On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

Meyer, Gunter H.

doi:10.1137/0705057

Cited by 118 publications

(32 citation statements)

References 5 publications

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“…Weighted conditions. In the late sixties, Meyer extended condition (C10) where df (x) −1 is allowed to go to infinity at most linearly in x ; see Theorem 1.1 of [41]. More precisely, he proved that a local diffeomorphism f : R n → R n which has a locally Lipschitz continuous Fréchet derivative such that df (x) −1 ≤ a x +b for all x ∈ R n for some a, b > 0 is a global homeomorphism.…”

Section: 23mentioning

confidence: 99%

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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Section: 23mentioning

confidence: 99%

On global inverse theorems

Gutú¹

2016

TMNA

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“…The results of §3 contain some new global existence theorems for (2) and also include the well-known Hadamard-Levy theorem (see [7] and [9]) as well as a recent generalization by Meyer [10]. §4 essentially covers and extends the results of Ficken [6].…”

mentioning

confidence: 90%

“…For a review of earlier work about this method see, for example, the introduction of Ficken [6] who then proceeds to develop certain results about the global solvability of H(t, y) = x by using the local solvability provided by the implicit function theorem. Other results are due to Davidenko [3]; see also Yakovlev [14] and, more recently, Davis [4], and Meyer [10].…”

mentioning

confidence: 93%

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Local Mapping Relations and Global Implicit Function Theorems

Rheinboldt¹

1969

Transactions of the American Mathematical Society

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“…One "follows" a "path" which leads from the boundary to some one or more of the fixed points. For some contemporary papers using similar methods, see [19], [23]. This has been called the Newton method.…”

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confidence: 99%

The homotopy continuation method: numerically implementable topological procedures

Alexander¹,

Yorke²

1978

Trans. Amer. Math. Soc.

144

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Abstract. The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.In 1963, Hirsch [13] gave a short-and by now classic-proof of the Brouwer Fixed Point Theorem by a generic argument. This theorem is usually proved by some kind of degree argument and degree genetically counts inverse images of points. Hirsch directly looked at the inverse image of a generic point. The idea that one could replace degree arguments by looking at the inverse images of points of maps was made the theme of a book by J. Milnor [24] and later by V. Guillemin and A. Pollack [12, especially Chapters 2,3] and Hirsch [14, especially Chapter 5]. We offer these books as general references for transversality and Sard's theorem.Without knowledge of Hirsch's paper, H. Scarf [31] in 1967 used much the same ideas to numerically approximate a Brouwer fixed point. One "follows" a "path" which leads from the boundary to some one or more of the fixed points. For some contemporary papers using similar methods, see [19], [23]. This has been called the Newton method. B. C. Eaves [8] in 1972 developed a slightly different approach to the same end. His idea was to homotope one of a set of standard maps to the map in question; by foUowing the fixed points of the changing maps, the fixed-point-set of the map in question could be located. It is this version we call the homotopy continuation method and

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On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

Cited by 118 publications

References 5 publications

On global inverse theorems

On global inverse theorems

Local Mapping Relations and Global Implicit Function Theorems

The homotopy continuation method: numerically implementable topological procedures

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