非线性泛函分析及其应用：非线性单调算子（第2B卷）
目 录内容简介
Preface to Part II/B
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 Ka6anov Method for Stationary Conservation Laws
25.14. The Abstract Ka6anov 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 Differeh，：e 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
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 Ka6anov Method for Stationary Conservation Laws
25.14. The Abstract Ka6anov 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 Differeh，：e 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
目 录内容简介
这部书讲清楚了泛函分析理论对数学其他领域的应用。例如，第2A卷讲述线性单调算子。他从椭圆型方程的边值问题出发，讲问题的古典解，由于具体物理背景的需要，问题须作进一步推广，而需要讨论问题的广义解。这种方法背后的分析原理是什么？其实就是完备化思想的一个应用！将古典问题所依赖的连续函数空间，完备化成为Sobolev空间，则可讨论问题的广义解。在这种讨论中间，我们可以看到Hilbert空间的作用。书中不仅有这种理论讨论，而且还讲了怎样计算问题的近似解（Ritz方法）。
这部书讲清楚了分析理论在诸多领域（如物理学、化学、生物学、工程技术和经济学等等）的广泛应用。例如，第3卷讲解变分方法和优化，它从函数极值问题开始，讲到变分问题及其对于Euler微分方程和Hammerstein积分方程的应用；讲到优化理论及其对于控制问题（如庞特里亚金极大值原理）、统计优化、博弈论、参数识别、逼近论的应用；讲了凸优化理论及应用；讲了极值的各种近似计算方法。比如第4卷，讲物理应用，写作原理是：由物理事实到数学模型；由数学模型到数学结果；再由数学结果到数学结果的物理解释；最后再回到物理事实。
再次，该书由浅入深地讲透了基本理论的发展历程及走向，它既讲清楚了所涉及学科的具体问题，也讲清楚了其背后的数学原理及其作用。数学理论讲得也非常深入，例如，不动点理论，就从Banach不动点定理讲到Schauder不动点定理，以及Bourbaki—Kneser不动点定理等等。
这套书的写作起点很低，具备本科数学水平就可以读；应用都是从最简单情形入手，应用领域的读者也可以读；全书材料自足，各部分又尽可能保持独立；书后附有极其丰富的参考文献及一些文献评述；该书文字优美，引用了许多大师的格言，读之你会深受启发。这套书的优点不胜枚举，每个与数理学科相关的人，搞理论的，搞应用的，搞研究的，搞教学的，都可读该书，哪怕只是翻一翻，都不会空手而返！
这部书讲清楚了分析理论在诸多领域（如物理学、化学、生物学、工程技术和经济学等等）的广泛应用。例如，第3卷讲解变分方法和优化，它从函数极值问题开始，讲到变分问题及其对于Euler微分方程和Hammerstein积分方程的应用；讲到优化理论及其对于控制问题（如庞特里亚金极大值原理）、统计优化、博弈论、参数识别、逼近论的应用；讲了凸优化理论及应用；讲了极值的各种近似计算方法。比如第4卷，讲物理应用，写作原理是：由物理事实到数学模型；由数学模型到数学结果；再由数学结果到数学结果的物理解释；最后再回到物理事实。
再次，该书由浅入深地讲透了基本理论的发展历程及走向，它既讲清楚了所涉及学科的具体问题，也讲清楚了其背后的数学原理及其作用。数学理论讲得也非常深入，例如，不动点理论，就从Banach不动点定理讲到Schauder不动点定理，以及Bourbaki—Kneser不动点定理等等。
这套书的写作起点很低，具备本科数学水平就可以读；应用都是从最简单情形入手，应用领域的读者也可以读；全书材料自足，各部分又尽可能保持独立；书后附有极其丰富的参考文献及一些文献评述；该书文字优美，引用了许多大师的格言，读之你会深受启发。这套书的优点不胜枚举，每个与数理学科相关的人，搞理论的，搞应用的，搞研究的，搞教学的，都可读该书，哪怕只是翻一翻，都不会空手而返！
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