Differentiability in Fréchet spaces and delay differential equations
In infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search...
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| Format: | Article |
| Language: | English |
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University of Szeged
2019-02-01
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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=6838 |
| _version_ | 1797830505204285440 |
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| author | Hans-Otto Walther |
| author_facet | Hans-Otto Walther |
| author_sort | Hans-Otto Walther |
| collection | DOAJ |
| description | In infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search for a state space in which solutions are unique and differentiable with respect to initial data leads to smoothness hypotheses on the vector functional $f$ in an equation of the general form
\begin{equation*}
x'(t)=f(x_t)\in\mathbb{R}^n,\quad\text{with}\ x_t(s)=x(t+s)\quad\text{for}\ s\leq 0,
\end{equation*}
which have implications (a) on the nature of the delay (which is hidden in $f$) and (b) on the type of continuous differentiability which is present. We find the appropriate {\it strong} kind of continuous differentiability and show that there is a continuous semiflow of continuously differentiable solution operators on a Fréchet manifold, with local invariant manifolds at equilibria. |
| first_indexed | 2024-04-09T13:37:12Z |
| format | Article |
| id | doaj.art-514135d9d8974b4082bc1f1c148747f3 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:37:12Z |
| publishDate | 2019-02-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-514135d9d8974b4082bc1f1c148747f32023-05-09T07:53:09ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752019-02-0120191314410.14232/ejqtde.2019.1.136838Differentiability in Fréchet spaces and delay differential equationsHans-Otto Walther0University of Giessen, Giessen, GermanyIn infinite-dimensional spaces there are non-equivalent notions of continuous differentiability which can be used to derive the familiar results of calculus up to the Implicit Function Theorem and beyond. For autonomous differential equations with variable delay, not necessarily bounded, the search for a state space in which solutions are unique and differentiable with respect to initial data leads to smoothness hypotheses on the vector functional $f$ in an equation of the general form \begin{equation*} x'(t)=f(x_t)\in\mathbb{R}^n,\quad\text{with}\ x_t(s)=x(t+s)\quad\text{for}\ s\leq 0, \end{equation*} which have implications (a) on the nature of the delay (which is hidden in $f$) and (b) on the type of continuous differentiability which is present. We find the appropriate {\it strong} kind of continuous differentiability and show that there is a continuous semiflow of continuously differentiable solution operators on a Fréchet manifold, with local invariant manifolds at equilibria.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6838fréchet spacefrechet differentiabilitydelay differential equationunbounded delaysemiflowinvariant manifolds |
| spellingShingle | Hans-Otto Walther Differentiability in Fréchet spaces and delay differential equations Electronic Journal of Qualitative Theory of Differential Equations fréchet space frechet differentiability delay differential equation unbounded delay semiflow invariant manifolds |
| title | Differentiability in Fréchet spaces and delay differential equations |
| title_full | Differentiability in Fréchet spaces and delay differential equations |
| title_fullStr | Differentiability in Fréchet spaces and delay differential equations |
| title_full_unstemmed | Differentiability in Fréchet spaces and delay differential equations |
| title_short | Differentiability in Fréchet spaces and delay differential equations |
| title_sort | differentiability in frechet spaces and delay differential equations |
| topic | fréchet space frechet differentiability delay differential equation unbounded delay semiflow invariant manifolds |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6838 |
| work_keys_str_mv | AT hansottowalther differentiabilityinfrechetspacesanddelaydifferentialequations |