隱函數和解映射（簡體書）

Setting up equations and solving them has long been so important that， in popular imagination， it has virtually come to describe what mathematical analysis and its applications are all about. A central issue in the subject is whether the solution to an equation involving parameters may be viewed as a function of those parameters， and if so， what properties that function might have. This is addressed by the classical theory ofimplicit functions， which began with single real variables and progressed through multiple variables to equations in infinite dimensions， such as equations associated with inLegral and differential operators.隱函數定理是分析的最主要定理之一，是偏微分方程和數值分析的最基本工具。本書在經典框架及其外研究隱函數的本質，主要側重於研究變分問題解映射的性質。本書自稱體系，並將大量散落的材料綜合起來，旨在提供一個研究這門學科的參考書籍。第一章以一種學生和本科生微積分的老師新聞樂見的方式講述經典隱函數定理，以下的章節在難度上逐漸增加，將隱映射看作是一種關聯定義的，而非方程定義的。書中講述了數值分析和優化中的應用。本書是本學科學術上的巨大成果，註定會成為這門學科的一本標準參考書。目次：方程定義的隱函數；變分問題的隱函數定理；集值解映射的一般性質；一般導數的常規性質；無限維中的一般性；在數值變分分析中的應用。
讀者對象：數學專業的學生、老師和科研人員。

Preface
Acknowledgements

Chapter 1. Functions defined implicitly by equations
1A. The classical inverse function theorem
1B. The classical implicit function theorem
1C. Calmness
1D. Lipschitz continuity
1E. Lipschitz invertibility from approximations
1F. Selections of multi-valued inverses
1G. Selections from nonstrict differentiability

Chapter 2. Implicit function theorems for variational problems
2A. Generalized equations and variational problems
2B. Implicit function theorems for generalized equations
2C. Ample parameterization and parametric robustness
2D. Semidifferentiable functions
2E. Variational inequalities with polyhedral convexity
2F. Variational inequalities with monotonicity
2G. Consequences t'or optimization

Chapter 3. Regularity properties of set-valued solution mappings
3A. Set convergence
3B. Continuity of set-valued mappings
3C. Lipschitz continuity of set valued mappings
3D. Outer Lipschitz continuity
3E. Aubin property， metric regularity and linear openness
3F. Implicit mapping theorems with metric regularity
3G. Strong metric regularity
3H. Calmness and metric subregularity
3L Strong metric subregularity

Chapter 4. Regularity properties through generalized derivatives
4A. Graphical differentiation
4B. Derivative crkeria for the Aubin property
4C. Characterization of strong metric subregularity
4D. Applications to parameterized constraint systems
4E. Isolated calmness for variational inequalities
4F. Single-valued localizations for variational inequalities
4G. Special nonsmooth inverse function theorems
4H. Results utilizing coderivatives

Chapter 5. Regularity in infinite dimensions
5A. Openness and positively homogeneous mappings
5B. Mappings with closed and convex graphs
5C. Sublinear mappings
5D. The theorems of Lyusternik and Graves
5E. Metric regularity in metric spaces
5F. Strong metric regularity and implicit function theorems
5G. The Bartle-Graves theorem and extensions

Chapter 6. Applications in numerical variational analysis
6A. Radius theorems and conditioning
6B. Constraints and feasibility
6C. Iterative processes for generalized equations
6D. An implicit function theorem for Newton's iteration
6E. Galerkin's method for quadratic minimization
6F. Approximations in optimal control

References
Notation
Index