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Inverse Function Theorem in Fréchet Spaces

Milen Ivanov, Nadia Zlateva

Abstract

We consider the classical Inverse Function Theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from Variational Analysis when applied to a directionally differentiable function, produce very general surjectivity result and, if injectivity can be ensured, Inverse Function Theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations.

Inverse Function Theorem in Fréchet Spaces

Abstract

We consider the classical Inverse Function Theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from Variational Analysis when applied to a directionally differentiable function, produce very general surjectivity result and, if injectivity can be ensured, Inverse Function Theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations.

Paper Structure

This paper contains 7 sections, 14 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local surjectivity
  4. Proof of the main result
  5. Local injectivity
  6. Implicit Map Theorem
  7. Application to differential equations

Key Result

Theorem 1.1

Let $(X,\|\cdot\|_n)$ be a Fréchet space and $(Y,\|\cdot\|_n)$ be a non-exotic Fréchet space and let $f:X\to Y$ be a continuous function. Let $U$ be a nonempty open subset of $X$ such that $f$ is injective and directionally differentiable on $U$; and there are $d\in \mathbb{N}$ and $c_n>0$ such that Then for each $x\in U$ there exists an open $V\ni x$ such that $f$ is invertible on $V$ and

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5: Corollary 3, LOEV
  • Lemma 2.6
  • Lemma 3.1
  • Proposition 3.2
  • Theorem 3.3
  • ...and 6 more