Surjection and inversion for locally Lipschitz maps between Banach spaces

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 tools, we obtain the existence and uniqueness of solution for certain nonlinear equations defined by locally Lipschitz mappings. Along the way, we also obtain a global surjection theorem for locally Lipschitz maps in terms of pseudo-Jacobians.