On some generalizations of Hadamard's inversion theorem beyond differentiability

TL;DR

This paper extends Hadamard's inversion theorem to continuous, possibly non-Lipschitz functions in finite-dimensional spaces, providing new conditions for global invertibility using nonsmooth analysis tools.

Contribution

It introduces sufficient conditions for global invertibility of nonsmooth functions via strict estimators and coderivatives, generalizing classical differentiability-based results.

Findings

Established conditions for global invertibility of nonsmooth functions.

Provided quantitative estimates of the inverse's Lipschitz constant.

Extended Hadamard's theorem beyond differentiability assumptions.

Abstract

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the existence of global Lipschitz continuous inverse by translating its metric assumptions in terms of nonsmooth analysis constructions. This exploration focuses on continuous, but possibly not locally Lipschitz mappings, acting in finite-dimensional Euclidean spaces. As a result, sufficient conditions for global invertibility are formulated by means of strict estimators, *-difference of convex compacta and regular/basic coderivatives. These conditions qualify the global inverse as a Lipschitz continuous mapping and provide quantitative estimates of its Lipschitz constant in terms of the above constructions.