An Inverse Mapping Theorem in Fréchet-Montel Spaces

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.

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References

  1. 1.

    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)

  2. 2.

    Cibulka, R., Fabian, M.: On primal regularity estimates for set-valued mappings. J. Math. Anal. Appl. 438, 444–464 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cibulka, R., Fabian, M.: On Nash-Moser-Ekeland inverse mapping theorem. Vietnam J. Math. 47, 527–545 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cibulka, R., Fabian, M., Kruger, A.Y.: On semiregularity of mappings. J. Math. Anal. Appl. 473, 811–836 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cibulka, R., Roubal, T.: Ioffe-type criteria in extended quasi-metric spaces, J. Convex Anal. 27. to appear (2020)

  6. 6.

    Dontchev, A. L., Rockafellar, R. T.: Implicit functions and solution mappings, 2nd edn. Springer, Dordrecht (2014)

    MATH  Google Scholar 

  7. 7.

    Dunford, N., Schwartz, J.: Linear operators, Part I: General theory. Interscience Publ., London (1958)

    MATH  Google Scholar 

  8. 8.

    Ekeland, I.: An inverse function theorem in Frechet́ spaces. Ann. Inst. H. Poincaré, 28, 91–105 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)

    MathSciNet  Article  Google Scholar 

  10. 10.

    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)

    Book  Google Scholar 

  11. 11.

    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)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    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)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ivanov, M., Zlateva, N.: A simple case of Nash-Moser-Ekeland theorem. arXiv:1809.0063v2 [math.FA] (2018)

  14. 14.

    Ivanov, M., Zlateva, N.: A simple case within Nash-Moser-Ekeland theory. C. R. Acad. Bulg. Sci. 72, 152–157 (2019)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ivanov, M., Zlateva, N.: Surjectivity in Fréchet spaces. J. Opt. Theory Appl. 182, 265–284 (2019)

    Article  Google Scholar 

  16. 16.

    Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8, 457–459 (1941)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Köthe, G.: Topological vector spaces. Springer-Verlag, New York (1969)

    MATH  Google Scholar 

  18. 18.

    Rådström, H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3, 165–169 (1952)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Seierstad, A.: Open mapping theorems for directionally differentiable functions. In: No. 2004, 21 Memorandum, Department of Economics, University of Oslo (2004)

Download references

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.

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Correspondence to Radek Cibulka.

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Dedicated to the 65th birthday of Alex Kruger

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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.

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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

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Keywords

  • Fréchet-Montel space
  • Right directional derivative
  • Nash-Moser inverse mapping theorem
  • Local surjection
  • Directional semiregularity
  • Nonlinear image

Mathematics Subject Classification (2010)

  • 58C15
  • 47J07
  • 46T20