Mild solutions, variation of constants formula, and linearized stability for delay differential equations
The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there...
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| Format: | Article |
| Language: | English |
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University of Szeged
2023-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9993 |
| _version_ | 1797352372417069056 |
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| author | Junya Nishiguchi |
| author_facet | Junya Nishiguchi |
| author_sort | Junya Nishiguchi |
| collection | DOAJ |
| description | The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution. |
| first_indexed | 2024-03-08T13:15:30Z |
| format | Article |
| id | doaj.art-92eb23d91d30453aa410e78d507db28f |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-03-08T13:15:30Z |
| publishDate | 2023-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-92eb23d91d30453aa410e78d507db28f2024-01-18T08:28:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752023-08-0120233217710.14232/ejqtde.2023.1.329993Mild solutions, variation of constants formula, and linearized stability for delay differential equationsJunya Nishiguchi0Mathematical Science Group, Advanced Institute for Materials Research (AIMR), Tohoku University, 2-1-1, Katahira, Aoba-ku, Sendai, 980-8577, JapanThe method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9993delay differential equationsdiscontinuous history functionsfundamental matrix solutionvariation of constants formulaprinciple of linearized stabilitypoincaré–lyapunov theorem |
| spellingShingle | Junya Nishiguchi Mild solutions, variation of constants formula, and linearized stability for delay differential equations Electronic Journal of Qualitative Theory of Differential Equations delay differential equations discontinuous history functions fundamental matrix solution variation of constants formula principle of linearized stability poincaré–lyapunov theorem |
| title | Mild solutions, variation of constants formula, and linearized stability for delay differential equations |
| title_full | Mild solutions, variation of constants formula, and linearized stability for delay differential equations |
| title_fullStr | Mild solutions, variation of constants formula, and linearized stability for delay differential equations |
| title_full_unstemmed | Mild solutions, variation of constants formula, and linearized stability for delay differential equations |
| title_short | Mild solutions, variation of constants formula, and linearized stability for delay differential equations |
| title_sort | mild solutions variation of constants formula and linearized stability for delay differential equations |
| topic | delay differential equations discontinuous history functions fundamental matrix solution variation of constants formula principle of linearized stability poincaré–lyapunov theorem |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9993 |
| work_keys_str_mv | AT junyanishiguchi mildsolutionsvariationofconstantsformulaandlinearizedstabilityfordelaydifferentialequations |