On an extension of a global implicit function theorem
We study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible. Generalizing known results, we provide sufficient criteria which...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2022-05-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.309/ |
| _version_ | 1797651542069739520 |
|---|---|
| author | Berger, Thomas Haller, Frédéric |
| author_facet | Berger, Thomas Haller, Frédéric |
| author_sort | Berger, Thomas |
| collection | DOAJ |
| description | We study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible. Generalizing known results, we provide sufficient criteria which are easy to check. These conditions essentially rely on the existence of diffeomorphisms between the respective projections of the set of zeros and appropriate Banach spaces, as well as a corresponding growth bound. The projections further allow to consider cases where the global implicit function is not defined on all of the open subset corresponding to the first variable. |
| first_indexed | 2024-03-11T16:16:19Z |
| format | Article |
| id | doaj.art-7014cc647e934e41ab20f31935cc33e1 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:19Z |
| publishDate | 2022-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-7014cc647e934e41ab20f31935cc33e12023-10-24T14:19:30ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-05-01360G543945010.5802/crmath.30910.5802/crmath.309On an extension of a global implicit function theoremBerger, Thomas0Haller, Frédéric1Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, GermanyFachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, GermanyWe study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible. Generalizing known results, we provide sufficient criteria which are easy to check. These conditions essentially rely on the existence of diffeomorphisms between the respective projections of the set of zeros and appropriate Banach spaces, as well as a corresponding growth bound. The projections further allow to consider cases where the global implicit function is not defined on all of the open subset corresponding to the first variable.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.309/ |
| spellingShingle | Berger, Thomas Haller, Frédéric On an extension of a global implicit function theorem Comptes Rendus. Mathématique |
| title | On an extension of a global implicit function theorem |
| title_full | On an extension of a global implicit function theorem |
| title_fullStr | On an extension of a global implicit function theorem |
| title_full_unstemmed | On an extension of a global implicit function theorem |
| title_short | On an extension of a global implicit function theorem |
| title_sort | on an extension of a global implicit function theorem |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.309/ |
| work_keys_str_mv | AT bergerthomas onanextensionofaglobalimplicitfunctiontheorem AT hallerfrederic onanextensionofaglobalimplicitfunctiontheorem |