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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the singularities of nonlinear Fredholm operators of positive index

Authors: M. S. Berger and R. A. Plastock
Journal: Proc. Amer. Math. Soc. 79 (1980), 217-221
MSC: Primary 58B15; Secondary 47H99
MathSciNet review: 565342
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Abstract: The singular set $ B = \{ x\vert F'(x)$ is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces $ {X_1}$ and $ {X_2}$) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative $ F'(x)$, it is shown that the singular set B is nonempty. Furthermore, when the Banach spaces are infinite dimensional, B cannot be the countable union of compact sets nor can $ {F^{ - 1}}(F(B))$ contain isolated points.

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Keywords: Nonlinear Fredholm operator, fiber bundle map, higher homotopy groups
Article copyright: © Copyright 1980 American Mathematical Society