Exercises in Analysis: Part 2: Nonlinear Analysis

1272 Pages·2016·33.65 MB·English

by  Leszek Gasinski

Problem Books in Mathematics
Leszek Gasiński
Nikolaos S. Papageorgiou
Exercises in
Analysis
Part 2: Nonlinear Analysis
Problem Books in Mathematics
SeriesEditors:
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH03755
USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/714

Leszek Gasin´ski • Nikolaos S. Papageorgiou
Exercises in Analysis
Part 2: Nonlinear Analysis
123
LeszekGasin´ski NikolaosS.Papageorgiou
FacultyofMathematics DepartmentofMathematics
andComputerScience NationalTechnicalUniversity
JagiellonianUniversity Athens,Greece
Krako´w,Poland
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-319-27815-5 ISBN978-3-319-27817-9 (eBook)
DOI10.1007/978-3-319-27817-9
LibraryofCongressControlNumber:2015958668
MathematicsSubjectClassification(2010):00A07;46-XX;30LXX,28-XX,60G46
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
This is the second part of the project initiated with “Exercises in
Analysis. Part 1.” In that volume we focused on some classical
tools from Analysis. Specifically we considered Metric Spaces, Mea-
sure Theory, the interplay between Measure Theory and Topology,
and Functional Analysis (Banach Spaces). In the present volume we
turn our attention to the main topics of Nonlinear Analysis, which are
useful in applications. So, we deal with the following subjects:
1. Function Spaces
2. Nonlinear and Multivalued Maps
3. Smooth and Nonsmooth Calculus
4. Degree Theory and Fixed Point Theory
5. Variational and Topological Methods
Each one of the above topics is a separate chapter. Each chapter
startswithacomprehensivepresentationofthetheoryandthenhason
the average 200 problems together with their solutions. The problems
are marked with (cid:2), (cid:2)(cid:2), and (cid:2)(cid:2)(cid:2) according to the level of difficulty of
the problem ((cid:2) for easy, (cid:2)(cid:2) for reasonable, and (cid:2)(cid:2)(cid:2) for difficult). We
believe that people using tools of nonlinear analysis will find useful
information either in the summary of the theory or in the problems.
The topics studied in this volume cover a large part of what is known
as Nonlinear Analysis.
Once again the authors express their gratitude to the editor Mrs.
ElizabethLoewforhercooperationandstrongmoralsupportinrealiz-
ingthisdifficultprojectandforKrystynaGasin´skaandRafa(cid:3)lKawafor
preparing the beautiful drawings of branches decorating the solutions
of the problems in both volumes.
Krako´w, Poland Leszek Gasin´ski
Athens, Greece Nikolaos S. Papageorgiou
V

Contents
1 Function Spaces 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Lebesgue–Bochner Spaces . . . . . . . . . . . . . 9
1.1.3 BV-Functions, Absolutely Continuous Func-
tions, Spaces of Measures . . . . . . . . . . . . . 15
1.1.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . 30
1.1.5 Auxiliary Notions . . . . . . . . . . . . . . . . . 42
1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 255
2 Nonlinear and Multivalued Maps 257
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 257
2.1.1 Compact, Completely Continuous, and Proper
Maps . . . . . . . . . . . . . . . . . . . . . . . . 257
2.1.2 Multifunctions . . . . . . . . . . . . . . . . . . . 265
2.1.3 Maximal Monotone Maps and Generalizations . 283
2.1.4 Accretive Maps . . . . . . . . . . . . . . . . . . . 293
2.1.5 Miscellaneous Results . . . . . . . . . . . . . . . 298
2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 300
2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 340
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 487
3 Smooth and Nonsmooth Calculus 489
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 489
3.1.1 Gˆateaux and Fr´echet Derivatives . . . . . . . . . 489
3.1.2 Convex Functionals and Variational Inequalities 502
VII
VIII CONTENTS
3.1.3 Locally Lipschitz Functions . . . . . . . . . . . . 517
3.1.4 Γ-Convergence and Relaxation . . . . . . . . . . 522
3.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 535
3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 575
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 731
4 Degree Theory and Fixed Point Theory 733
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 733
4.1.1 Degree Theory . . . . . . . . . . . . . . . . . . . 733
4.1.2 Metric Fixed Point Theory . . . . . . . . . . . . 755
4.1.3 Topological Fixed Point Theory . . . . . . . . . . 760
4.1.4 Order Fixed Point Theory . . . . . . . . . . . . . 769
4.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 778
4.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 817
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 955
5 Variational and Topological Methods 957
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 957
5.1.1 Minimization Methods . . . . . . . . . . . . . . . 957
5.1.2 Minimax Methods for Critical Points . . . . . . . 970
5.1.3 Morse Theory: Critical Groups . . . . . . . . . . 989
5.1.4 Dirichlet Elliptic Problems . . . . . . . . . . . . 996
5.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 1005
5.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 1055
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 1229
List of Symbols 1231
Index 1243
Chapter 1
Function Spaces
1.1 Introduction
1.1.1 Lp-Spaces
Definition 1.1
Let (Ω,Σ,μ) be a measure space, 1 (cid:2) p < +∞ and
(cid:2) (cid:3) (cid:4)
Lp(Ω) = f: Ω −→ R : f is measurable and |f(ω)|pdμ < +∞ .
Ω
Consider the equivalence relation ∼ defined by
f ∼ g if and only if f(ω) = g(ω) for μ-a.a. ω ∈ Ω
and then let Lp(Ω) = Lp(Ω)/∼. On Lp(Ω) we consider the norm
(cid:5)(cid:3) (cid:6)
1
p
(cid:6)f(cid:6) = |f(ω)|pdμ .
p
Ω
For p = +∞, we define
(cid:2)
L∞(Ω) = f: Ω −→ R : f is measurable and there exists M > 0,
(cid:4)
(cid:7) (cid:8)
such that μ ω ∈ Ω : |f(ω)| > M = 0 .
©Springer International Publishing Switzerland 2016 1
L. Gasin´ski, N.S. Papageorgiou, Exercises in Analysis: Part 2,
ProblemBooksin Mathematics, DOI 10.1007/978-3-319-27817-9 1