Local Mapping Relations and Global Implicit Function Theorems

Rheinboldt, Werner C.

doi:10.2307/1994907

Cited by 21 publications

(36 citation statements)

References 8 publications

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“…This will be achieved by means of a continuation property, much in the spirit of Rheinboldt [27] and Plastock [22]. Our result will apply to a wide class of path-connected metric spaces, which we introduce now.…”

Section: Continuation Property On Metric Spacesmentioning

confidence: 97%

Global homeomorphisms and covering projections on metric spaces

Gutú

Jaramillo

2007

Math. Ann.

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Abstract. For a large class of metric spaces with nice local structure, which includes Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we give sufficient conditions for a local homeomorphism to be a covering projection. We first obtain a general condition in terms of a path continuation property. As a consequence, we deduce several conditions in terms of path-liftings involving a generalized derivative, and in particular we obtain an extension of Hadamard global inversion theorem in this context. Next we prove that, in the case of quasi-isometric mappings, some of these sufficient conditions are also necessary. Finally, we give some applications to the existence of global implicit functions.

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Section: Continuation Property On Metric Spacesmentioning

confidence: 97%

Global homeomorphisms and covering projections on metric spaces

Gutú

Jaramillo

2007

Math. Ann.

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“…On the other hand, Plastock [13] obtained conditions for a function f to be globally equivalent to a projection, in the particular case that f is a nonlinear Fredholm map between Banach spaces. Plastock used a powerful continuation method (the "method of line lifting"), which had proved to be also quite useful in problems of global inversion of functions (see e.g., [12], [15] and [7]). …”

Section: Introductionmentioning

confidence: 99%

Fibrations on Banach manifolds

Gutú¹,

Jaramillo²

2004

Pacific J. Math.

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“…In the late sixties, the article of Rheinboldt [54] appeared in the literature where a general theory is established for global implicit function theorems in terms of the continuation property. The continuation property was influenced in part by the so-called continuation method in numerical analysis.…”

Section: 31mentioning

confidence: 99%

“…In particular, the continuation property guarantees the existence of the lifting defined on whole interval I = [0, 1]. Various applications to global implicit and inversion theorems are presented in [54] essentially for normed linear spaces where P is taken as the set of all smooth paths on Y . As discussed below, in this case much more can be said.…”

Section: 31mentioning

confidence: 99%

“…For the proof, he used the prolongation of local inverses of f along lines. Soon after, Plastock [48] introduced a limiting property for lines, called condition L, analogous to the continuation property used by Rheinboldt [54] in a more abstract context. Plastock proved that a local homeomorphism f satisfies condition L if and only if it is a global homeomorphism.…”

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confidence: 99%

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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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Local Mapping Relations and Global Implicit Function Theorems

Cited by 21 publications

References 8 publications

Global homeomorphisms and covering projections on metric spaces

Global homeomorphisms and covering projections on metric spaces

Fibrations on Banach manifolds

On global inverse theorems

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