Fixed points and differential equations with asymptotically constant or periodic solutions
Cooke and Yorke developed a theory of biological growth and epidemics based on an equation $x'(t)=g(x(t))-g(x(t-L))$ with the fundamental property that $g$ is an arbitrary locally Lipschitz function. They proved that each solution either approaches a constant or $\pm \infty$ on its maximal righ...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2004-05-01
|
| 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=186 |
| _version_ | 1797830816617725952 |
|---|---|
| author | Theodore Burton |
| author_facet | Theodore Burton |
| author_sort | Theodore Burton |
| collection | DOAJ |
| description | Cooke and Yorke developed a theory of biological growth and epidemics based on an equation $x'(t)=g(x(t))-g(x(t-L))$ with the fundamental property that $g$ is an arbitrary locally Lipschitz function. They proved that each solution either approaches a constant or $\pm \infty$ on its maximal right-interval of definition. They also raised a number of interesting questions and conjectures concerning the determination of the limit set, periodic solutions, parallel results for more general delays, and stability of solutions. Although their paper motivated many subsequent investigations, the basic questions raised seem to remain unanswered.
We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control. |
| first_indexed | 2024-04-09T13:42:09Z |
| format | Article |
| id | doaj.art-f2c93d388c7a4d76a6b7bc8565fedb98 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:42:09Z |
| publishDate | 2004-05-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-f2c93d388c7a4d76a6b7bc8565fedb982023-05-09T07:52:57ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752004-05-0120041113110.14232/ejqtde.2004.1.11186Fixed points and differential equations with asymptotically constant or periodic solutionsTheodore Burton0Northwest Research Institute, Port Angeles, WA, U.S.A.Cooke and Yorke developed a theory of biological growth and epidemics based on an equation $x'(t)=g(x(t))-g(x(t-L))$ with the fundamental property that $g$ is an arbitrary locally Lipschitz function. They proved that each solution either approaches a constant or $\pm \infty$ on its maximal right-interval of definition. They also raised a number of interesting questions and conjectures concerning the determination of the limit set, periodic solutions, parallel results for more general delays, and stability of solutions. Although their paper motivated many subsequent investigations, the basic questions raised seem to remain unanswered. We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=186 |
| spellingShingle | Theodore Burton Fixed points and differential equations with asymptotically constant or periodic solutions Electronic Journal of Qualitative Theory of Differential Equations |
| title | Fixed points and differential equations with asymptotically constant or periodic solutions |
| title_full | Fixed points and differential equations with asymptotically constant or periodic solutions |
| title_fullStr | Fixed points and differential equations with asymptotically constant or periodic solutions |
| title_full_unstemmed | Fixed points and differential equations with asymptotically constant or periodic solutions |
| title_short | Fixed points and differential equations with asymptotically constant or periodic solutions |
| title_sort | fixed points and differential equations with asymptotically constant or periodic solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=186 |
| work_keys_str_mv | AT theodoreburton fixedpointsanddifferentialequationswithasymptoticallyconstantorperiodicsolutions |