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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These include: • Limits and Continuity • Linear Functional Analysis • Nonlinear Functional Analysis (existence and approximation of solutions to equations and inclusions) • Tangents and Normals • Differentiation of Maps • Gradients of ...
We did not restrict our exposition to the framework of finite dimensional vector-spaces, since set- valued analysis is also useful for solving problems involving partial differential equations or inclusions. But whenever the proofs of ...
... 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 away from the restriction that a map is bijective when we want to solve an equation.
Problems with constraints also yield specific set-valued maps: Solving the equation f(x) = y, where the solution x is ... inequalities (also called "generalized equations" by some authors), which are again inclusions in disguise.
So, the control system governed by the family of parametrized differential equations x'(t) = f(x(t),u(t)) where u(t) € U(x(t)) is actually governed by the differential inclusion x'(t) 6 F(x(t)) 6. Optimization provides examples of ...