This monograph explores the concept of the Brouwer degree and its continuing impact on the development of important areas of nonlinear analysis. The authors define the degree using an analytical approach proposed by Heinz in 1959 and further developed by Mawhin in 2004, linking it to the Kronecker index and employing the language of differential forms. The chapters are organized so that they can be approached in various ways depending on the interests of the reader. Unifying this structure is the central role the Brouwer degree plays in nonlinear analysis, which is illustrated with existence, surjectivity, and fixed point theorems for nonlinear mappings. Special attention is paid to the computation of the degree, as well as to the wide array of applications, such as linking, differential and partial differential equations, difference equations, variational and hemivariational inequalities, game theory, and mechanics. Each chapter features bibliographic and historical notes, and the final chapter examines the full history. Brouwer Degree will serve as an authoritative reference on the topic and will be of interest to professional mathematicians, researchers, and graduate students.
Author(s): George Dinca, Jean Mawhin
Series: Progress in Nonlinear Differential Equations and Their Applications , 95
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 447
City: Cham
Tags: Kronecker Index, Brouwer Degree, Fixed Points, Zeros, Bifurcation, Leray-Schauder Degree
Preface
Contents
1 The Kronecker Index and the Brouwer Degree
1.1 The Kronecker Index
1.1.1 Definition
1.1.2 The Winding Number in R2
1.1.3 The Gauss Linking Number of Two Curves in R3
1.1.4 The Case Where f : ∂D Rn →Sn-1
1.1.5 A Minimum Problem from Ginzburg-Landau's Theory
1.1.6 The Kronecker Integral as a Volume Integral
1.1.7 Historical Notes
1.1.8 Bibliographical Notes
1.2 The Brouwer Degree
1.2.1 Smooth Mappings
1.2.2 Continuous Mappings
1.2.3 The Case Where f : ∂D →Rn {0}
1.2.4 The Case of an Unbounded Set D
1.2.5 Uniqueness of the Degree
1.2.6 The Brouwer Index
1.2.7 Generalized Implicit Function Theorem
1.2.8 Historical Notes
1.2.9 Bibliographical Notes
1.3 Degree in Finite-Dimensional Vector Spaces
1.3.1 Composition with Linear Isomorphisms
1.3.2 Definitions and Basic Properties
1.3.3 The Brouwer Index
1.3.4 Reduction Formulas
1.3.5 Computation of the Index in Degenerate Case
1.3.6 Historical Notes
1.3.7 Bibliographical Notes
1.4 Retracts, Fixed Points, Vector Fields, Linking
1.4.1 Retracts and Retractions
1.4.2 Fixed Point Theorems
1.4.3 Toward a Fixed Point Index in Retracts of Rn
1.4.4 Fixed Point Theorems in Cones
1.4.5 The Hairy Ball Theorem
1.4.6 Linking and Critical Points
1.4.7 Historical Notes
1.4.8 Bibliographical Notes
2 Continuation, Existence and Bifurcation
2.1 Continuation Theorems
2.1.1 Extended Homotopy Invariance Property
2.1.2 The Leray–Schauder Continuation Theorem
2.1.3 The Method of a Priori Bounds
2.1.4 The Leray–Schauder Alternative
2.1.5 The Case of an Unbounded Set
2.1.6 Historical Notes
2.1.7 Bibliographical Notes
2.2 Zeros, Surjectivity and Fixed Points of Mappings
2.2.1 Perturbations of Linear Mappings
2.2.2 Surjectivity of Some Coercive Mappings
2.2.3 Monotone Mappings
2.2.4 Boundary Conditions for the Existence of a Zero
2.2.5 Fixed Points of Expansive-Compressive-Type Mappings
2.2.6 Fixed Points of Poincaré–Miranda-Type Mappings
