The Schröder equation and asymptotic properties of linear delay differential equations

J. Cermák, Technical University of Brno, Brno, Czech Republic

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 6. (2004), pp. 1-8.

Abstract: We study the asymptotic behaviour of solutions of the differential equation
$$
\dot{x}(t)=-p(t)[x(t)-kx(t-\tau (t))]+q(t),\qquad t\in I=[t_0,\infty),
$$
where $k\ne 0$ is a scalar, $p$ is a positive function and $\tau$ is a positive and unbounded delay. Our aim is to show that under some asymptotic bounds on $q$ the behaviour (as $t\to\infty$) of all solutions of this equation can be estimated via a solution of the Schr\" oder equation
$$
\varphi (t-\tau (t))=\lambda\varphi (t),\qquad t\in I
$$
with a suitable positive parameter $\lambda$.

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