非线性泛函分析及其应用（第1卷）：不动点定理
目 录内容简介
Preface to the Second Corrected Printing
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PRINCIPLES
CHAPTER 1
The Banach Fixed-Point Theorem and lterative Methods
1.1. The Banach Fixed-Point Theorem
1.2. Continuous Dependence on a Parameter
1.3. The Significance of the Banach…
查看完整
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PRINCIPLES
CHAPTER 1
The Banach Fixed-Point Theorem and lterative Methods
1.1. The Banach Fixed-Point Theorem
1.2. Continuous Dependence on a Parameter
1.3. The Significance of the Banach…
查看完整
目 录内容简介
首先，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）。
其次，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…
查看完整
其次，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…
查看完整
目 录内容简介
Preface to the Second Corrected Printing
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PRINCIPLES
CHAPTER 1
The Banach Fixed-Point Theorem and lterative Methods
1.1. The Banach Fixed-Point Theorem
1.2. Continuous Dependence on a Parameter
1.3. The Significance of the Banach Fixed-Point Theorem
1.4. Applications to Nonlinear Equations
1.5. Accelerated Convergence and Newtons Method
1.6. The Picard-Lindel6fTheorem
1.7. The Main Theorem for Iterative Methods for Linear Operator
Equations
1.8. Applications to Systems of Linear Equations
1.9. Applications to Linear Integral Equations
CHAPTER 2
The Schauder Fixed-Point Theorem and Compactness
2.1. Extension Theorem
2.2. Retracts
2.3. The Brouwer Fixed-Point Theorem
2.4. Existence Principle for Systems of Equations
2.5. Compact Operators
2.6. The Schauder Fixed-Point Theorem
2.7. Peanos Theorem
2.8. Integral Equations with Small Parameters
2.9. Systems of Integral Equations and Semilinear Differential
Equations
2.10. A General Strategy
2.11. Existence Principle for Systems of Inequalities
APPLICATIONS OF THE FUNDAMENTAL
FIXED-POINT PRINCIPLES
CHAPTER 3
Ordinary Differential Equations in B-spaces
3.1. Integration of Vector Functions of One Real Variable t
3.2. Differentiation of Vector Functions of One Real Variable t
3.3. Generalized Picard-Lindeltf Theorem
3.4. Generalized Peano Theorem
3.5. Gronwalrs Lemma
3.6. Stability of Solutions and Existence of Periodic Solutions
3.7. Stability Theory and Plane Vector Fields, Electrical Circuits,
Limit Cycles
3.8. Perspectives
CHAPTER 4
Differential Calculus and the Implicit Function Theorem
4.1. Formal Differential Calculus
4.2. The Derivatives of Frtchet and Giteaux
4.3. Sum Rule, Chain Rule, and Product Rule
4.4. Partial Derivatives
4.5. Higher Differentials and Higher Derivatives
4.6. Generalized Taylors Theorem
4.7. The Implicit Function Theorem
4.8. Applications of the Implicit Function Theorem
4.9. Attracting and Repelling Fixed Points and Stability
4.10. Applications to Biological Equilibria
4.11. The Continuously Differentiable Dependence of the Solutions of
Ordinary Differential Equations in B-spaces on the Initial Values
and on the Parameters
4.12. The Generalized Frobenius Theorem and Total Differential
Equations
4.13. Diffeomorphisms and the Local Inverse Mapping Theorem
4.14. Proper Maps and the Global Inverse Mapping Theorem
4.15. The Surjective Implicit Function Theorem
4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank
Theorem
4.17. A Look at Manifolds
4.18. Submersions and a Look at the Sard-Smale Theorem
4.19. The Parametrized Sard Theorem and Constructive Fixed-Point
Theory
CHAPTER 5
Newtons Method
5.1. A Theorem on Local Convergence
5.2. The Kantorovi Semi-Local Convergence Theorem
CHAPTER 6
Continuation with Respect to a Parameter
6.1. The Continuation Method for Linear Operators
6.2. B-spaces of H61der Continuous Functions
6.3. Applications to Linear Partial Differential Equations
6.4. Functional-Analytic Interpretation of the Existence Theorem and
its Generalizations
6.5. Applications to Semi-linear Differential Equations
6.6. The Implicit Function Theorem and the Continuation Method
6.7. Ordinary Differential Equations in B-spaces and the Continuation
Method
6.8. The Leray-Schauder Principle
6.9. Applications to Quasi-linear Elliptic Differential Equations
CHAPTER 7
Positive Operators
7. I. Ordered B-spaces
