A generalization of the inverse mapping theorem in infinite dimensions

TL;DR

This paper generalizes the inverse mapping theorem to include weaker non-expansiveness conditions and non-smooth maps, extending its applicability to PDE systems and implicit function theorems.

Contribution

It introduces a new version of the inverse mapping theorem based on property ${ m A}$, broadening its scope beyond classical smoothness assumptions.

Findings

Generalized inverse mapping theorem applicable to non-smooth maps

Extensions to implicit function and PDE existence theorems

Applicable to infinite-dimensional spaces

Abstract

We present a generalization of the inverse mapping theorem, where variations of a weaker non-expansiveness property (referred to as property ${\sf A}$) replace the key $\mathsf{C}^1$ condition. We also obtain inverse mapping theorems that can be applied to non-smooth maps. Also as a by-product of the generalized inverse mapping theorem, we prove generalizations of the implicit function theorem and existence and uniqueness theorem of abstract PDE systems as well.