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

 

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 $X$ be a real reflexive Banach space with dual $X^{*}$ and $G\subset X$open and bounded and such that $0\in G.$  Let $T:X\supset D(T)\to 2^{X^{*}}$be maximal monotone with $0\in D(T)$ and $0\in T(0),$ and $C:X\supset D(C)\to X^{*}$ with $0\in D(C)$ and $C(0)\neq 0.$ A general and more unified eigenvalue theory is developed for the pair of operators $(T,C).$  Further conditions are given for the existence of a pair $(\lambda ,x) \in (0,\infty )\times (D(T+C)\cap \partial G)$ such that

\begin{displaymath}(**)\quad\qquad\qquad\qquad\qquad\qquad\qquad Tx+\lambda Cx\owns 0.\quad\qquad\qquad\qquad\qquad\qquad\qquad\end{displaymath}

The ``implicit" eigenvalue problem, with $C(\lambda ,x)$ in place of $\lambda Cx,$ 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 $T,~C.$ 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: http://dx.doi.org/10.1090/S0002-9947-05-03761-X
PII: S 0002-9947(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.