On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces
Authors:
Athanassios G. Kartsatos and Igor V. Skrypnik
Journal:
Trans. Amer. Math. Soc. 358 (2006), 38513881
MSC (2000):
Primary 47H14, 47H07, 47H11
Published electronically:
July 26, 2005
MathSciNet review:
2219002
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Abstract: Let be a real reflexive Banach space with dual and open and bounded and such that Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators Further conditions are given for the existence of a pair such that
The ``implicit" eigenvalue problem, with in place of is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.
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Additional Information
Athanassios G. Kartsatos
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 336205700
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/S000299470503761X
PII:
S 00029947(05)03761X
Keywords:
Maximal monotone operators,
$(S_{+})$mappings,
Browder's degree,
Skrypnik's degree,
degree for sums of densely defined mappings,
nonlinear eigenvalue problems
Received by editor(s):
May 6, 2003
Received by editor(s) in revised form:
June 3, 2004
Published electronically:
July 26, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
