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SeriesEditors
StevenG.Krantz,WashingtonUniversity,St.Louis
ShrawanKumar,UniversityofNorthCarolinaatChapelHill
JanNekova´ˇr,Universite´ PierreetMarieCurie,Paris
Forfurthervolumes:
http://www.springer.com/series/4842
Pavel Drábek • Jaroslav Milota
Methods of Nonlinear Analysis
Applications to Differential Equations
Second Edition
Pavel Drábek Jaroslav Milota
Faculty of Applied Sciences Faculty of Information Technology
Department of Mathematics Department of Applied Mathematics
University of West Bohemia Czech Technical University
Plzeň Praha 6
Czech Republic Czech Republic
ISBN 978-3-0348-0386-1 ISBN 978-3-0348-0387-8 (eBook)
DOI 10.1007/978-3-0348-0387-8
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Library of Congress Control Number: 2012955049
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Preliminaries
1.1 Elements of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Properties of Linear and Nonlinear Operators
2.1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2 Compact Linear Operators . . . . . . . . . . . . . . . . . . . . . . 78
2.3 Contraction Principle . . . . . . . . . . . . . . . . . . . . . . . . . 93
3 Abstract Integral and Differential Calculus
3.1 Integration of Vector Functions . . . . . . . . . . . . . . . . . . . . 109
3.2 Differential Calculus in Normed Linear Spaces. . . . . . . . . . . . 126
3.2A Newton Method . . . . . . . . . . . . . . . . . . . . . . . . 144
4 Local Properties of Differentiable Mappings
4.1 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . 149
4.2 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . 156
4.3 Local Structure of Differentiable Maps, Bifurcations . . . . . . . . 166
4.3A Differentiable Manifolds, Tangent Spaces and Vector
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.3B Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 207
4.3C Integration on Manifolds. . . . . . . . . . . . . . . . . . . . 220
5 Topological Methods
5.1 Brouwer Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . 243
5.1A Contractible Sets . . . . . . . . . . . . . . . . . . . . . . . . 250
5.2 Schauder Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . 252
5.2A Fixed Point Theorems for Noncompact Operators . . . . . 257
5.3 Classical Solutions of PDEs, Functional Setting . . . . . . . . . . . 264
5.4 Classical Solutions, Applications of Fixed Point Theorems . . . . . 268
5.5 Weak Solutions of PDEs, Functional Setting . . . . . . . . . . . . . 271
5.6 Weak Solutions of PDEs, Applications of Fixed Point Theorems. . 277
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vi Contents
5.7 Brouwer Topological Degree . . . . . . . . . . . . . . . . . . . . . . 283
5.7A Brouwer Topological Degree on Manifolds . . . . . . . . . . 293
5.8 Leray-Schauder Topological Degree . . . . . . . . . . . . . . . . . . 313
5.8A Global Bifurcation Theorem . . . . . . . . . . . . . . . . . . 331
5.8B Topological Degree for Generalized Monotone Operators . . 339
5.9 Weak Solutions of PDEs, Applications of Degree Theory . . . . . . 346
5.9A Weak Solutions of PDEs, Application of the Degree of
Generalized Monotone Mappings . . . . . . . . . . . . . . . 353
6 Monotonicity Methods
6.1 Theory of Monotone Operators . . . . . . . . . . . . . . . . . . . . 361
6.1A Browder and Leray–Lions Theorem. . . . . . . . . . . . . . 375
6.2 Weak Solutions of PDEs, Application of Monotone Operators . . . 381
6.2A Weak Solutions of PDEs, Application of Leray–Lions
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
6.3 Supersolutions, Subsolutions, Monotone Iterations . . . . . . . . . 391
6.3A Minorant Principle and Krein–Rutman Theorem . . . . . . 400
6.3B Supersolutions, Subsolutions and Topological Degree . . . . 413
6.4 Maximum Principle for ODEs . . . . . . . . . . . . . . . . . . . . . 421
6.5 Maximum Principle for PDEs . . . . . . . . . . . . . . . . . . . . . 428
7 Variational Methods
7.1 Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
7.2 Global Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
7.2A Supersolutions, Subsolutions and Global Extrema. . . . . . 464
7.2B Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . 467
7.3 Weak Solutions of PDEs, Applications of Global Minimizers . . . . 477
7.4 Mountain Pass Theorem . . . . . . . . . . . . . . . . . . . . . . . . 479
7.4A PseudogradientVector Fields in Banach Spaces . . . . . . . 489
7.5 Weak Solutions of PDEs, Applications of Mountain Pass
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
7.6 Saddle Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 499
7.6A Linking Theorem . . . . . . . . . . . . . . . . . . . . . . . . 508
7.7 Weak Solutions of PDEs, Applications of Saddle Point Theorem . 515
7.7A Weak Solutions of PDEs, Applications of General Saddle
Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 521
7.8 Relative Extrema and Lagrange Multipliers . . . . . . . . . . . . . 527
7.8A Lusternik–Schnirelmann Method . . . . . . . . . . . . . . . 540
7.8B Krasnoselski Potential Bifurcation Theorem . . . . . . . . . 554
8 Some Applications to Partial Differential Equations
8.1 Linear Evolution Equations and Semigroups . . . . . . . . . . . . . 565
8.2 Semilinear Evolution Equations . . . . . . . . . . . . . . . . . . . . 575
8.3 Linearization of Quasilinear PDEs and Fixed Point Theorems . . . 585
8.4 Equations on Riemann Manifolds . . . . . . . . . . . . . . . . . . . 595
Contents vii
Summary of Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
Typical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Comparison of Bifurcation Results . . . . . . . . . . . . . . . . . . . . . . . 617
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
Preface
In this second edition we have made minor revisions and added new material to
Chapter 8 and partly also to Chapters 2, 3 and 6. We quote from the Preface to
the first edition in order to clarify the original motivation behind publication of
this book:
“Therearemanybooksandmonographsdevotedtothemethodsofnonlinear
analysis and their applications. Typically, such a book is either dedicated to a
particulartopicandtreatsdetailswhicharedifficulttounderstandforastudent,or
itdealswithanapplicationtocomplicatednonlinearpartialdifferentialequations
in which a lot of technicalities are involved. In both cases it is very difficult for a
studenttogetorientedinthiskindofmaterialandtopickuptheideasunderlying
the main tools for treating the problems in question. The purpose of this book
is to describe the basic methods of nonlinear analysis and to illustrate them on
simple examples. Our aim is to motivate each method considered, to explain it
in a general form but in the simplest possible abstract framework, and finally, to
showitsapplication(typicallytoboundaryvalueproblemsforelementaryordinary
or partial differential equations). To keep the text free of technical details and
makeit accessible also to beginners we did notformulate some key assertionsand
illustrative examples in the most general form.
