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Proceedings of the American Mathematical Society

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

 

 

An extension of the Rabinowitz bifurcation
theorem to Lipschitz potential operators
in Hilbert spaces


Authors: Alexander Ioffe and Efim Schwartzman
Journal: Proc. Amer. Math. Soc. 125 (1997), 2725-2732
MSC (1991): Primary 58E05; Secondary 49K99
DOI: https://doi.org/10.1090/S0002-9939-97-04061-6
MathSciNet review: 1415327
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Abstract: The main result of the paper is an extension of the bifurcation theorem of Rabinowitz to equations $Ax + {\varphi _{\lambda }(x)\ }= \lambda x$ with $\varphi $ continuous jointly in $(\lambda ,x)$ and $ {\varphi _{\lambda }(\cdot )\ }$ of class $C^{1,1}$. We also prove a bifurcation theorem for critical points of the function $g_{\lambda }(x)$ which is just continuous and changes at $x=0$ an isolated minimum (in $x$) to isolated maximum when $\lambda $ passes, say, zero. The proofs of the theorems, as well as the the theorems themselves, are new, in certain important aspects, even when applied to smooth functions.


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Additional Information

Alexander Ioffe
Affiliation: Department of Mathematics The Technion Haifa 32000, Israel
Email: ioffe@math.technion.ac.il

Efim Schwartzman
Affiliation: Department of Mathematics The Technion Haifa 32000, Israel

DOI: https://doi.org/10.1090/S0002-9939-97-04061-6
Keywords: Critical point, $C^{1, 1}$-function, bifurcation, modulus of regularity, Palais-Smail condition
Received by editor(s): February 5, 1996
Additional Notes: The research was supported by the US-Israel BSF grant 90-00455 and by the Israel Ministry of Science and Technology grant 3501–1–91
Communicated by: Hal L. Smith
Article copyright: © Copyright 1997 American Mathematical Society