An Inverse Mapping Theorem in Fréchet-Montel Spaces


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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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).

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

Mathematics Subject Classification (2010)

  • 58C15
  • 47J07
  • 46T20