The Schröder equation and asymptotic properties of linear delay differential equations
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...
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Format: | Article |
Language: | English |
Published: |
University of Szeged
2004-08-01
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Series: | Electronic Journal of Qualitative Theory of Differential Equations |
Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=193 |
Summary: | 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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ISSN: | 1417-3875 1417-3875 |