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非线性泛函分析及其应用·第1卷：不动点定理
作者：（德）宰德勒 著
出版社：世界图书出版公司
出版时间：2009-08-01
ISBN：9787510005190
定价：¥99.00
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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！
作者简介
暂缺《非线性泛函分析及其应用·第1卷：不动点定理》作者简介
目录
Preface to the Second Corrected Printing
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PI~INCIPLES
CHAPTER I The Banach Fixed-Point Theorem and Iterative 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 Newton's Method
1.6. The Picard-Lindel6f Theorem
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. Peano's 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-Lindeirf Theorem
3.4. Generalized Peano Theorem
3.5. Gronwall's 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 Frrchet and G~teaux
4.3. Sum Rule, Chain Rule, and Product Rule
4.4. Partial Derivatives
4.5. Higher Differentials and Higher Derivatives
4.6. Generalized Taylor's 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
……
Preface to the First Printing
Introduction
FUNDAMENTAL FIXED-POINT PI~INCIPLES
CHAPTER I The Banach Fixed-Point Theorem and Iterative 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 Newton's Method
1.6. The Picard-Lindel6f Theorem
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. Peano's 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-Lindeirf Theorem
3.4. Generalized Peano Theorem
3.5. Gronwall's 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 Frrchet and G~teaux
4.3. Sum Rule, Chain Rule, and Product Rule
4.4. Partial Derivatives
4.5. Higher Differentials and Higher Derivatives
4.6. Generalized Taylor's 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
……
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