Asymptotic behaviour of solutions of some differential equations with an unbounded delay
We investigate the asymptotic properties of all solutions of the functional 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 and $\tau (t)$ is an unbounded delay. Under certain restrictions we relate asymptotic behaviour of s...
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| Format: | Article |
| Language: | English |
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University of Szeged
2000-01-01
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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=31 |
| _version_ | 1797830758959677440 |
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| author | Jan Čermák |
| author_facet | Jan Čermák |
| author_sort | Jan Čermák |
| collection | DOAJ |
| description | We investigate the asymptotic properties of all solutions of the functional 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 and $\tau (t)$ is an unbounded delay. Under certain restrictions we relate asymptotic behaviour of solutions $x(t)$ of this equation to the behaviour of a solution $\varphi (t)$ of the auxiliary functional nondifferential equation
$$\varphi (t)=|k|\,\varphi (t-\tau (t)),\qquad t\in I.$$ |
| first_indexed | 2024-04-09T13:42:19Z |
| format | Article |
| id | doaj.art-188d51c5c75645f2a8a34ceb8b3993e3 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:42:19Z |
| publishDate | 2000-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-188d51c5c75645f2a8a34ceb8b3993e32023-05-09T07:52:56ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752000-01-01199921810.14232/ejqtde.1999.5.231Asymptotic behaviour of solutions of some differential equations with an unbounded delayJan Čermák0Technical University of Brno, Brno, Czech RepublicWe investigate the asymptotic properties of all solutions of the functional 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 and $\tau (t)$ is an unbounded delay. Under certain restrictions we relate asymptotic behaviour of solutions $x(t)$ of this equation to the behaviour of a solution $\varphi (t)$ of the auxiliary functional nondifferential equation $$\varphi (t)=|k|\,\varphi (t-\tau (t)),\qquad t\in I.$$http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=31 |
| spellingShingle | Jan Čermák Asymptotic behaviour of solutions of some differential equations with an unbounded delay Electronic Journal of Qualitative Theory of Differential Equations |
| title | Asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| title_full | Asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| title_fullStr | Asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| title_full_unstemmed | Asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| title_short | Asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| title_sort | asymptotic behaviour of solutions of some differential equations with an unbounded delay |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=31 |
| work_keys_str_mv | AT jancermak asymptoticbehaviourofsolutionsofsomedifferentialequationswithanunboundeddelay |