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

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 C2, or even C1, or even Fréchet-differentiable.

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 C2, ni même C1, ni même différentiable au sens de Fréchet.

DOI : 10.1016/j.anihpc.2010.11.001
Keywords: 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},
     mrnumber = {2765512},
     zbl = {1256.47037},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2010.11.001/}
}
TY  - JOUR
AU  - Ekeland, Ivar
TI  - An inverse function theorem in Fréchet spaces
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 91
EP  - 105
VL  - 28
IS  - 1
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2010.11.001/
DO  - 10.1016/j.anihpc.2010.11.001
LA  - en
ID  - AIHPC_2011__28_1_91_0
ER  - 
%0 Journal Article
%A Ekeland, Ivar
%T An inverse function theorem in Fréchet spaces
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 91-105
%V 28
%N 1
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2010.11.001/
%R 10.1016/j.anihpc.2010.11.001
%G en
%F AIHPC_2011__28_1_91_0
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. https://www.numdam.org/articles/10.1016/j.anihpc.2010.11.001/

[1] Serge Alinhac, Patrick Gérard, Opérateurs Pseudo-différentiels et Théorème de Nash–Moser, Interéditions et Éditions du CNRS, Paris (1991), Grad. Stud. Math. vol. 82, Amer. Math. Soc., Rhode Island (2000) | MR | Zbl

[2] Vladimir I. Arnol'D, Small divisors, Dokl. Akad. Nauk CCCP 137 (1961), 255-257, Dokl. Akad. Nauk CCCP 138 (1961), 13-15 | MR

[3] Vladimir I. Arnol'D, Small divisors I, Izvestia Akad. Nauk CCCP 25 (1961), 21-86

[4] Vladimir I. Arnol'D, Small divisors II, Ouspekhi Math. Nauk 18 (1963), 81-192

[5] Massimiliano Berti, Philippe Bolle, Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Ration. Mech. Anal. 195 (2010), 609-642 | MR | Zbl

[6] Nassif Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Math. vol. 107, Cambridge Univ. Press (1993) | MR | Zbl

[7] Ivar Ekeland, Sur les problèmes variationnels, C. R. Acad. Sci. Paris 275 (1972), 1057-1059, C. R. Acad. Sci. Paris 276 (1973), 1347-1348 | MR | Zbl

[8] Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443-474 | MR | Zbl

[9] Richard Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (1) 7 (1982), 65-222 | MR | Zbl

[10] Andrei N. Kolmogorov, On the conservation of quasi-periodic motion for a small variation of the Hamiltonian function, Dokl. Akad. Nauk CCCP 98 (1954), 527-530 | MR

[11] Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Natl. Acad. Sci. USA 47 (1961), 1824-1831 | MR | Zbl

[12] Jürgen Moser, A rapidly convergent iteration method and nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa 20 (1966), 266-315 | EuDML | Numdam | MR | Zbl

[13] John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63 | MR | Zbl

[14] R.T. Rockafellar, Conjugate Duality and Optimization, SIAM/CBMS Monograph Ser. vol. 16, SIAM Publications (1974) | MR | Zbl

[15] Jacob Schwartz, On Nash's implicit functional theorem, Comm. Pure Appl. Math. 13 (1960), 509-530 | MR | Zbl

