Inverse Function Theorem in Fr\'echet Spaces

TL;DR

This paper extends the classical Inverse Function Theorem to Fréchet spaces using modern techniques, providing general surjectivity results and conditions for Lipschitz continuity of the inverse, with applications to differential equations.

Contribution

It introduces a new approach combining geometrisation of tame estimates and variational analysis for inverse function theorems in Fréchet spaces, generalizing previous results.

Findings

Established a general surjectivity result for inverse functions in Fréchet spaces.

Derived conditions under which the inverse function exhibits Lipschitz-like continuity.

Presented an application to differential equations demonstrating the theorem's utility.

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.