Inversion of Nonsmooth Maps between Banach Spaces

Abstract

We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.

Surjection and inversion for locally Lipschitz maps between Banach spaces

“…Again from Corollary 2.18 of [21] we have that Jf satisfies the chain rule condition. Furthermore, Jf also satisfies the strong chain rule condition.…”

“…Furthermore, by Theorem 2.3.10 (Chain Rule II) of [5], we have that if f strictly differentiable, in particular C 1 , then Jf satisfies the strong chain rule condition. [21] we have that Jf satisfies the chain rule condition. Furthermore, taking into account that ∂f (x) is a closed convex subset of L(X, Y ) and using Theorem 5.2 in [30] we deduce that Jf satisfies in fact the strong chain rule condition.…”

“…Let f ∶ U → Y be a continuous map. Some important examples of a derivative-like objects for continuous maps can be included in a general frame so called pseudo-Jacobians (see [21]). The definition of a pseudo-Jacobian of f at a point x involves an approximation of the "scalarized" functions y * ○ f , through all directions y * ∈ Y by means of upper Dini directional derivatives and a sublinearization of the approximations by a set of operators.…”

“…The following examples of pseudo-Jacobians for a continuous function f ∶ U → Y between Banach spaces are explained with detail in [21]; see also [24] for the finitedimensional theory of pseudo-Jacobians.…”

“…Further results along this line have been obtained in [14] and [10] in the more general setting of maps between metric spaces. More recently, in [21], the authors consider the notion of pseudo-Jacobian for a continuous map between Banach spaces, which is an extension of pseudo-Jacobian matrices of Jeyakumar and Luc to this setting, and obtain different global inversions results in this context.…”