Springer Science & Business Media, Mar 2, 2009 - Science - 461 pages
"An elegantly written, introductory overview of the field, with a near perfect choice of what to include and what not, enlivened in places by historical tidbits and made eminently readable throughout by crisp language. It has succeeded in doing the near-impossible—it has made a subject which is generally inhospitable to nonspecialists because of its ‘family jargon’ appear nonintimidating even to a beginning graduate student."
—The Journal of the Indian Institute of Science
"The book under review gives a comprehensive treatment of basically everything in mathematics that can be named multivalued/set-valued analysis. It includes...results with many historical comments giving the reader a sound perspective to look at the subject...The book is highly recommended for mathematicians and graduate students who will find here a very comprehensive treatment of set-valued analysis."
"I recommend this book as one to dig into with considerable pleasure when one already knows the subject...‘Set-Valued Analysis’ goes a long way toward providing a much needed basic resource on the subject."
—Bulletin of the American Mathematical Society
"This book provides a thorough introduction to multivalued or set-valued analysis...Examples in many branches of mathematics, given in the introduction, prevail [upon] the reader the indispensability [of dealing] with sequences of sets and set-valued maps...The style is lively and vigorous, the relevant historical comments and suggestive overviews increase the interest for this work...Graduate students and mathematicians of every persuasion will welcome this unparalleled guide to set-valued analysis."
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This is not always the solution, for, by so doing, many important structural properties may be unfortunately lost; others are useless artifacts, making life more difficult rather than more simple. These points of view, which were widely ...
... Differential Inclusions 383 10.1 The Viability Theorem 387 10.1.1 Solutions to Differential Inclusions 388 10.1.2 ... Semi-Groups 399 10.4 Filippov's Theorem 400 10.5 Derivatives of the Solution Map 403 Bibliographical Comments 411 ...
First, we encounter set-valued maps each time we face ill-posed problems or inverse problems, i.e., problems for which either the existence of a solution or its uniqueness is not guaranteed for some data: Set-valued maps allow us to get ...
They also arise when we wish to treat a problem qualitatively, by looking for solutions common to a set of data, ... Problems with constraints also yield specific set-valued maps: Solving the equation f(x) = y, where the solution x is ...
Optimization provides examples of problems where uniqueness of the solution is naturally lacking: Let W be a function from X x Y to R. We consider the family of minimization problems VyeF, V(y) := mfW(x,y) parametrized by parameters y.