7.2. Monotone Increasing Operators
7.3. The Abstract Gronwall Lemma and its Applications to Integral
Inequalities
7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability
7.5. Applications
7.6. Minorant Methods and Positive Eigensolutions
7.7. Applications
7.8. The Krein-Rutman Theorem and its Applications
7.9. Asymptotic Linear Operators
7.10. Main Theorem for Operators of Monotone Type
7.11. Application to a Heat Conduction Problem
7.12. Existence of Three Solutions
7.13. Main Theorem for Abstract Hammerstein Equations in Ordered
B-spaces
7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation,
Stability, and the Nonlinear Krein-Rutman Theorem
7.15. Applications to Hammerstein Integral Equations
7.16. Applications to Semi-linear Elliptic Boundary-Value Problems
7.17. Application to Elliptic Equations with Nonlinear Boundary
Conditions
7.18. Applications to Boundary Initial-Value Problems for Parabolic
Differential Equations and Stability
CHAPTER 8
Analytic Bifurcation Theory
8.1. A Necessary Condition for Existence of a Bifurcation Point
8.2. Analytic Operators
8.3. An Analytic Majorant Method
8.4. Fredholm Operators
8.5. The Spectrum of Compact Linear Operators
（Riesz-Schauder Theory）
8.6. The Branching Equations of Ljapunov-Schmidt
8.7. The Main Theorem on the Generic Bifurcation From Simple Zeros
8.8. Applications to Eigenvalue Problems
8.9. Applications to Integral Equations
8.10. Application to Differential Equations
8.11. The Main Theorem on Generic Bifurcation for Multiparametric
Operator Equations——The Bunch Theorem
8.12. Main Theorem for Regular Semi-linear Equations
8.13. Parameter-Induced Oscillation
8.14. Self-Induced Oscillations and Limit Cycles
8.15. Hopf Bifurcation
8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros
8.17. Stability of Bifurcation Solutions
8.18. Generic Point Bifurcation
CHAPTER 9
Fixed Points of Multivalued Maps
9.1. Generalized Banach Fixed-Point Theorem
9.2. Upper and Lower Semi-continuity of Multivalued Maps
9.3. Generalized Schauder Fixed-Point Theorem
9.4. Variational Inequalities and the Browder Fixed-Point Theorem
9.5. An Extremal Principle
9.6. The Minimax Theorem and Saddle Points
9.7. Applications in Game Theory
9.8. Selections and the Marriage Theorem
……
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
Index
^ 收 起
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PRINCIPLES
CHAPTER 1
The Banach Fixed-Point Theorem and lterative Methods
1.1. The Banach Fixed-Point Theorem
1.2. Continuous Dependence on a Parameter
1.3. The Significance of the Banach Fixed-Point Theorem
1.4. Applications to Nonlinear Equations
1.5. Accelerated Convergence and Newtons Method
1.6. The Picard-Lindel6fTheorem
1.7. The Main Theorem for Iterative Methods for Linear Operator
Equations
1.8. Applications to Systems of Linear Equations
1.9. Applications to Linear Integral Equations
CHAPTER 2
The Schauder Fixed-Point Theorem and Compactness
2.1. Extension Theorem
2.2. Retracts
2.3. The Brouwer Fixed-Point Theorem
2.4. Existence Principle for Systems of Equations
2.5. Compact Operators
2.6. The Schauder Fixed-Point Theorem
2.7. Peanos Theorem
2.8. Integral Equations with Small Parameters
2.9. Systems of Integral Equations and Semilinear Differential
Equations
2.10. A General Strategy
2.11. Existence Principle for Systems of Inequalities
APPLICATIONS OF THE FUNDAMENTAL
FIXED-POINT PRINCIPLES
CHAPTER 3
Ordinary Differential Equations in B-spaces
3.1. Integration of Vector Functions of One Real Variable t
3.2. Differentiation of Vector Functions of One Real Variable t
3.3. Generalized Picard-Lindeltf Theorem
3.4. Generalized Peano Theorem
3.5. Gronwalrs Lemma
3.6. Stability of Solutions and Existence of Periodic Solutions
3.7. Stability Theory and Plane Vector Fields, Electrical Circuits,
Limit Cycles
3.8. Perspectives
CHAPTER 4
Differential Calculus and the Implicit Function Theorem
4.1. Formal Differential Calculus
4.2. The Derivatives of Frtchet and Giteaux
4.3. Sum Rule, Chain Rule, and Product Rule
4.4. Partial Derivatives
4.5. Higher Differentials and Higher Derivatives
4.6. Generalized Taylors Theorem
4.7. The Implicit Function Theorem
4.8. Applications of the Implicit Function Theorem