In order to make the text self-contained, we decided to comment on several
notions and statements in footnotes. To place the material from the footnotes
in the text could disturb a more advanced reader and make the exposition more
complicated.Inordertoemphasizetheroleofthestatementsinourexpositionwe
identifythemasTheorem,Proposition,LemmaandCorollary.However,thereader
shouldbe awareofthe factthat this byno means expressesthe importance ofthe
statement within the whole of mathematics. So, several times, we call important
theorems Propositions, Lemmas or Corollaries.
Although the book should primarily serve as a textbook for students on the
graduate level, it can be a suitable source for scientists and engineers who have
need of modern methods of nonlinear analysis.
At this point we would like to include a few words about our good friend,
colleagueandmentorSvatoplukFuˇc´ıktowhomwededicatethisbook.Hisworkin
thefieldofnonlinearanalysisiswellrecognizedandalthoughhediedin1979atthe
ix
x Preface
age of 34, he ranks among the most important and gifted Czech mathematicians
of the 20th century.
WewouldliketothankMarieBenediktov´aandJiˇr´ıBenediktforanexcellent
typesettingofthisbookinLATEX2ε,excellentfiguresandillustrationsaswellasfor
their valuable comments which improved the quality of the text. Our thanks be-
longalso to Eva Faˇsangov´a,GabrielaHolubov´a, Eva Kaspˇr´ıkov´aand PetrStehl´ık
for their careful reading of the manuscript and useful comments which have de-
creasedthenumberofourmistakesandmadethistextmorereadable.Ourspecial
thanksbelongtoJiˇr´ıJarn´ıkforcorrectionofourEnglish,RalphChillandHerbert
Leinfelder for their improvements of the text and methodological advice.”
The exposition ofthe material in the secondedition is attwo levels, visually
differentiated by different font sizes. The basic material is contained in the body
ofthefirstsevenchapters.Themoreadvancedmaterialiscontainedinappendices
to a number of sections and is presented in a smaller font size. The last chapter
containsselectedspecial materialfromthetheoryofpartialdifferential equations.
Thebasicmaterialisindependentofthemoreadvancedmaterial,isself-contained,
and can be read by students new to the subject. It should prepare an undergrad-
uate student in mathematics to read scientific papers in nonlinear analysis and
to understand applications of the methods presented to more complex problems.
Each chapter contains a number of exercises which should provoke reader’s cre-
ativity and help develop his or her own style of approaching problems. However,
the exercises play an additional role. They carry some of the technical material
thatwasomittedinsimplifyingsomeofthebasicproofs.Theyarethusanorganic
partofthe expositionforgraduatestudentswhoalreadyhaveexperiencewiththe
methods of nonlinear analysis and are interested in generalizations.
In the second edition we organize the material differently than in the first
edition. We dedicate an extra chapter to monotonicity methods, where we newly
include the elements of maximum principles both for ordinary differential equa-
tionsandpartialdifferentialequations.Mostofthematerialaboutboundaryvalue
problemsforpartialdifferentialequationsisnowsubsumedintothebodyofChap-
ters 5–7 as an immediate application of the methods presented there. The last
Chapter 8 contains technically more involved material which was not included in
the first edition. It deals with semigroups,semilinear evolutionpartial differential
equations,applicationoffixedpointmethodstosolvesomequasilinearpartialdif-
ferential equations as well as some partial differential equations on manifolds. In
particular, to prepare the reader for subsections on evolution partial differential
equations we extend some parts of Chapters 2 and 3.
We would like to thank Sonˇa K¨onigsmarkov´afor the typing the new text in
the second edition and to Jiˇr´ıBenedikt for creating new illustrations. Our special
thanksbelongtotheeditorialstaffofBirkh¨auser,inparticulartoSylviaLotrovsky
and Thomas Hempfling, for the agreeable collaboration.
Plzenˇ–Praha, Pavel Dra´bek
September 2012 Jaroslav Milota
