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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Structure and dimension of global branches of solutions to multiparameter nonlinear equations


Authors: J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli
Journal: Trans. Amer. Math. Soc. 291 (1985), 383-435
MSC: Primary 58E07; Secondary 47H15
MathSciNet review: 800246
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Abstract: This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a selfcontained fashion, most of the recent results on such problems. Our theory applies whenever any generalized degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behavior of the sets of solutions, of bifurcation points and of continuation points.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0800246-0
PII: S 0002-9947(1985)0800246-0
Keywords: Global branches of solutions, multiparameter continuation problems, multiparameter bifurcation problems, covering dimension, essential maps, cantor manifolds
Article copyright: © Copyright 1985 American Mathematical Society