## Set-Valued Analysis"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." —Mathematical Reviews "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." —Zentralblatt Math |

### From inside the book

Results 1-5 of 43

Definition 1.3.1 Let

**X**and Y be metric spaces. A set-valued map F from

**X**to Y is characterized by its graph Graph(F), the subset of the product space

**X x**Y defined by Graph(F) := {{

**x**,y) £

**X**xY \ y

**G F**(

**x**)} We shall say that F(

**x**) is the ...

We then note that if

**x**€

**X**is such that V n € N, x e Ln

**x**€ Int(L„) then F is lower semicontinuous at

**x**. Indeed, let V C Y be any open subset such that F(

**x**) D V ^ 0. Then there exists n € N and y

**G F**(

**x**)nVW C F(

**x**)C\V Thus

**x**E Ln and ...

**X**-^Z defined by R(

**x**) := {ueF(

**x**) | f(

**x**,u) e G(

**x**)} (1.5) is lower semicontinuous with nonempty convex values. ... Indeed, if xn — >

**x**and if u belongs to S(

**x**) C F(

**x**), there exists a sequence of elements un

**G F**(xn) which converges to u ...

... to

**x**and z G G(

**x**). We have to approximate z by elements zn G G(xn). We introduce the following numbers: en := sup d(z,F(xn,y))/2 (1.8) yeH(xn) Now, let us choose for each y G H(xn) an element un(y)

**G F**(xn,y) satisfying ||«-«n(y)|| ...

Lemma 2.1.2 A set-valued map F is convex if and only if Vjci, x2 E Dom(F), VA e [0,1], XF(xi) + (1 - \)F(x2) C F(Xxl + (1 -

**X**)x2) It is a process if and only if V

**x**G X, V A > 0, XF(

**x**) = F{Xx) and 0

**G F**(0) and a convex process if and ...

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