On the singularities of nonlinear Fredholm operators of positive index

Berger, M. S.; Plastock, R. A.

doi:10.1090/S0002-9939-1980-0565342-5

On the singularities of nonlinear Fredholm operators of positive index
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by M. S. Berger and R. A. Plastock PDF

Proc. Amer. Math. Soc. 79 (1980), 217-221 Request permission

Abstract:

The singular set $B = \{ x|F’(x)$ 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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