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), 3851-3881

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 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:
https://doi.org/10.1090/S0002-9947-05-03761-X

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.