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Functional analysis, Operator theory
On an extension of a global implicit function theorem
Thomas Berger1 ; Frédéric Haller2
1 Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
2 Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 439-450.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/crmath.309
Classification: 26B10, 58C15

Thomas Berger 1; Frédéric Haller 2

1 Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
2 Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Thomas Berger; Frédéric Haller. On an extension of a global implicit function theorem. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 439-450. doi : 10.5802/crmath.309. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.309/

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