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Nonlinear mappings that are globally equivalent to a projection


Author: Roy Plastock
Journal: Trans. Amer. Math. Soc. 275 (1983), 373-380
MSC: Primary 58C25; Secondary 47H17
DOI: https://doi.org/10.1090/S0002-9947-1983-0678357-0
MathSciNet review: 678357
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Abstract: The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse, $ {F^ \downarrow }(x)$, to the derivative map $ {F^\prime }(x)$ and a global solution to the differential equation $ {P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})$. Both these problems are solved and the generalized Hadamard-Levy criterion

$\displaystyle \int_0^\infty {\mathop {\inf }\limits_{\vert x\vert < s} \left({1/\vert{F^ \downarrow }(x)\vert} \right)\,ds = \infty } $

is shown to be sufficient for $ F$ to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678357-0
Keywords: Fredholm map of positive index, fiber bundle map
Article copyright: © Copyright 1983 American Mathematical Society

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