4.9. Attracting and Repelling Fixed Points and Stability
4.10. Applications to Biological Equilibria
4.11. The Continuously Differentiable Dependence of the Solutions of
Ordinary Differential Equations in B-spaces on the Initial Values
and on the Parameters
4.12. The Generalized Frobenius Theorem and Total Differential
Equations
4.13. Diffeomorphisms and the Local Inverse Mapping Theorem
4.14. Proper Maps and the Global Inverse Mapping Theorem
4.15. The Surjective Implicit Function Theorem
4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank
Theorem
4.17. A Look at Manifolds
4.18. Submersions and a Look at the Sard-Smale Theorem
4.19. The Parametrized Sard Theorem and Constructive Fixed-Point
Theory
CHAPTER 5
Newtons Method
5.1. A Theorem on Local Convergence
5.2. The Kantorovi Semi-Local Convergence Theorem
CHAPTER 6
Continuation with Respect to a Parameter
6.1. The Continuation Method for Linear Operators
6.2. B-spaces of H61der Continuous Functions
6.3. Applications to Linear Partial Differential Equations
6.4. Functional-Analytic Interpretation of the Existence Theorem and
its Generalizations
6.5. Applications to Semi-linear Differential Equations
6.6. The Implicit Function Theorem and the Continuation Method
6.7. Ordinary Differential Equations in B-spaces and the Continuation
Method
6.8. The Leray-Schauder Principle
6.9. Applications to Quasi-linear Elliptic Differential Equations
CHAPTER 7
Positive Operators
7. I. Ordered B-spaces
7.2. Monotone Increasing Operators
7.3. The Abstract Gronwall Lemma and its Applications to Integral
Inequalities
7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability
7.5. Applications
7.6. Minorant Methods and Positive Eigensolutions
7.7. Applications
7.8. The Krein-Rutman Theorem and its Applications
7.9. Asymptotic Linear Operators
7.10. Main Theorem for Operators of Monotone Type
7.11. Application to a Heat Conduction Problem
7.12. Existence of Three Solutions
7.13. Main Theorem for Abstract Hammerstein Equations in Ordered
B-spaces
7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation,
Stability, and the Nonlinear Krein-Rutman Theorem
7.15. Applications to Hammerstein Integral Equations
7.16. Applications to Semi-linear Elliptic Boundary-Value Problems
7.17. Application to Elliptic Equations with Nonlinear Boundary
Conditions
7.18. Applications to Boundary Initial-Value Problems for Parabolic
Differential Equations and Stability
CHAPTER 8
Analytic Bifurcation Theory
8.1. A Necessary Condition for Existence of a Bifurcation Point
8.2. Analytic Operators
8.3. An Analytic Majorant Method
8.4. Fredholm Operators
8.5. The Spectrum of Compact Linear Operators
（Riesz-Schauder Theory）
8.6. The Branching Equations of Ljapunov-Schmidt
8.7. The Main Theorem on the Generic Bifurcation From Simple Zeros
8.8. Applications to Eigenvalue Problems
8.9. Applications to Integral Equations
8.10. Application to Differential Equations
8.11. The Main Theorem on Generic Bifurcation for Multiparametric
Operator Equations——The Bunch Theorem
8.12. Main Theorem for Regular Semi-linear Equations
8.13. Parameter-Induced Oscillation
8.14. Self-Induced Oscillations and Limit Cycles
8.15. Hopf Bifurcation
8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros
8.17. Stability of Bifurcation Solutions
8.18. Generic Point Bifurcation
CHAPTER 9
Fixed Points of Multivalued Maps
9.1. Generalized Banach Fixed-Point Theorem
9.2. Upper and Lower Semi-continuity of Multivalued Maps
9.3. Generalized Schauder Fixed-Point Theorem
9.4. Variational Inequalities and the Browder Fixed-Point Theorem
9.5. An Extremal Principle
9.6. The Minimax Theorem and Saddle Points
9.7. Applications in Game Theory
9.8. Selections and the Marriage Theorem
……
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
Index
^ 收 起
目 录内容简介
首先，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）。
其次，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，该书由浅人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！
^ 收 起
其次，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，该书由浅人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！
^ 收 起
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