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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365 9.4.2 The Intersection Lemma 369 9.4.3 Lipschitz Selections of Lipschitz Maps 372 9.5 Selections of Caratheodory maps 373 9.6 Caratheodory Parametrization 376 9.7 Measurable/Lipschitz Parametrization 379 10 Differential Inclusions ...
We also provide a useful technical lemma: Lemma 1.1.6 Let us consider a sequence of subsets Ln c Z of a metric space Z and a sequence of subsets Mn CY of a compact metric space Y. Let ip : Z x Y t-> R be an upper semicontinuous function ...
Lemma 1.1.9 Let us consider a sequence of subsets Kn contained in a bounded subset of a finite dimensional vector space X. Then co (Limsup^oo-Kn) = f] co I (J Kn N>0 \n>N Proof — The closed convex hull of the upper limit is obviously ...
Lemma 1.5.4 Let us assume that F is lower semicontinuous and that H is upper semicontinuous with compact images. Then the numbers en defined by (1.8) converge to 0. Proof— Since F is lower semicontinuous, Corollary 1.4.17 to the 52 1 ...
... F(xn,y))/2 < e/2 y£/f(a;n) i.e., our lemma is proved. □ For set-valued maps with non convex images, we deduce from Theorem 1.2.9 its continuous version: Theorem 1.5.5 Let G : X ~> Z be a closed lower semicontinuous set-valued map ...