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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Set theory, which developed from the theory of analytical functions, could prove useful to its elder sister and could show people of good will its qualities and richness" . This unfortunate situation led to two concepts of semiconti- ...
We shall prove an equivalent statement, called the Equilibrium Theorem, which provides the existence of an equilibrium of a set-valued map, a solution to the inclusion F(x) 3 0. Of course, for applications, we need not only to solve ...
... hull of the upper limit is obviously contained in the closed convex subset A := fl co I |J Kn N>0 \n>N We have to prove that it is equal to it when the dimension of X is finite and the subsets Kn are contained in a bounded set.
Hence we have to prove only the second claim. Let us consider a sequence xn € Kn such that a subsequence of elements A(xn) (again denoted A(xn)) converges weakly to some y in Y. We shall check that (£n)neN has a weak cluster point, ...
... Limsupn^00(L„nA-1(M„)) D (Limsup^^L^nA^Limmfn^ooMn) and Limsupn^00(LnnA_1(Mn)) D (Liminfn_00Ln)nA_1(Limsupri_+00Mn) Proof — The inclusion LimMn^ooiLnDA^iMn)) C (LimMn^ooL^nA^iLimmfn^Mn) being obvious, let us prove the other one, ...