  • Bednarczuk, Ewa; Brezhneva, Olga; Leśniewski, Krzysztof; Prusińska, Agnieszka; Tret’yakov, Alexey A. Towards Nonlinearity: The p-Regularity Theory, Entropy, Volume 27 (2025) no. 5, p. 518 | DOI:10.3390/e27050518
  • Giordano, Paolo; Luperi Baglini, Lorenzo Beyond Cauchy–Kowalewsky: a Picard–Lindelöf theorem for smooth PDE, Journal of Fixed Point Theory and Applications, Volume 27 (2025) no. 2 | DOI:10.1007/s11784-025-01184-5
  • Wang, Chong; Huang, Rongli; Bao, Jiguang On the second boundary value problem for a class of fully nonlinear flow III, Journal of Evolution Equations, Volume 24 (2024) no. 3 | DOI:10.1007/s00028-024-00983-6
  • Gangbo, Wilfrid; Jacobs, Matt; Kim, Inwon Well-posedness and regularity for a polyconvex energy, ESAIM: Control, Optimisation and Calculus of Variations, Volume 29 (2023), p. 67 | DOI:10.1051/cocv/2023041
  • Ivanov, M.; Quincampoix, M.; Zlateva, N. Metric Regularity for Set-Valued Maps in Fréchet-Montel Spaces. Implicit Mapping Theorem, Set-Valued and Variational Analysis, Volume 31 (2023) no. 2 | DOI:10.1007/s11228-023-00679-y
  • Zhang, Chuang-liang; Huang, Nan-jing On Ekeland's variational principle for interval-valued functions with applications, Fuzzy Sets and Systems, Volume 436 (2022), p. 152 | DOI:10.1016/j.fss.2021.10.003
  • Gutú, Olivia Global inversion for metrically regular mappings between Banach spaces, Revista Matemática Complutense, Volume 35 (2022) no. 1, p. 25 | DOI:10.1007/s13163-020-00380-w
  • Shannon, Chris On Lipschitz implicit function theorems in Banach spaces and applications, Journal of Mathematical Analysis and Applications, Volume 494 (2021) no. 2, p. 124589 | DOI:10.1016/j.jmaa.2020.124589
  • Ivanov, Milen; Zlateva, Nadia Inverse mapping theorem in Fréchet spaces, Journal of Optimization Theory and Applications, Volume 190 (2021) no. 1, p. 300 | DOI:10.1007/s10957-021-01885-0
  • Jourani, A.; Silva, F. J. Existence of Lagrange Multipliers under Gâteaux Differentiable Data with Applications to Stochastic Optimal Control Problems, SIAM Journal on Optimization, Volume 30 (2020) no. 1, p. 319 | DOI:10.1137/18m1223411
  • Cibulka, Radek; Fabian, Marián; Roubal, Tomáš An Inverse Mapping Theorem in Fréchet-Montel Spaces, Set-Valued and Variational Analysis, Volume 28 (2020) no. 1, p. 195 | DOI:10.1007/s11228-020-00536-2
  • Lu, Guangcun Morse theory methods for a class of quasi-linear elliptic systems of higher order, Calculus of Variations and Partial Differential Equations, Volume 58 (2019) no. 4 | DOI:10.1007/s00526-019-1577-1
  • Huang, Rongli; Ye, Yunhua On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I, International Mathematics Research Notices, Volume 2019 (2019) no. 18, p. 5539 | DOI:10.1093/imrn/rnx278
  • Ivanov, Milen; Zlateva, Nadia Surjectivity in Fréchet Spaces, Journal of Optimization Theory and Applications, Volume 182 (2019) no. 1, p. 265 | DOI:10.1007/s10957-019-01482-2
  • Jaramillo, Jesús A.; Lajara, Sebastián; Madiedo, Óscar Inversion of Nonsmooth Maps between Banach Spaces, Set-Valued and Variational Analysis, Volume 27 (2019) no. 4, p. 921 | DOI:10.1007/s11228-018-0499-y
  • Cibulka, Radek; Fabian, Marián On Nash–Moser–Ekeland Inverse Mapping Theorem, Vietnam Journal of Mathematics, Volume 47 (2019) no. 3, p. 527 | DOI:10.1007/s10013-019-00342-w
  • Huynh, Van Ngai; Théra, Michel Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions, Journal of Mathematical Analysis and Applications, Volume 457 (2018) no. 2, p. 1403 | DOI:10.1016/j.jmaa.2017.07.040
  • Baldi, Pietro; Floridia, Giuseppe; Haus, Emanuele Exact controllability for quasilinear perturbations of KdV, Analysis PDE, Volume 10 (2017) no. 2, p. 281 | DOI:10.2140/apde.2017.10.281
  • Baldi, Pietro; Haus, Emanuele A Nash–Moser–Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs, Journal of Functional Analysis, Volume 273 (2017) no. 12, p. 3875 | DOI:10.1016/j.jfa.2017.09.016
  • Cibulka, R.; Fabian, M. On primal regularity estimates for set-valued mappings, Journal of Mathematical Analysis and Applications, Volume 438 (2016) no. 1, p. 444 | DOI:10.1016/j.jmaa.2016.02.016
  • Secchi, Paolo On the Nash-Moser Iteration Technique, Recent Developments of Mathematical Fluid Mechanics (2016), p. 443 | DOI:10.1007/978-3-0348-0939-9_23
  • Huang, Rongli On the second boundary value problem for Lagrangian mean curvature flow, Journal of Functional Analysis, Volume 269 (2015) no. 4, p. 1095 | DOI:10.1016/j.jfa.2015.05.003
  • Stevens, Ben The Nash-Moser Iteration Technique with Application to Characteristic Free-Boundary Problems, Hyperbolic Conservation Laws and Related Analysis with Applications, Volume 49 (2014), p. 311 | DOI:10.1007/978-3-642-39007-4_13
  • Dontchev, Asen L.; Rockafellar, R. Tyrrell Metric Regularity in Infinite Dimensions, Implicit Functions and Solution Mappings (2014), p. 277 | DOI:10.1007/978-1-4939-1037-3_5
  • Penot, Jean-Paul Metric and Topological Tools, Calculus Without Derivatives, Volume 266 (2013), p. 1 | DOI:10.1007/978-1-4614-4538-8_1
  • Penot, Jean-Paul Elements of Differential Calculus, Calculus Without Derivatives, Volume 266 (2013), p. 117 | DOI:10.1007/978-1-4614-4538-8_2
  • Penot, Jean-Paul Elements of Convex Analysis, Calculus Without Derivatives, Volume 266 (2013), p. 187 | DOI:10.1007/978-1-4614-4538-8_3
  • Penot, Jean-Paul Elementary and Viscosity Subdifferentials, Calculus Without Derivatives, Volume 266 (2013), p. 263 | DOI:10.1007/978-1-4614-4538-8_4
  • Penot, Jean-Paul Circa-Subdifferentials, Clarke Subdifferentials, Calculus Without Derivatives, Volume 266 (2013), p. 357 | DOI:10.1007/978-1-4614-4538-8_5
  • Penot, Jean-Paul Limiting Subdifferentials, Calculus Without Derivatives, Volume 266 (2013), p. 407 | DOI:10.1007/978-1-4614-4538-8_6
  • Penot, Jean-Paul Graded Subdifferentials, Ioffe Subdifferentials, Calculus Without Derivatives, Volume 266 (2013), p. 463 | DOI:10.1007/978-1-4614-4538-8_7

Cité par 31 documents. Sources : Crossref