Nonlinear Functional Analysis and its Applications: II/B: Nonlinear Monotone Operators (Nonlinear Functional Analysis & Its Applications)

✍ Scribed by Zeidler E.

• Publisher: Springer
• Year: 1989
• Tongue: English
• Leaves: 751
• Edition: 1990
• Category: Library

✦ Synopsis

This is the second of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self -contained and accessible to the nonspecialist. Part II concerns the theory of monotone operators. It is divided into two subvolumes, II/A and II/B, which form a unit. The present Part II/A is devoted to linear monotone operators. It serves as an elementary introduction to the modern functional analytic treatment of variational problems, integral equations, and partial differential equations of elliptic, parabolic and hyperbolic type. This book also represents an introduction to numerical functional analysis with applications to the Ritz method along with the method of finite elements, the Galerkin methods, and the difference method. Many exercises complement the text. The theory of monotone operators is closely related to Hilbert's rigorous justification of the Dirichlet principle, and to the 19th and 20th problems of Hilbert which he formulated in his famous Paris lecture in 1900, and which strongly influenced the development of analysis in the twentieth century.

✦ Table of Contents

Front cover
David Hilbert (1862-1943)
Title page
Date-line
Dedication
Preface to Part II/B
Contents (Part II/B)
Contents (Part II/A)
GENERALIZATION TO NONLINEAR STATIONARY PROBLEMS
Basic Ideas of the Theory of Monotone Operators
CHAPTER 25. Lipschitz Continuous, Strongly Monotone Operators, the Projection Iteration Method, and Monotone Potential Operators
§25.1. Sequences of $k$-Contractive Operators
§25.2. The Projection Iteration Method for $k$-Contractive Operators
§25.3. Monotone Operators
§25.4. The Main Theorem on Strongly Monotone Operators, and the Projection Iteration Method
§25.5. Monotone and Pseudomonotone Operators, and the Calculus of Variations
§25.6. The Main Theorem on Monotone Potential Operators
§25.7. The Main Theorem on Pseudomonotone Potential Operators
§25.8. Application to the Main Theorem on Quadratic Variational Inequalities
§25.9. Application to Nonlinear Stationary Conservation Laws
§25.10. Projection Iteration Method for Conservation Laws
§25.11. The Main Theorem on Nonlinear Stationary Conservation Laws
§25.12. Duality Theory for Conservation Laws and Two-sided a posteriori Error Estimates for the Ritz Method
§25.13. The Kacanov Method for Stationary Conservation Laws
§25.14. The Abstract Kacanov Method for Variational Inequalities
CHAPTER 26. Monotone Operators and Quasi-Linear Elliptic Differential Equations
§26.1. Hemicontinuity and Demicontinuity
§26.2. The Main Theorem on Monotone Operators
§26.3. The Nemyckii Operator
§26.4. Generalized Gradient Method for the Solution of the Galerkin Equations
§26.5. Application to Quasi-Linear Elliptic Differential Equations of Order $2m$
§26.6. Proper Monotone Operators and Proper Quasi-Linear Elliptic Differential Operators
CHAPTER 27. Pseudomonotone Operators and Quasi-Linear Elliptic Differential Equations
§27.1. The Conditions ($M$) and ($S$), and the Convergence of the Galerkin Method
§27.2. Pseudomonotone Operators
§27.3. The Main Theorem on Pseudomonotone Operators
§27.4. Application to Quasi-Linear Elliptic Differential Equations
§27.5. Relations Between Important Properties of Nonlinear Operators
§27.6. Dual Pairs of B-Spaces
§27.7. The Main Theorem on Locally Coercive Operators
§27.8. Application to Strongly Nonlinear Differential Equations
CHAPTER 28. Monotone Operators and Hammerstein Integral Equations
§28.1. A Factorization Theorem for Angle-Bounded Operators
§28.2. Abstract Hammerstein Equations with Angle-Bounded Kernel Operators
§28.3. Abstract Hammerstein Equations with Compact Kernel Operators
§28.4. Application to Hammerstein Integral Equations
§28.5. Application to Semilinear Elliptic Differential Equations
CHAPTER 29. Noncoercive Equations, Nonlinear Fredholm Alternatives, Locally Monotone Operators, Stability, and Bifurcation
§29.1. Pseudoresolvent, Equivalent Coincidence Problems, and the Coincidence Degree
§29.2. Fredholm Alternatives for Asymptotically Linear, Compact Perturbations of the Identity
§29.3. Application to Nonlinear Systems of Real Equations
§29.4. Application to Integral Equations
§29.5. Application to Differential Equations
§29.6. The Generalized Antipodal Theorem
§29.7. Fredholm Alternatives for Asymptotically Linear ($S$)-Operators
§29.8. Weak Asymptotes and Fredholm Alternatives
§29.9. Application to Semilinear Elliptic Differential Equations of the Landesman-Lazer Type
§29.10. The Main Theorem on Nonlinear Proper Fredholm Operators
