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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A local bifurcation theorem for $ C\sp 1$-Fredholm maps


Authors: P. M. Fitzpatrick and Jacobo Pejsachowicz
Journal: Proc. Amer. Math. Soc. 109 (1990), 995-1002
MSC: Primary 58E07; Secondary 47H15
MathSciNet review: 1009988
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Abstract: The Krasnosel'skii Bifurcation Theorem is generalized to $ {C^1}$-Fredholm maps. Let $ X$ and $ Y$ be Banach spaces, $ F:R \times X \to Y$ be $ {C^1}$-Fredholm of index 1 and $ F(\lambda ,0) \equiv 0$ . If $ I \subseteq R$ is a closed, bounded interval at whose endpoints $ \frac{{\partial F}}{{\partial x}}\frac{{\partial F}}{{\partial x}}(\lambda ,0)$ is invertible, and the parity of $ \frac{{\partial F}}{{\partial x}}(\lambda ,0)$ on $ I$ is -1, then $ I$ contains a bifurcation point of the equation $ F(\lambda ,x) = 0$. At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1009988-5
PII: S 0002-9939(1990)1009988-5
Keywords: Bifurcation, nonlinear Fredholm, parity
Article copyright: © Copyright 1990 American Mathematical Society