On global inverse theorems

Abstract

Abstract. Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of f at every x, namely, f is a local homeomorphism or f is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of f , ideally expecting to conclude that f is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of Graves Theorem is obtained for a suitable submersion f with image in a Banach space: given r > 0 and x 0 in the domain of f we give a radius ̺(r) > 0, closely related to the hypothesis of the Hadamard Theorem, such that B̺(f (x 0 )) ⊂ f (Br (x 0 )).

Inversion of Nonsmooth Maps between Banach Spaces

“…These kinds of sufficient conditions for global invertibility were first considered by Hadamard [9] for maps between finite-dimensional spaces, and then by Lévy [17] in the case of maps between Hilbert spaces, and have been widely used since then. We refer to the recent survey of Gutú [7] for an extensive and detailed information about these and other conditions for global inversion of smooth maps between Banach spaces and, more generally, between Finsler manifolds.…”

On an Application of a Higher Order Inversion Theorem to Certain Integral Equations and Boundary Value Problems

“…In [8], there are given some sufficient conditions for a mapping from the Banach space to the Hilbert space to be a diffeomorphism. As well, the homeomorphism between Banach spaces is considered in [3], [9] and [13].…”

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

“…Analogous conditions for global invertibility have been also obtained in the more general setting where f : X → Y is a C 1 -map between Finsler manifolds, mainly in connection with covering maps, the path-lifting property, and other related topological properties. We refer to the survey of Gutú [6] for an extensive and detailed information about global invertibility of smooth maps between Finsler manifolds.…”