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。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。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。

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。

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-lteration 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 posterior.i Error Estimates for the Ritz Method

 25.13.The Kacanov Method for Stationary Conservation Laws

 25.14.The Abstract Kacanov 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