隐函数和解映射

PrelaceAcknowledgementsChapter 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 1E Selections of multi．valued inverses 1G．Selections from nonstrict differentiabilityChapter 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 2E Variational inequalities with monotonicity 2G．Consequences for optimizationChapter 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 3I．Strong metric subregularityChapter 4．Regularity properties through generalized derivatives 4A．Graphical differentiation 4B．Derivative criteria 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 Iocalizations for variational inequalities 4G．Special nonsmooth inverse function theorems 4H．Results utilizing coderivativesChapter 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 extensionsChapter 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 NotationIndex

隐函数定理是分析的最主要定理之一，是偏微分方程和数值分析的最基本工具。邓契夫等编著的《隐函数和解映射》在经典框架及其外研究隐函数的本质，主要侧重于研究变分问题解映射的性质。本书自称体系，并将大量散落的材料综合起来，旨在提供一个研究这门学科的参考书籍。第一章以一种学生和本科生微积分的老师新闻乐见的方式讲述经典隐函数定理，以下的章节在难度上逐渐增加，将隐映射看作是一种关联定义的，而非方程定义的。书中讲述了数值分析和优化中的应用。本书是本学科学术上的巨大成果，注定会成为这门学科的一本标准参考书。