On the singularities of nonlinear fredholm operators of positive index

M. S. Berger; R. A. Plastock

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

On the singularities of nonlinear fredholm operators of positive index

M. S. Berger, R. A. Plastock

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

The singular set B = {x|F¹(x) is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces X₁ and X₂) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative F¹(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.

Original language	English (US)
Pages (from-to)	217-221
Number of pages	5
Journal	Proceedings of the American Mathematical Society
Volume	79
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9939-1980-0565342-5
State	Published - Jun 1980
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

Keywords

Fiber bundle map
Higher homotopy groups
Nonlinear fredholm operator

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10.1090/S0002-9939-1980-0565342-5

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abstract = "The singular set B = \{x|F1(x) is not surjective\} of a nonlinear Fredholm operator F of positive index (between Banach spaces X1 and X2) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fr{\'e}chet derivative F1(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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year = "1980",

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doi = "10.1090/S0002-9939-1980-0565342-5",

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AU - Berger, M. S.

AU - Plastock, R. A.

PY - 1980/6

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N2 - The singular set B = {x|F1(x) is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces X1 and X2) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative F1(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.

AB - The singular set B = {x|F1(x) is not surjective} of a nonlinear Fredholm operator F of positive index (between Banach spaces X1 and X2) is investigated. Under the assumption that the mapping is proper and has a locally Lipschitzian Fréchet derivative F1(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.

KW - Fiber bundle map

KW - Higher homotopy groups

KW - Nonlinear fredholm operator

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 2

ER -

On the singularities of nonlinear fredholm operators of positive index

Abstract

All Science Journal Classification (ASJC) codes

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