Abstract
Influenced by a recent note by M. Ivanov and N. Zlateva, we prove a statement in the style of Nash-Moser-Ekeland theorem for mappings from a Fréchet-Montel space with values in any Fréchet space (not necessarily standard). The mapping under consideration is supposed to be continuous and directionally differentiable (in particular Gateaux differentiable) with the derivative having a right inverse. We also consider an approximation by a graphical derivative and by a linear operator in the spirit of Graves’ theorem. Finally, we derive corollaries of the abstract results in finite dimensions. We obtain, in particular, sufficient conditions for the directional semiregularity of a mapping defined on a (locally) convex compact set in directions from a locally conic set; and also conditions guaranteeing that the nonlinear image of a convex set contains a prescribed ordered interval.
Similar content being viewed by others
References
Aubin, J. -P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Advances in mathematics. Supplementary studies, 160–232 (1981)
Cibulka, R., Fabian, M.: On primal regularity estimates for set-valued mappings. J. Math. Anal. Appl. 438, 444–464 (2016)
Cibulka, R., Fabian, M.: On Nash-Moser-Ekeland inverse mapping theorem. Vietnam J. Math. 47, 527–545 (2019)
Cibulka, R., Fabian, M., Kruger, A.Y.: On semiregularity of mappings. J. Math. Anal. Appl. 473, 811–836 (2019)
Cibulka, R., Roubal, T.: Ioffe-type criteria in extended quasi-metric spaces, J. Convex Anal. 27. to appear (2020)
Dontchev, A. L., Rockafellar, R. T.: Implicit functions and solution mappings, 2nd edn. Springer, Dordrecht (2014)
Dunford, N., Schwartz, J.: Linear operators, Part I: General theory. Interscience Publ., London (1958)
Ekeland, I.: An inverse function theorem in Frechet́ spaces. Ann. Inst. H. Poincaré, 28, 91–105 (2011)
Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)
Fabian, M., Hájek, P., Habala, P., Montesinos, V., Zizler, V.: Banach space theory: The basis for linear and non-linear analysis. Springer Verlag, CMS Books in Mathematics, New York (2011)
Glicksberg, I.L.: A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points. Proc. Amer. Math. Soc. 3, 170–174 (1952)
Huynh, V.N., Théra, M.: Ekeland’s inverse function theorem in graded Fréchet spaces revisited for multifunctions. J. Math. Anal. Appl. 457, 1403–1421 (2018)
Ivanov, M., Zlateva, N.: A simple case of Nash-Moser-Ekeland theorem. arXiv:1809.0063v2 [math.FA] (2018)
Ivanov, M., Zlateva, N.: A simple case within Nash-Moser-Ekeland theory. C. R. Acad. Bulg. Sci. 72, 152–157 (2019)
Ivanov, M., Zlateva, N.: Surjectivity in Fréchet spaces. J. Opt. Theory Appl. 182, 265–284 (2019)
Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)
Köthe, G.: Topological vector spaces. Springer-Verlag, New York (1969)
Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)
Seierstad, A.: Open mapping theorems for directionally differentiable functions. In: No. 2004, 21 Memorandum, Department of Economics, University of Oslo (2004)
Acknowledgements
While our note was more or less ready, we found on Arxiv a new version of the preprint [13], the paper [14], where the authors prove the special case of our Theorem 1 when f is defined on whole space X. This time, they considered the common concept of Gateaux differentiability (not that from [13]). We would like to thank anonymous referees for their valuable comments leading to substantial improvements of our note.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the 65th birthday of Alex Kruger
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported by the project InteCom, no. CZ.02.1.01/0.0/0.0/17_ 048/0007267.
The second author was supported by the grant of GAČR 17-00941S and by RVO: 67985840.
The third author was supported by the Grant Agency of the Czech Republic, grant no. 20-11164L.
Rights and permissions
About this article
Cite this article
Cibulka, R., Fabian, M. & Roubal, T. An Inverse Mapping Theorem in Fréchet-Montel Spaces. Set-Valued Var. Anal 28, 195–208 (2020). https://doi.org/10.1007/s11228-020-00536-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-020-00536-2
Keywords
- Fréchet-Montel space
- Right directional derivative
- Nash-Moser inverse mapping theorem
- Local surjection
- Directional semiregularity
- Nonlinear image