An implicit function theorem for non-smooth maps between Fr\'echet spaces

TL;DR

This paper establishes a Nash-Moser type inverse function theorem for non-smooth maps between Fréchet spaces, avoiding Newton's method and quadratic convergence, thus broadening applicability to less smooth maps.

Contribution

It introduces a new inverse function theorem for tame maps between Fréchet spaces that does not rely on the map being twice differentiable, unlike previous results.

Findings

Proves an inverse function theorem for non-smooth maps between Fréchet spaces.

Avoids the use of Newton's method and quadratic convergence.

Applicable to maps not requiring C^2 smoothness.

Abstract

We prove an inverse function theorem of Nash-Moser type for maps between Fr\'echet spaces satisfying tame estimates. In contrast to earlier proofs, we do not use the Newton method, that is, we do not use quadratic convergence to overcome the lack of derivatives. In fact, our theorem holds when the map to be inverted is not C^2