Nonlinear spectral theory

421 Pages·2004·1.638 MB·English

by  Jurgen Appell, Espedito De Pascale, Alfonso Vignoli

Nonlinear Spectral Theory
Jürgen Appell
Espedito De Pascale
Alfonso Vignoli
Walter de Gruyter
de Gruyter Series in Nonlinear Analysis and Applications 10
Editors
A.Bensoussan (Paris)
R.Conti (Florence)
A.Friedman (Minneapolis)
K.-H.Hoffmann (Munich)
L.Nirenberg (New York)
A. Vignoli (Rome)
Managing Editor
J.Appell (Würzburg)

Jürgen Appell
Espedito De Pascale
Alfonso Vignoli
Nonlinear Spectral Theory
≥
Walter de Gruyter · Berlin · New York
Authors
JürgenAppell EspeditoDePascale AlfonsoVignoli
UniversitätWürzburg Universita` dellaCalabria Universita` diRoma“TorVergata”
MathematischesInstitut DipartimentodiMatematica DepartimentodiMatematica
AmHubland PonteBucci,Cubo31B ViadellaRicercaScientifica
97074Würzburg,Germany 87036Arcavacata/Rende,Italy 00133Roma,Italy
E-mail: E-mail: E-mail:
[email protected] [email protected] [email protected]
wuerzburg.de
Mathematics Subject Classification 2000: 47-02; 34B15, 34G20, 34L16, 35J60, 35P30, 45C05,
45G05,45G10,47A10,47A12,47A75,47H10,47H14,47H30,47J05,47J10,47J25,47L07
Keywords:spectrum,resolvent,nonlinearoperator,eigenvalue,zero-epioperator,stablysolvableop-
erator,homogeneousoperator,monotoneoperator,numericalrange,bifurcationproblem,nonlinear
Fredholmalternative,boundaryvalueproblem,p-Laplaceoperator
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LibraryofCongressCataloging-in-PublicationData
Appell,Jürgen.
Nonlinearspectraltheory/JürgenAppell,EspeditoDePas-
cale,AlfonsoVignoli.
p. cm.(cid:1)(DeGruyterseriesinnonlinearanalysisand
applications;10)
Includesbibliographicalreferencesandindex.
ISBN3-11-018143-6(acid-freepaper)
1. Spectral theory (Mathematics) 2. Nonlinear theories.
I. De Pascale, Espedito. II. Vignoli, Alfonso, 1940(cid:1)
III.Title. IV.Series.
QA320.A67 2004
515(cid:2).7222(cid:1)dc22 2004043900
ISBN 3-11-018143-6
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Für Kristina in Liebe und Dankbarkeit
Jürgen
A mia moglieAdele e ai miei figli Luigi, Egidio,
Giovanna, Emanuele, MariaVittoria e Stefano
Espedito
Alfonso desea dedicar este libro a su adorada
esposa Lucilla

Preface
Thismonographistheoutcomeoftheauthors’workatspectraltheoryfornonlinear
operators (or nonlinear spectral theory, for short) during the last 10 years. It could
nothavebeenrealizedwithoutseveralmutualvisitsandinvitations. Wegratefullyac-
knowledgegenerousfinancialsupportofseveralvisitsofthefirstauthorinItalybythe
ConsiglioNazionaledelleRicerche(CNR),theIstitutoNazionalediAltaMatematica
(INdAM),andtheMinisterodell’Istruzione,dell’UniversitàedellaRicerca(MIUR,
fondi PRIN del programma “Analisi Reale ed Analisi Funzionale”). Visiting pro-
fessorships of the second and third author in Germany have been supported by the
Deutsche Forschungsgemeinschaft (DFG) and Deutscher Akademischer Austausch-
dienst (DAAD).We are particularly indebted to the DFG for supporting the project
“NonlinearSpectralandEigenvalueTheory”(grant#Ap40/15-1)between1999and
2001, and to the VolkswagenStiftung (grant # I/77 562) for supporting a sabbatical
semesterofthefirstauthorattheUniversityofRome.
Weoweagreatdebttoseveralfriends. Amongthem,weexpressourmostsincere
gratitudetoElenaGiorgieriandMartinVäthwhocarefullyreadapreliminaryversion
of the manuscript and, through their constant advice and benevolent criticism, con-
siderably improved the presentation. We are also indebted to many colleagues and
students who have attended our lectures and seminar talks on this subject and have
suffered, not always in silence, through awkward presentations. It is a pleasure to
acknowledgethegreathelpgivenusbydeGruyterVerlag, inparticularbyManfred
KarbeandIreneZimmermann.
Würzburg,Arcavacata,Rome Theauthors

Contents
Preface vii
Introduction 1
1 SpectraofBoundedLinearOperators 11
1.1 Thespectrumofaboundedlinearoperator . . . . . . . . . . . . . . . 11
1.2 Compactandα-contractivelinearoperators . . . . . . . . . . . . . . 16
1.3 Subdivisionofthespectrum . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Essentialspectraofboundedlinearoperators . . . . . . . . . . . . . 30
1.5 Notes,remarksandreferences . . . . . . . . . . . . . . . . . . . . . 33
2 SomeCharacteristicsofNonlinearOperators 40
2.1 Somemetriccharacteristicsofnonlinearoperators . . . . . . . . . . . 40
2.2 Alistofexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Compactandα-contractivenonlinearoperators . . . . . . . . . . . . 51
2.4 Specialsubsetsofscalars . . . . . . . . . . . . . . . . . . . . . . . . 61
2.5 Notes,remarksandreferences . . . . . . . . . . . . . . . . . . . . . 64
3 InvertibilityofNonlinearOperators 67
3.1 Properandray-properoperators . . . . . . . . . . . . . . . . . . . . 67
3.2 Coerciveandray-coerciveoperators . . . . . . . . . . . . . . . . . . 73
3.3 Furtherpropertiesofnonlinearoperators . . . . . . . . . . . . . . . . 76
3.4 Themappingspectrum . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Excursion: topologicaldegreetheory . . . . . . . . . . . . . . . . . . 85
3.6 Notes,remarksandreferences . . . . . . . . . . . . . . . . . . . . . 92
4 TheRhodiusandNeubergerSpectra 94
4.1 TheRhodiusspectrum . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.2 Fréchetdifferentiableoperators . . . . . . . . . . . . . . . . . . . . . 96
4.3 TheNeubergerspectrum . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4 Specialclassesofoperators . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Notes,remarksandreferences . . . . . . . . . . . . . . . . . . . . . 107
5 TheKachurovskijandDörfnerSpectra 109
5.1 Lipschitzcontinuousoperators . . . . . . . . . . . . . . . . . . . . . 109
5.2 TheKachurovskijspectrum . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 TheDörfnerspectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 118