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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203 5.4.2 Localization of Inverse Images 206 5.4.3 The Equilibrium Map 207 5.4.4 Local Injectivity 209 5.5 Qualitative Solutions 210 5.6 Higher Order Derivatives 215 6 Epiderivatives of Extended Functions 219 6.1 Contingent ...
6.6.1 Second Order Epiderivatives of Moreau-Yosida Approximations 263 7 Graphical & Epigraphical Convergence 265 7.1 Graphical Limits 267 7.1.1 Definitions 267 7.1.2 Graphical Convergence of Closed Convex Processes 269 7.1.3 Monotone ...
... with a function V the set-valued map Vj defined by VT(x) := [V(x),+oo[ whose graph is the epigraph of V. The graphs of the derivatives of such set-valued maps Vf are the epigraphs of functions which are called epiderivatives .
As far as optimization is concerned, the Fermat Rule can be extended to any function by using these epiderivatives. Since they enjoy a rich calculus, we obtain in this way many necessary conditions for a minimum.
We shall state some existence theorems, show that the set of solutions depends continuously upon the initial data, describe some properties of Lyapunov functions (which, thanks to the concept of epiderivatives, can even be taken lower ...