§29.11. Locally Strictly Monotone Operators
§29.12. Locally Regularly Monotone Operators, Minima, and Stability
§29.13. Application to the Buckling of Beams
§29.14. Stationary Points of Functionals
§29.15. Application to the Principle of Stationary Action
§29.16. Abstract Statical Stability Theory
§29.17. The Continuation Method
§29.18. The Main Theorem of Bifurcation Theory for Fredholm Operators of Variational Type
§29.19. Application to the Calculus of Variations
§29.20. A General Bifurcation Theorem for the Euler Equations and Stability
§29.21. A Local Multiplicity Theorem
§29.22. A Global Multiplicity Theorem
GENERALIZATION TO NONLINEAR NONSTATIONARY PROBLEMS
CHAPTER 30. First-Order Evolution Equations and the Galerkin Method
§30.1. Equivalent Formulations of First-Order Evolution Equations
§30.2. The Main Theorem on Monotone First-Order Evolution Equations
§30.3. Proof of the Main Theorem
§30.4. Application to Quasi-Linear Parabolic Differential Equations of Order $2m$
§30.5. The Main Theorem on Semibounded Nonlinear Evolution Equations
§30.6. Application to the Generalized Korteweg-de Vries Equation
CHAPTER 31. Maximal Accretive Operators, Nonlinear Nonexpansive Semigroups, and First-Order Evolution Equations
§31.1. The Main Theorem
§31.2. Maximal Accretive Operators
§31.3. Proof of the Main Theorem
§31.4. Application to Monotone Coercive Operators on B-Spaces
§31.5. Application to Quasi-Linear Parabolic Differential Equations
§31.6. A Look at Quasi-Linear Evolution Equations
§31.7. A Look at Quasi-Linear Parabolic Systems Regarded as Dynamical Systems
CHAPTER 32. Maximal Monotone Mappings
§32.1. Basic Ideas
§32.2. Definition of Maximal Monotone Mappings
§32.3. Typical Examples for Maximal Monotone Mappings
§32.4. The Main Theorem on Pseudomonotone Perturbations of Maximal Monotone Mappings
§32.5. Application to Abstract Hammerstein Equations
§32.6. Application to Hammerstein Integral Equations
§32.7. Application to Elliptic Variational Inequalities
§32.8. Application to First-Order Evolution Equations
§32.9. Application to Time-Periodic Solutions for Quasi-Linear Parabolic Differential Equations
§32.10. Application to Second-Order Evolution Equations
§32.11. Regularization of Maximal Monotone Operators
§32.12. Regularization of Pseudomonotone Operators
§32.13. Local Boundedness of Monotone Mappings
§32.14. Characterization of the Surjectivity of Maximal Monotone Mappings
§32.15. The Sum Theorem
§32.16. Application to Elliptic Variational Inequalities
§32.17. Application to Evolution Variational Inequalities
§32.18. The Regularization Method for Nonuniquely Solvable Operator Equations
§32.19. Characterization of Linear Maximal Monotone Operators
§32.20. Extension of Monotone Mappings
§32.21. 3-Monotone Mappings and Their Generalizations
§32.22. The Range of Sum Operators
§32.23. Application to Hammerstein Equations
§32.24. The Characterization of Nonexpansive Semigroups in H-Spaces
CHAPTER 33. Second-Order Evolution Equations and the Galerkin Method
§33.1. The Original Problem
§33.2. Equivalent Formulations of the Original Problem
§33.3. The Existence Theorem
§33.4. Proof of the Existence Theorem
§33.5. Application to Quasi-Linear Hyperbolic Differential Equations
§33.6. Strong Monotonicity, Systems of Conservation Laws, and Quasi-Linear Symmetric Hyperbolic Systems
§33.7. Three Important General Phenomena
§33.8. The Formation of Shocks
§33.9. Blowing-Up Effects
§33.10. Blow-Up of Solutions for Semilinear Wave Equations
§33.11. A Look at Generalized Viscosity Solutions of Hamilton-Jacobi Equations
GENERAL THEORY OF DISCRETIZATION METHODS
CHAPTER 34. Inner Approximation Schemes, $A$-Proper Operators, and the Galerkin Method
§34.1. Inner Approximation Schemes
§34.2. The Main Theorem on Stable Discretization Methods with Inner Approximation Schemes
§34.3. Proof of the Main Theorem
§34.4. Inner Approximation Schemes in H-Spaces and the Main Theorem on Strongly Stable Operators
§34.5. Inner Approximation Schemes in B-Spaces
§34.6. Application to the Numerical Range of Nonlinear Operators
CHAPTER 35. External Approximation Schemes, $A$-Proper Operators, and the Difference Method
§35.1. External Approximation Schemes
§35.2. Main Theorem on Stable Discretization Methods with External Approximation Schemes
§35.3. Proof of the Main Theorem
§35.4. Discrete Sobolev Spaces
§35.5. Application to Difference Methods
§35.6. Proof of Convergence
CHAPTER 36. Mapping Degree for $A$-Proper Operators
§36.1. Definition of the Mapping Degree
§36.2. Properties of the Mapping Degree
§36.3. The Antipodal Theorem for $A$-Proper Operators
§36.4. A General Existence Principle
Appendix
References
List of Symbols
List of Theorems
List of the Most Important Definitions
List of Schematic Overviews
List of Important Principles
Index
Back cover