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Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal O’Regan VOLUME 9 NONLINEAR ANALYSIS SERIES IN MATHEMATICAL ANALYSIS AND APPLICATIONS Series in Mathematical Analysis and Applications (SIMAA) is edited by Ravi P. Agarwal, Florida Institute of Technology, USA and Donal O’Regan, National University of Ireland, Galway, Ireland. The series is aimed at reporting on new developments in mathematical analysis and applications of a high standard and or current interest. Each volume in the series is devoted to a topic in analysis that has been applied, or is potentially applicable, to the solutions of scientific, engineering and social problems. Volume 1 Method of Variation of Parameters for Dynamic Systems V. Lakshmikantham and S.G. Deo Volume 2 Integral and Integrodifferential Equations: Theory, Methods and Applications Edited by Ravi P. Agarwal and Donal O’Regan Volume 3 Theorems of Leray-Schauder Type and Applications Donal O’Regan and Radu Precup Volume 4 Set Valued Mappings with Applications in Nonlinear Analysis Edited by Ravi P. Agarwal and Donal O’Regan Volume 5 Oscillation Theory for Second Order Dynamic Equations Ravi P. Agarwal, Said R. Grace, and Donal O’Regan Volume 6 Theory of Fuzzy Differential Equations and Inclusions V. Lakshmikantham and Ram N. Mohapatra Volume 7 Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations V. Lakshmikantham, S. Koksal, and Raymond Bonnett Volume 8 Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems Leszek Gasin´ski and Nikolaos S. Papageorgiou Volume 9 Nonlinear Analysis Leszek Gasin´ski and Nikolaos S. Papageorgiou Series in Mathematical Analysis and Applications Edited by Ravi P. Agarwal and Donal O’Regan VOLUME 9 NONLINEAR ANALYSIS Leszek Gasiński Nikolaos S. Papageorgiou Boca Raton London New York Singapore Published in 2005 by Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-484-3 (Hardcover) International Standard Book Number-13: 978-1-58488-484-2 (Hardcover) Library of Congress Card Number 2005045529 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. 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Library of Congress Cataloging-in-Publication Data Gasinski, Leszek. Nonlinear analysis / Leszek Gasinski, Nikolaos S. Papageorgiou. p. cm. -- (Series in mathematical analysis and applications ; v. 9) Includes bibliographical references and index. ISBN 1-58488-484-3 1. Nonlinear functional analysis. 2. Nonlinear operators. I. Papageorgiou, Nikolaos Socrates. II. Title. III. Series. QA321.5.G37 2005 515'.7--dc22 2005045529 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group and the CRC Press Web site at is the Academic Division of T&F Informa plc. http://www.crcpress.com To Prof. Zdzis(cid:195)law Denkowski Contents 1 Hausdorff Measures and Capacity 1 1.1 Measure Theoretical Background . . . . . . . . . . . . . . . . 3 1.2 Covering Results . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Hausdorff Measure and Hausdorff Dimension . . . . . . . . . 22 1.4 Differentiation of Hausdorff Measures . . . . . . . . . . . . . 44 1.5 Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . . . 52 1.6 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2 Lebesgue-Bochner and Sobolev Spaces 107 2.1 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . 108 2.2 Lebesgue-Bochner Spaces and Evolution Triples . . . . . . . 127 2.3 Compactness Results . . . . . . . . . . . . . . . . . . . . . . 150 2.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2.5 Inequalities and Embedding Theorems . . . . . . . . . . . . . 213 2.6 Fine Properties of Functions and BV-Functions . . . . . . . 239 2.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 3 Nonlinear Operators and Young Measures 265 3.1 Compact and Fredholm Operators . . . . . . . . . . . . . . . 266 3.2 Operators of Monotone Type . . . . . . . . . . . . . . . . . . 303 3.3 Accretive Operators and Semigroups of Operators . . . . . . 343 3.4 The Nemytskii Operator and Integral Functions . . . . . . . 405 3.5 Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . 427 3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 4 Smooth and Nonsmooth Analysis and Variational Principles 467 4.1 Differential Calculus in Banach Spaces . . . . . . . . . . . . 468 4.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . 488 4.3 Haar Null Sets and Locally Lipschitz Functions . . . . . . . 501 4.4 Duality and Subdifferentials . . . . . . . . . . . . . . . . . . 512 4.5 Integral Functionals and Subdifferentials . . . . . . . . . . . 558 4.6 Variational Principles . . . . . . . . . . . . . . . . . . . . . . 578 4.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 vii viii 5 Critical Point Theory 607 5.1 Deformation Results . . . . . . . . . . . . . . . . . . . . . . . 