2.2.7 Uniqueness of Zeros and of Fixed Points
2.2.8 Historical Notes
2.2.9 Bibliographical Notes
2.3 Characteristic Values, Index and Bifurcation
2.3.1 Characteristic Values of Couples of Linear Mappings
2.3.2 Multiplicity of a Characteristic Value
2.3.3 The Leray–Schauder Formula for the Brouwer Index
2.3.4 Bifurcation Points
2.3.5 Bifurcation Through Linearization
2.3.6 Global Structure of Bifurcation Branches
2.3.7 Bifurcation from Infinity
2.3.8 Historical Notes
2.3.9 Bibliographical Notes
2.4 Set-Valued Mappings
2.4.1 The Kakutani Fixed Point Theorem
2.4.2 The von Neumann Minimax Theorem
2.4.3 The Nash Equilibrium of a Non-cooperative Game
2.4.4 Historical Notes
2.4.5 Bibliographical Notes
3 Infinite-Dimensional Problems
3.1 Fixed Point Theorems
3.1.1 The Ky Fan Fixed Point Theorem
3.1.2 The Tychonov and Schauder Fixed Point Theorems
3.1.3 Historical Notes
3.1.4 Bibliographical Notes
3.2 The KKM Theorem and Variational Inequalities
3.2.1 The KKM Theorem and Its Generalizations
3.2.2 Variational Inequalities
3.2.3 Variational Inequalities for Monotone Operators
3.2.4 Hemivariational Inequalities
3.2.5 Historical Notes
3.2.6 Bibliographical Notes
3.3 Equations in Reflexive Banach Spaces
3.3.1 The Surjectivity of Monotone Coercive Operators
3.3.2 A Monotone Quasilinear Dirichlet Problem
3.3.3 A Non Monotone Quasilinear Dirichlet Problem
3.3.4 Historical Notes
3.3.5 Bibliograhical Notes
3.4 Stationary Navier-Stokes Equations
3.4.1 The Problem
3.4.2 Variational Formulation
3.4.3 Existence of a Solution
3.4.4 Uniqueness of the Solution
3.4.5 Historical Notes
3.4.6 Bibliographical Notes
3.5 Toward the Leray-Schauder Degree
3.5.1 Extending the Brouwer Degree to Normed Spaces
3.5.2 The Schauder Projection and Approximation Theorems
3.5.3 Definition of the Leray-Schauder Degree
3.5.4 Justification of the Definition
3.5.5 Historical Notes
3.5.6 Bibliographical Notes
4 Difference Equations
4.1 Periodic Solutions of First Order Equations
4.1.1 Periodic Solutions
4.1.2 Bounded Nonlinearities
4.1.3 Lower and Upper Solutions
4.1.4 Multiplicity Results of the Ambrosetti-Prodi Type
4.1.5 Historical Notes
4.1.6 Bibliographical Notes
4.2 Applications to Population Dynamics
4.2.1 Nonlinearities Bounded from Below or Above
4.2.2 Lotka-Volterra-Type Systems
4.2.3 Historical Notes
4.2.4 Bibliographical Notes
4.3 The Dirichlet Problem for Second Order Equations
4.3.1 Formulation and Spectrum of the Linear Part
4.3.2 Bifurcation from the Trivial Solution
4.3.3 Multiple Solutions Near Resonance
4.3.4 Nonlinearities Bounded from Below or Above
4.3.5 Lower and Upper Solutions
4.3.6 Multiplicity Results of the Ambrosetti-Prodi Type
4.3.7 Historical Notes
4.3.8 Bibliographical Notes
5 Periodic Solutions of Differential Systems
5.1 The Poincaré Operator
5.1.1 Initial Value and Periodic Problems
5.1.2 Bounded Perturbations of Some Linear Systems
5.1.3 The Stampacchia Method
5.1.4 The Krasnosel'skii-Perov Existence Theorem
5.1.5 Historical Notes
5.1.6 Bibliographical Notes
5.2 Guiding Functions
5.2.1 Definition and Preliminaries
5.2.2 Systems with Continuable Solutions
5.2.3 Systems with Non-continuable Solutions
5.2.4 Generalized Guiding Functions
5.2.5 Historical Notes
5.2.6 Bibliographical Notes
5.3 Evolution Complementarity Systems
5.3.1 Equivalent Formulations
5.3.2 The Cauchy Problem
5.3.3 The Poincaré Operator
5.3.4 Guiding Functions
5.3.5 Historical Notes
