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 Fermat's Rule, which remains the main strategy for obtaining necessary conditions of op- timality, from mathematical programming to calculus of variations to optimal control). Fermat also was the first to discover the ...
If we try to adapt to the set-valued case the two equivalent definitions of continuity of single-valued maps, we obtain two notions which are no longer equivalent! 2In his famous thesis, judged at the time far too much abstract; ...
We obtain in this way a variety of closed cones made of what we call tangent vectors. The most popular of these tangent cones is for the time the contingent cone introduced in the thirties by Bouligand, (which is the upper limit of ...
Since they enjoy a rich calculus, we obtain in this way many necessary conditions for a minimum. This can be done by transferring the set-valued differential calculus to what can be called an epidifferential calculus.
1.2.1 Direct Images We begin with the direct images of upper limits: We easily obtain equalities when / is proper. We recall that a continuous single-valued map from a metric space X to a metric space Y is proper if and only if one of ...