608 5.2 Minimax Theorems . . . . . . . . . . . . . . . . . . . . . . . 642 5.3 Structure of the Critical Set . . . . . . . . . . . . . . . . . . 654 5.4 Multiple Critical Points . . . . . . . . . . . . . . . . . . . . . 661 5.5 Lusternik-Schnirelman Theory and Abstract Eigenvalue Prob- lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 6 Eigenvalue Problems and Maximum Principles 707 6.1 Linear Elliptic Operators . . . . . . . . . . . . . . . . . . . . 708 6.2 The Partial p-Laplacian . . . . . . . . . . . . . . . . . . . . . 732 6.3 The Ordinary p-Laplacian . . . . . . . . . . . . . . . . . . . 759 6.4 Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . 775 6.5 Comparison Principles . . . . . . . . . . . . . . . . . . . . . . 788 6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797 7 Fixed Point Theory 803 7.1 Metric Fixed Point Theory . . . . . . . . . . . . . . . . . . . 804 7.2 Topological Fixed Point Theory . . . . . . . . . . . . . . . . 821 7.3 Partial Order and Fixed Points . . . . . . . . . . . . . . . . . 833 7.4 Fixed Points of Multifunctions . . . . . . . . . . . . . . . . . 877 7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 891 Appendix 895 A.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895 A.2 Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 899 A.3 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . 908 A.4 Calculus and Nonlinear Analysis . . . . . . . . . . . . . . . . 912 List of Symbols 915 References 925 Preface Linear functional analysis deals with infinite dimensional topological vector spaces(whichmixinafruitfulwaythelinear(algebraic)structurewithtopo- logical one) and the linear operators acting between them. The effort was to extend standard results of linear analysis to an infinite dimensional con- text. Thefirsthalfofthetwentiethcenturyismarkedbyintensivetheoretical investigations in this area, which were also accompanied by detailed treat- ment of linear mathematical models. With the exception of a short period during the 1930’s (compact operators and Leray-Schauder degree), nonlinear operators were out of the emerging picture. However, mounting evidence from diverse other fields such as physics, engineering, economics, biology and others suggested that there should be an effort to extend the linear theory to various kinds of nonlinear operators. Systematic efforts in this direction startedintheearly1960’sandmarkthebeginningofwhatisknowntodayas “NonlinearAnalysis.” Sincethenseveraltheorieshavebeendevelopedinthis respectandtodaysomeofthemarewellestablishedapproachingtheirlimits, while others are still the object of intense research activity. It is not a co- incidence that simultaneously with the advent of nonlinear analysis, we have the appearance of nonsmooth analysis and of multivalued analysis, both of which were motivated by concrete needs in applied areas such as control theory, optimization, game theory and economics. Their development provided nonlinear analysis with new concepts, tools and theories that enriched the subject considerably. Today nonlinear analysis is a well established mathe- maticaldiscipline,whichischaracterizedbyaremarkablemixtureofanalysis, topology and applications. It is exactly the fact that the subject combines in abeautifulwaythesethreeitemsthatmakesitattractivetomathematicians. Thenotionsandtechniquesofnonlinearanalysisprovidetheappropriatetools to develop more realistic and accurate models describing various phenomena. This gives nonlinear analysis a rather interdisciplinary character. Today the more theoretically inclined nonmathematician (engineer, economist, biologist orchemist)needsaworkingknowledgeofatleastapartofnonlinearanalysis in order to be able to conduct a complete qualitative analysis of his models. This supports a high demand for books on nonlinear analysis. Of course the subject is big (vast is maybe a more appropriate word) and no single book can cover all its theoretical and applied parts. In this volume, we have fo- cused on those topics of nonlinear analysis which are pertinent to the theory ofboundaryvalueproblemsandtheirapplicationssuchascontroltheoryand calculus of variations. ix x InChapter1wedealwithHausdorffmeasuresandcapacities,whichprovide themeanstoestimatethe“size”or“dimension”of“thin”or“highlyirregular” sets. The recent development of fractal geometry and its uses in a variety of applied areas (such as Brownian motion of particles, turbulence in fluids, geographical coastlines and surfaces etc) renewed the interest on Hausdorff measures,whichforalongperiodwereatopicofsecondaryimportancewithin measuretheory. InthischapterwealsohaveourfirstencounterwithLipschitz