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

 

The index of a critical point for densely defined operators of type $(S_+)_L$ in Banach spaces


Authors: Athanassios G. Kartsatos and Igor V. Skrypnik
Journal: Trans. Amer. Math. Soc. 354 (2002), 1601-1630
MSC (2000): Primary 47H11
Published electronically: October 3, 2001
MathSciNet review: 1873020
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Abstract: The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type $(S_{+})_{L}$ acting from a real, reflexive and separable Banach space $X$ into $X^{*}.$ This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in $l^{p},~p>2,$ illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type $(S_{+})).$ Applications to nonlinear Dirichlet problems have appeared elsewhere.


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Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida,Tampa, Florida 33620-5700
Email: hermes@math.usf.edu

Igor V. Skrypnik
Affiliation: Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine
Email: skrypnik@iamm.ac.donetsk.ua

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02934-8
PII: S 0002-9947(01)02934-8
Keywords: Reflexive separable Banach space, operators of type $(S_{+})_{L},$ degree theory for $(S_{+})_{L}$-operators, index of a critical point for $(S_{+})_{L}$-operators, bifurcation
Received by editor(s): September 30, 1998
Published electronically: October 3, 2001
Article copyright: © Copyright 2001 American Mathematical Society