**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.

Communicated by L. Hatvani.
| Received on 2003-09-30 Appeared on 2004-08-31 |

**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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