Implicit Functions and Solution Mappings

A View from Variational Analysis

  • Asen L. Dontchev
  • R. Tyrrell Rockafellar

Table of contents

  1. Front Matter
    Pages i-xxviii
  2. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 1-68
  3. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 69-142
  4. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 143-212
  5. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 213-276
  6. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 277-362
  7. Asen L. Dontchev, R. Tyrrell Rockafellar
    Pages 363-454
  8. Back Matter
    Pages 455-466

About this book


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 second edition of Implicit Functions and Solution Mappings presents an updated and more complete picture of the field by including solutions of problems that have been solved since the first edition was published, and places old and new results in a broader perspective. 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. Updates to this edition include new sections in almost all chapters, new exercises and examples, updated commentaries to chapters and an enlarged index and references section.

From reviews of the first edition:

“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 Borwein, IEEE Control Systems Magazine, February, 2012)

“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 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)


Graphical differentiation Implicit functions Metric regularity in infinite dimensions Radius theorems and conditioning Solution mappings The linear-quadratic regulator problem

Authors and affiliations

  • Asen L. Dontchev
    • 1
  • R. Tyrrell Rockafellar
    • 2
  1. 1.Mathematical ReviewsAnn ArborUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media New York 2014
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-1036-6
  • Online ISBN 978-1-4939-1037-3
  • Series Print ISSN 1431-8598
  • Series Online ISSN 2197-1773
  • Buy this book on publisher's site