An inverse function theorem in Fréchet spaces
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 91-105.

Je présente un théorème d'inversion pour des applications différentiables entre espaces de Fréchet, qui contient le théorème classique de Nash et Moser. Contrairement à ce dernier, la démonstration donnée ici ne repose pas sur l'algorithme itératif de Newton, mais sur le théorème de convergence dominée de Lebesgue et le principe variationnel d'Ekeland. Comme conséquence, les hypothèses sont substantiellement affaiblies : on ne demande pas que l'application F à inverser soit de classe ${C}^{2}$, ni même ${C}^{1}$, ni même différentiable au sens de Fréchet.

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be ${C}^{2}$, or even ${C}^{1}$, or even Fréchet-differentiable.

DOI : https://doi.org/10.1016/j.anihpc.2010.11.001
Mots clés : Inverse function theorem, Implicit function theorem, Fréchet space, Nash–Moser theorem
@article{AIHPC_2011__28_1_91_0,
author = {Ekeland, Ivar},
title = {An inverse function theorem in Fr\'echet spaces},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {91--105},
publisher = {Elsevier},
volume = {28},
number = {1},
year = {2011},
doi = {10.1016/j.anihpc.2010.11.001},
zbl = {1256.47037},
mrnumber = {2765512},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2010.11.001/}
}
Ekeland, Ivar. An inverse function theorem in Fréchet spaces. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 1, pp. 91-105. doi : 10.1016/j.anihpc.2010.11.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.11.001/

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