The Schröder equation and asymptotic properties of linear delay differential equations

• Main Author: Jan Čermák
• Format: Article
• Language: English
• Published: University of Szeged 2004-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=193

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 $\lambda$.