On Nash–Moser–Ekeland Inverse Mapping Theorem

Abstract

We present a criterion for local surjectivity of mappings between graded Fréchet spaces in the spirit of a well-known criterion in Banach spaces. As applications, we get “hard inverse mapping theorem” in the flavor of Nash–Moser. The technology of proofs was strongly influenced by a recent paper of Ekeland.

This is a preview of subscription content, access via your institution.

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, pp 160–232. Academic Press, New York (1981)

  2. 2.

    Cibulka, R., Fabian, M., Ioffe, A.D.: On primal regularity estimates for single-valued mappings. J. Fixed Point Theory Appl. 17, 187–208 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Fabian, M., Preiss, D.: A generalization of the interior mapping theorem of Clarke and Pourciau. Comment. Math. Univ. Carol. 28, 311–324 (1987)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Hájek, P., Johanis, M.: Smooth Analysis in Banach Spaces. Series in Nonlinear Analysis and Applications, vol. 19. De Gruyter, Berlin (2014)

    MATH  Google Scholar 

  7. 7.

    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  MATH  Google Scholar 

  8. 8.

    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ioffe, A.D.: Variational Analysis of Regular Mappings: Theory and Applications. Springer Monographs in Mathematics. Springer International Publishing (2017)

  10. 10.

    Penot, J.-P.: Calculus without Derivatives. Graduate Texts in Mathematics, vol. 266. Springer-Verlag, New York (2013)

    Book  Google Scholar 

Download references

Funding

The second author was supported by the grant of GAČR 17-00941S and by RVO: 67985840.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Radek Cibulka.

Additional information

Dedicated to the 80th birthday of Alex D. Ioffe

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cibulka, R., Fabian, M. On Nash–Moser–Ekeland Inverse Mapping Theorem. Vietnam J. Math. 47, 527–545 (2019). https://doi.org/10.1007/s10013-019-00342-w

Download citation

Keywords

  • Fréchet space
  • Nash–Moser inverse mapping theorem
  • Local surjection
  • Metric regularity

Mathematics Subject Classification (2010)

  • 58C15
  • 47J07
  • 46T20