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 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.2000 Mathematics Subject Classification. 47J07, 58E05, 49J52.

Metric regularity, pseudo-Jacobians and global inversion theorems on Finsler manifolds

“…Further results along this line have been obtained in [9] and [4] in the more general setting of mappings between metric spaces. In a different direction, for global inversion results in terms of Palais-Smale conditions, we refer to [21], [7] and [10], and references therein.…”

On Nonsmooth Global Implicit Function Theorems for Locally Lipschitz Functions from Banach Spaces to Euclidean Spaces

“…Several works have been done to overcome these difficulties. For example, the papers [9,10] provided conditions for surjectivity and inversion of locally Lipschitz functions between Banach spaces under assumptions formulated in terms of pseudo-Jacobian.…”