基本信息·出版社：世界图书出版公司 ·页码：466 页 ·出版日期：2009年08月 ·ISBN：7510005213/9787510005213 ·条形码：9787510005213 ·版本：第1版 ...
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非线性泛函分析及其应用,第2B卷,非线性单调算子 |
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非线性泛函分析及其应用,第2B卷,非线性单调算子 |
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基本信息·出版社：世界图书出版公司
·页码：466 页
·出版日期：2009年08月
·ISBN：7510005213/9787510005213
·条形码：9787510005213
·版本：第1版
·装帧：平装
·开本：24
·正文语种：英语
·外文书名：Nolinear Functional Analysis&It's Application Vol 2/B Nonlinear Monotone Operators
内容简介 《非线性泛函分析及其应用,第2B卷,非线性单调算子》内容简介：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。”
编辑推荐 《非线性泛函分析及其应用,第2B卷,非线性单调算子》的写作起点很低，具备本科数学水平就可以读。
目录 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
……
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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