# Surjection and inversion for locally Lipschitz maps between Banach spaces

## TL;DR

This paper investigates the conditions under which non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces are globally invertible, extending classical results using advanced variational and nonsmooth analysis tools.

## Contribution

It introduces a new framework combining the Chang Palais-Smale condition and pseudo-Jacobians to establish global invertibility criteria for nonsmooth maps in Banach spaces.

## Key findings

- Established a version of Hadamard's integral condition for nonsmooth maps.
- Proved existence and uniqueness of solutions for nonlinear equations in Banach spaces.
- Extended classical invertibility results to the nonsmooth, infinite-dimensional setting.

## Abstract

We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.