5.3.6 Bibliographical Notes
6 Two-Dimensional Problems
6.1 Lower and Upper Solutions for Second Order Equations
6.1.1 Definitions
6.1.2 Existence Theorem
6.1.3 The Brouwer Degree of the Poincaré Operator
6.1.4 Historical Notes
6.1.5 Bibliographical Notes
6.2 Stability and Index of Periodic Solutions
6.2.1 Planar Periodic Systems
6.2.2 Second Order Differential Equations
6.2.3 Second Order Equations with Convex Nonlinearity
6.2.4 Second Order Equations with Periodic Nonlinearity
6.2.5 Historical Notes
6.2.6 Bibliographical Notes
6.3 Planar Differential Systems
6.3.1 Autonomous Systems
6.3.2 Nonresonant Forced Systems
6.3.3 Polar-Type Coordinates
6.3.4 Resonant Forced Systems
6.3.5 Asymmetric Piecewise-Linear Oscillators
6.3.6 Historical Notes
6.3.7 Bibliographical Notes
6.4 Computing the Degree in Dimension Two
6.4.1 The Kronecker Index on a Closed Simple Curve
6.4.2 Factorized Multilinear Mappings
6.4.3 Using Sturm Sequences
6.4.4 Holomorphic Functions
6.4.5 Historical Notes
6.4.6 Bibliographical Notes
6.5 Application to Stability and Control
6.5.1 Stability Conditions and a Zero Exclusion Principle
6.5.2 The First Schur Transform of a Polynomial
6.5.3 Routh–Hurwitz Stable Family of Polynomials
6.5.4 The Second Schur Transform of a Polynomial
6.5.5 Schur–Cohn Stable Family of Polynomials
6.5.6 Historical Notes
6.5.7 Bibliographical Notes
7 The Degree of Some Classes of Mappings
7.1 Homogeneous Polynomial Mappings
7.1.1 Cartesian Products of Mappings
7.1.2 Homogeneous Polynomial Mappings
7.1.3 Historical Notes
7.1.4 Bibliographical Notes
7.2 Orientation-Preserving Mappings
7.2.1 Definitions and Main Properties
7.2.2 Monotone Mappings
7.2.3 Holomorphic Mappings
7.2.4 Quaternionic Monomials
7.2.5 Historical Notes
7.2.6 Bibliographical Notes
7.3 Symmetrical Mappings
7.3.1 Odd Mappings
7.3.2 Elliptic Differential Operators
7.3.3 The Lusternik-Schnirelmann Covering Theorem
7.3.4 Measure of Non-compactness of the Unit Sphere
7.3.5 The Krasnosel'skii Genus
7.3.6 S1-Equivariant Mappings
7.3.7 A Reduction Formula for S1-Equivariant Mappings
7.3.8 Historical Notes
7.3.9 Bibliographical Notes
7.4 One-to-One and Composed Mappings
7.4.1 Invariance of Domain and of Dimension
7.4.2 The Banach-Mazur Theorem
7.4.3 The Leray Product Formula
7.4.4 The Jordan-Brouwer Separation Theorem
7.4.5 Degree of One-to-One Mappings
7.4.6 Deformation of an Elastic Body
7.4.7 A Class of Mappings with Orientation-Preserving Character
7.4.8 Historical Notes
7.4.9 Bibliographical Notes
8 History of the Brouwer Fixed Point Theorem
8.1 Discovery, Publication and Anticipations
8.1.1 Brouwer
8.1.2 Hadamard
8.1.3 Poincaré
8.1.4 Bohl
8.1.5 Bibliographical Notes
8.2 New Proofs and Infinite-Dimensional Extensions
8.2.1 Alexander, Birkhoff and Kellogg
8.2.2 Schauder and Tychonoff
8.2.3 Sperner
8.2.4 Knaster, Kuratowski and Mazurkiewicz
8.2.5 Bibliographical Notes
8.3 Game Theory, Economics and Computation
8.3.1 von Neumann and Kakutani
8.3.2 Nash
8.3.3 Scarf
8.3.4 Kellogg, Li and Yorke
8.3.5 Bibliographical Notes
8.4 Topology or Not Topology
8.4.1 Hammerstein and Golomb
8.4.2 Cinquini, Miranda and Scorza-Dragoni
8.4.3 Bibliographical Notes
8.5 Variational Inequalities, Simple Proofs, Applications
8.5.1 Hartman, Stampacchia and Karamardian
8.5.2 The Quest for Elementary Proofs
8.5.3 All-Out Applications
8.5.4 Bibliographical Notes
Bibliography
Index of Names
Index
