Implicit Functions and Solution Mappings
A View from Variational Analysis
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Beschreibung
The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This book treats the implicit function paradigm in the classical framework and beyond, focusing largely on properties of solution mappings of variational problems.
The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided.
This valuable book is a major achievement and is sure to become a standard reference on the topic.
This text on the implicit function paradigm is a self-contained reference that unifies a collection of many results scattered throughout other literature. It contains numerous exercises and a large amount of historical information.
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, andif so,what propertiesthat functionmighthave.Thisisaddressedbytheclassical theory of implicit functions, which began with single real variables and progressed through multiple variables to equations in in?nite dimensions, such as equations associated with integral and differential operators. A major aim of the book is to lay out that celebrated theory in a broader way than usual, bringing to light many of its lesser known variants, for instance where standard assumptions of differentiability are relaxed. However, another major aim is to explain how the same constellation of ideas, when articulated in a suitably expanded framework, can deal successfully with many other problems than just solving equations.
An attractive blend of modern topics and classical results
The authors have included a nice selection of exercises for classroom use
Ideal reference work which contains a large amount of historical material and references
Autor*in
Assen L. Dontchev
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From the reviews:
“This book is devoted to the theory of inverse and implicit functions and some of its modifications for solution mappings in variational problems. … The book is targeted to a broad audience of researchers, teachers and graduate students. It can be used as well as a textbook as a reference book on the topic. Undoubtedly, it will be used by mathematicians dealing with functional and numerical analysis, optimization, adjacent branches and also by specialists in mechanics, physics, engineering, economics, and so on.” (Peter Zabreiko, Zentralblatt MATH, Vol. 1178, 2010)
“The book starts with the classical framework of implicit function theorem and then is largely focusing on properties of solution mapping of variational problems. … The book ends with a rich list of references, a glossary of notation, and a subject index. This references reflect from one side the authors’s contribution to this topic and from other side the contributions of many other researchers all over the world. Certainly this wonderful work will be included in many libraries all over the world.” (Marian Mureşan, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)
“The present monograph will be a most welcome and valuable addition. … This book will save much time and effort, both for those doing research in variational analysis and for students learning the field. This important contribution fills a gap in the existing literature.” (Stephen M. Robinson, Mathematical Reviews, Issue 2010 g)
“The book commences with a helpful context-setting preface followed by six chapters. Each chapter starts with a useful preamble and concludes with a careful and instructive commentary, while a good set of references, a notation guide, and a somewhat brief index complete this study. … I unreservedly recommended this book to all practitioners and graduate students interested in modern optimization theory or control theory or to those just engaged by beautiful analysis cleanly described.” (Jonathan Michael Browein, IEEE Control Systems Magazine, February, 2012)
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