Table Of ContentSeries 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 Gasi´nski
Nikolaos S. Papageorgiou
Boca Raton London New York Singapore
Published in 2005 by
Chapman & Hall/CRC
Taylor & Francis Group
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© 2005 by Taylor & Francis Group, LLC
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International Standard Book Number-10: 1-58488-484-3 (Hardcover)
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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
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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), nonlin-
ear 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 the-
ory, 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 com-
pactness 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 de-
velop 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. WestartwiththeGˆateauxandFr´echetderivatives. 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 the-
ory 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.
