Global inversion and covering maps on length spaces

Garrido, Isabel; Gutú, Olivia; Jaramillo, Jesús Á.

doi:10.1016/j.na.2010.04.069

Cited by 14 publications

(16 citation statements)

References 19 publications

(26 reference statements)

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“…Later on, Ioffe obtained in [17] a global inversion result for a continuous map f which is locally one-to-one, using an analog of Hadamard integral condition, defined in terms of the so-called constant surjection of f at every point. 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.…”

Section: Introductionmentioning

confidence: 86%

Surjection and inversion for locally Lipschitz maps between Banach spaces

Gutú

Jaramillo

2019

Journal of Mathematical Analysis and Applications

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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.

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Section: Introductionmentioning

confidence: 86%

Surjection and inversion for locally Lipschitz maps between Banach spaces

Gutú

Jaramillo

2019

Journal of Mathematical Analysis and Applications

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“…This has been noted by John [29] for Banach spaces and generalized for length spaces in terms of D − x f in [19]. A direct proof for Finsler manifolds can be done in terms of µ(x) using the same arguments; see proof of Lemma 16 in the appendix.…”

Section: Coercivity and The Hadamard Integral Conditionmentioning

confidence: 92%

“…As a consequence, an extension of the Hadamard Theorem is obtained in terms of a metric version of µ(x), a kind of lower scalar Dini derivate. In [19] an estimation of the domain of invertibility around a point is provided for a local homeomorphism between length metric space, inspired by the aforementioned work of John [29]. In finite-dimensional case, analogous results were obtained by Pourciau [49,50] by means of the Clarke generalized Jacobian of f , recently extended by Jaramillo et al [27] for locally Lipschitz mappings between finite-dimensional Finsler manifolds.…”

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confidence: 88%

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On global inverse theorems

Gutú¹

2016

TMNA

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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 )).

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“…The problem of finding sufficient conditions for a local diffeomorphism to be a global one has been investigated by many authors in the framework of Banach spaces, cf. [14] and references therein. But it has not been the subject of study for more general Fréchet spaces.…”

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confidence: 99%

A Global Diffeomorphism Theorem for Fréchet Spaces

Eftekharinasab

2020

J Math Sci

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We give sufficient conditions for a C 1 c -local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fréchet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.2010 Mathematics Subject Classification. 57R50;46A04;46A50 .

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Global inversion and covering maps on length spaces

Cited by 14 publications

References 19 publications

Surjection and inversion for locally Lipschitz maps between Banach spaces

Surjection and inversion for locally Lipschitz maps between Banach spaces

On global inverse theorems

A Global Diffeomorphism Theorem for Fréchet Spaces

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