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Surjectivity in Fréchet spaces

Milen Ivanov, Nadia Zlateva

Abstract

We prove surjectivity result in Fréchet spaces of Nash-Moser type. That is, with uniform estimates over all semimorms. Our method works for functions which are only continuous and Gâteaux differentiable like in the recent result of Ekeland. We present the results in multi-valued setting exploring the relevant notions of map regularity.

Surjectivity in Fréchet spaces

Abstract

We prove surjectivity result in Fréchet spaces of Nash-Moser type. That is, with uniform estimates over all semimorms. Our method works for functions which are only continuous and Gâteaux differentiable like in the recent result of Ekeland. We present the results in multi-valued setting exploring the relevant notions of map regularity.

Paper Structure

This paper contains 6 sections, 18 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Statements of the Main Results
  3. Preliminaries
  4. LOEV Principle
  5. Proofs of the Main Results
  6. Conclusions

Key Result

Theorem 2

Let $(X,\|\cdot\|_n)$ and $(Y,|\cdot|_n)$ be Fréchet spaces and let be a multi-valued map with closed graph. Let $U\subset X$ and $V\subset Y$ be open and such that $\mathrm{Gr}\, F\cap (U\times V)\neq \emptyset$. Assume that for some $\mathbf{s}\in \mathbb{R}_+^\infty$ and some non-empty set $C\subset Y$ it holds that Then, for all $(x,y)\in\mathrm{Gr}\, F$ such that $x+\Pi_{\mathbf{s}}(X)\sub

Theorems & Definitions (35)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • proof
  • ...and 25 more