper deals with the non-autonomous linear delay differential equation

$\stackrel{˙}{x}\left(t\right)=c\left(t\right)\left(x\left(t\right)-px\left(\lambda t\right)\right),\phantom{\rule{1.em}{0ex}}0<\lambda <1,\phantom{\rule{1.em}{0ex}}p\ne 0,\phantom{\rule{1.em}{0ex}}t>0,$

where $p$ and $\lambda$ are real scalars and $c$ is a continuous and non-oscillatory function defined on $\left(0,\infty \right)$. The equation is referred to as pantograph equation, since in a simplified version it models the collection of current by the pantograph of an electric locomotive. The asymptotic properties of the solutions are in focus. The following condition on the growth of $c$ is imposed:

$\underset{t\to \infty }{lim sup}\frac{\lambda \phantom{\rule{0.166667em}{0ex}}c\left(\lambda t\right)}{c\left(t\right)}<1\phantom{\rule{0.166667em}{0ex}}·$

The main result of the paper says that if $c\in {C}^{1}\left(\left(0,\infty \right)\right)$ fulfills this condition and is eventually positive, then there exist real constants $L$ and $\rho$, where $\rho >0$, and a continuous periodic function $g$ of period $log{\lambda }^{-1}$ such that

$x\left(t\right)=L{x}^{*}\left(t\right)+{t}^{k}g\left(logt\right)+O\left({t}^{{\kappa }_{r}-\rho }\right)\phantom{\rule{0.166667em}{0ex}}·$

Here ${\kappa }_{r}$ is the real part of the possible complex $\kappa$ such that ${\lambda }^{k}=1/p$, and ${x}^{*}$ is the solution of the considered equation, such that

${x}^{*}\left(t\right)\sim exp\left({\int }_{\overline{t}}^{t}c\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds\right)\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{1.em}{0ex}}t\to \infty$

(the existence of such a sulution ${x}^{*}$ is proved in the paper). Though it is natural to distinguish the cases of the eventually positive and the eventually negative $c$, it is shown that a resembling asymptotic formula is valid also in the case of $c$ eventually negative. Finally, using a transformation approach these results are generalized to equations with a general form of the delay.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K06 Linear functional-differential equations