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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 (cid:1)(cid:1)Printedonacid-freepaperwhichfallswithintheguidelinesoftheANSI toensurepermanenceanddurability. 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 BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie;detailedbiblio- graphicdataisavailableintheInternetat(cid:3)http://dnb.ddb.de(cid:4). (cid:1) Copyright2004byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany. Allrightsreserved,includingthoseoftranslationintoforeignlanguages.Nopartofthisbook may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writingfromthepublisher. PrintedinGermany. Coverdesign:ThomasBonnie,Hamburg Typesetusingtheauthors’TEXfiles:I.Zimmermann,Freiburg Printingandbinding:Hubert&Co.GmbH&Co.Kg,Göttingen 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