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  • 非线性泛函分析及其应用,第1卷,不动点定理[平装]
  • 共1个商家     78.20元~78.20
  • 作者:宰德勒(作者)
  • 出版社:世界图书出版公司;第1版(2009年8月1日)
  • 出版时间:
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  • ISBN:9787510005190

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    《非线性泛函分析及其应用,第1卷,不动点定理》的写作起点很低,具备本科数学水平就可以读。

    作者简介

    作者:(德国)宰德勒

    目录

    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 Newton's 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. 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-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 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
    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
    Newton's 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

    序言

    自1932年,波兰数学家Banach发表第一部泛函分析专著“Theorie des operations lineaires”以来,这一学科取得了巨大的发展,它在其他领域的应用也是相当成功。如今,数学的很多领域没有了泛函分析恐怕寸步难行,不仅仅在数学方面,在理论物理方面的作用也具有同样的意义,M.Reed和B.Simon的“Methods of Modern MathematicalPhysjcs”在前言中指出:“自1926年以来,物理学的前沿已与日俱增集中于量子力学,以及奠定于量子理论的分支:原子物理、核物理固体物理、基本粒子物理等,而这些分支的中心数学框架就是泛函分析。”所以,讲述泛函分析的文献已浩如烟海。而每个时代,都有这个领域的代表性作品。例如上世纪50年代,F.Riesz和Sz.-Nagy的《泛函分析讲义》(中译版,科学出版社,1985),就是那个时代的一部具有代表性的著作;而60年代,N.Dunford和J.Schwartz的三大卷“Linear Operators”则是泛函分析学发展到那个时代的主要成果和应用的一个较全面的总结。泛函分析一经产生,它的发展就受到量子力学的强有力的推动,上世纪70年代,M.Reed和B.Simon的4卷“Methods 0f M0dern Mathematical Physics”是泛函分析对于量子力学应用的一个很好的总结。

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