and locally Lipschitz functionals which will be examined again in Chapter 4. At this point we prove the celebrated “Rademacher’s theorem.” Chapter2dealswithcertainclassesoffunctionspaces,whicharisenaturally in the study of boundary value problems. These are the Lebesgue-Bochner spaces (the suitable spaces for the analysis of evolution equations) and the Sobolev spaces (the suitable spaces for weak solutions of elliptic equations). We conduct a detailed study of these spaces with special emphasis on compactness and embedding results. Also using the tool of Hausdorff measures and capacities, we investigate the fine properties of Sobolev functions and also introduce and study functionals of bounded variation which are useful in theoretical mechanics. InChapter3,wedealwithcertainlargeclassesofnonlinearoperatorswhich arise often in applications. We examine compact operators for which we develop in parallel the corresponding linear theory, with one of the main results being the spectral theorem for compact self-adjoint operators on a Hilbert space. We also investigate nonlinear operators of monotone type which have their roots in the calculus of variations and exhibit remarkable surjectivity properties. Monotone operators lead to accretive operators, the two families beingidenticalinthecontextofHilbertspaces. Accretiveoperatorsareclosely connectedwiththegenerationtheoryofsemigroupsofoperators. Wealsoex- aminebothlinearandnonlinearsemigroups. Semigroupsarebasictoolsinthe studyofevolutionequations. Inaddition,weexaminetheNemytskii operator which is a nonlinear operator encountered in almost all problems. Finally, in the last section of the chapter, we discuss Young measures which provide the right framework to examine the limit behavior of the minimizing sequence of variational problems which do not have a solution. Young measures are used in optimal control and in the calculus of variations in connection with the so-called “relaxation method.” Chapter 4 presents the calculus of smooth and of certain broad classes of nonsmoothfunctions. WestartwiththeGâteauxandFréchetderivatives. We discuss the generic differentiability of continuous convex functions (Mazur’s theorem) and extend Rademacher’s theorem to locally Lipschitz functions between certain Banach spaces by using the notion of Haar-null sets. Then we pass to nondifferentiable functions and develop the duality properties and subdifferential theory of convex functions and the generalized subdifferential oflocallyLipschitzfunctions. Wealsoexamineintegralfunctionalsanddiscuss the celebrated Ekeland variational principle establishing its equivalence with some other geometric results of nonlinear analysis. xi InChapter5wepresentthecriticalpointtheoryofC1-functionsdefinedon a Banach space. This theory is in the core of the variational methods used in the study of boundary value problems. We follow the deformation approach whichleadstominimaxcharacterizationsofthecriticalvalues. Wealsostudy the structure of the set of critical points and derive results on the existence of multiple critical points. Next we present the Lusternik-Schnirelman theory which extends to nonlinear eigenvalue problems the corresponding linear theory of R. Courant. Chapter 6 uses the abstract results of Chapter 5 as well as results from earlierchapterstodevelopthespectrumoflinearellipticdifferentialoperators, ofthepartialp-Laplacian(withDirichletandNeumannboundaryconditions) and of the scalar and vector ordinary p-Laplacian (with Dirichlet, Neumann and periodic boundary conditions). We also present linear and nonlinear maximum principles and comparison results, which are useful tools in the study of boundary value problems. FinallyinChapter7wehavegatheredsomebasicfixedpointtheorems. We present results from metric fixed point theory, from topological fixed point theory and fixed point results based on the partial order induced by a closed, convex pointed cone. We also indicate how many of these results can be extended to multifunctions (set-valued functions). Wehavetriedtomakethevolumeself-contained. Forthisreasonattheend ofthebookwehaveincludedaratherextendedappendixforeasyreferenceof the general results used in the book. Nevertheless, within the test whenever we are in the need of using some results not proved in the book, we also give exact references where the interested reader can find additional information. Nowthattheprojecthasreacheditsconclusion,wewouldliketothankthe good people of CRC Press (especially Mrs. Jessica Vakili) for their help and kindcooperationduringthepreparationofthisbook. Wewouldliketothank the two editors of this series, Prof. R.P. Agarwal and Prof. D.O’Regan, for supporting this effort.