Set-valued function: Difference between revisions

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{{Short description|Function whose values are sets (mathematics)}}

{{About|set-valued functions as considered in variational analysis|multi-valued functions as they are considered in complex analysis|multivalued function}}

{{About||multi-valued functions of mathematical analysis|Multivalued function|functions whose arguments are sets|Set function}}

[[File:Multivalued_function.svg|right|frame|This diagram represents a multi-valued, but not a proper (single-valued) [[Function (mathematics)|function]], because the element 3 in ''X'' is associated with two elements, ''b'' and ''c'', in ''Y''.]]

A '''set-valued function''', (oralso called a '''correspondence''' or '''set-valued [[Relation (mathematics)|relation]]''', is a mathematical [[Function (mathematics)|function]] that maps elements from one set, knownthe as[[___domain theof a function|___domain of the function]], to setssubsets of elements in another set.<ref name=":02">{{Cite book |last1=Aliprantis |first1=Charalambos D. |url=https://books.google.com/books?id=Ma31CAAAQBAJ |title=Infinite Dimensional Analysis: A Hitchhiker's Guide |last2=Border |first2=Kim C. |date=2013-03-14 |publisher=Springer Science & Business Media |isbn=978-3-662-03961-8 |pages=523 |language=en}}</ref><ref name=":0">{{Cite book |last1=Wriggers |first1=Peter |url=https://books.google.com/books?id=R4lqCQAAQBAJ |title=New Developments in Contact Problems |last2=Panatiotopoulos |first2=Panagiotis |date=2014-05-04 |publisher=Springer |isbn=978-3-7091-2496-3 |pages=29 |language=en}}</ref> Set-valued functions are used in a variety of mathematical fields, including [[Mathematical optimization|optimization]], [[control theory]] and [[game theory]].

Set-valued functions are also known as [[multivalued function]]s in some references,<ref>{{Cite book |last=Repovš |first=Dušan |title=Continuous selections of multivalued mappings |date=1998 |publisher=Kluwer Academic |others=Pavel Vladimirovič. Semenov |isbn=0-7923-5277-7 |___location=Dordrecht |oclc=39739641}}</ref> but this article and the article [[Multivalued function]] follow the authors who make a distinction.

== TypesDistinction offrom multivalued functions ==

[[File:Multivalued_functions_illustration.svg|thumb|right|600px|Illustration distinguishing multivalued functions from set-valued relations according to the criterion in page 29 of ''New Developments in Contact Problems'' by Wriggers and Panatiotopoulos (2014).]]

Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called ''set-valued relations'') by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a [[Function (mathematics)|function]].<ref name=":0" /> Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.<ref name=":0" />

Alternatively, a [[multivalued function]] is a set-valued function {{mvar|f}} that has a further [[continuous function|continuity]] property, namely that the choice of an element in the set <math>f(x)</math> defines a corresponding element in each set <math>f(y)</math> for {{mvar|y}} close to {{mvar|x}}, and thus defines [[locally]] an ordinary function.

== ExamplesExample ==

The [[argmax]] of a function is in general, multivalued. For example, <math>\operatorname{argmax}_{x \in \mathbb{R}} \cos(x) = \{2 \pi k\mid k \in \mathbb{Z}\}</math>.

== Set-valued analysis ==

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There exist set-valued extensions of the following concepts from point-valued analysis: [[Continuous (mathematics)|continuity]], [[Differentiation (mathematics)|differentiation]], [[Integral|integration]],<ref>{{cite journal |last=Aumann |first=Robert J. |author-link=Robert Aumann |year=1965 |title=Integrals of Set-Valued Functions |journal=[[Journal of Mathematical Analysis and Applications]] |volume=12 |issue=1 |pages=1–12 |doi=10.1016/0022-247X(65)90049-1 |doi-access=free}}</ref> [[implicit function theorem]], [[Contraction mapping|contraction mappings]], [[measure theory]], [[Fixed-point theorem|fixed-point theorems]],<ref name="kakutani">{{cite journal |last=Kakutani |first=Shizuo |author-link=Shizuo Kakutani |year=1941 |title=A generalization of Brouwer's fixed point theorem |journal=[[Duke Mathematical Journal]] |volume=8 |issue=3 |pages=457–459 |doi=10.1215/S0012-7094-41-00838-4}}</ref> [[Optimization (mathematics)|optimization]], and [[topological degree theory]]. In particular, [[Equation|equations]] are generalized to [[Inclusion (set theory)|inclusions]], while differential equations are generalized to [[Differential inclusion|differential inclusions]].

== Examples ==

The [[argmax]] of a function is in general, multivalued. For example, <math>\operatorname{argmax}_{x \in \mathbb{R}} \cos(x) = \{2 \pi k\mid k \in \mathbb{Z}\}</math>.

== Types of multivalued functions ==

One can distinguish multiple concepts generalizing [[Continuity (mathematics)|continuity]], such as the [[closed graph]] property and [[Hemicontinuity|upper and lower hemicontinuity]]{{efn|Some authors use the term ‘semicontinuous’ instead of ‘hemicontinuous’.}}. There are also various generalizations of [[Measure (mathematics)|measure]] to multifunctions.

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Set-valued functions arise in [[Optimal control|optimal control theory]], especially [[Differential inclusion|differential inclusions]] and related subjects as [[game theory]], where the [[Kakutani fixed-point theorem]] for set-valued functions has been applied to prove existence of [[Nash equilibrium|Nash equilibria]]. This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.

Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the [[Michael selection theorem]], which provides another characterisation of [[paracompact]] spaces.<ref>{{cite journal |author=Ernest Michael |author-link=Ernest Michael |date=Mar 1956 |title=Continuous Selections. I |url=http://www.renyi.hu/~descript/papers/Michael_1.pdf |journal=Annals of Mathematics |series=Second Series |volume=63 |pages=361–382 |doi=10.2307/1969615 |jstor=1969615 |number=2 |hdl=10338.dmlcz/119700}}</ref><ref>{{cite journal |author1=Dušan Repovš |author1-link=Dušan Repovš |author2=P.V. Semenov |year=2008 |title=Ernest Michael and theory of continuous selections |journal=Topology Appl. |volume=155 |pages=755–763 |arxiv=0803.4473 |doi=10.1016/j.topol.2006.06.011 |number=8|s2cid=14509315 }}</ref> Other selection theorems, like Bressan-Colombo directional continuous selection, [[Kuratowski and Ryll-Nardzewski measurable selection theorem]], Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in [[optimal control]] and the theory of [[Differential inclusion|differential inclusions]].

== Notes ==

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== Further reading ==

* K. Deimling, ''[https://books.google.com/books?id=D9pgTAujcKcC&printsec=frontcover#v=onepage&q&f=false Multivalued Differential Equations]'', Walter de Gruyter, 1992

* C. D. Aliprantis and K. C. Border, ''Infinite dimensional analysis. Hitchhiker's guide'', Springer-Verlag Berlin Heidelberg, 2006

* J. Andres and L. Górniewicz, ''[https://books.google.com/books?id=PanqCAAAQBAJ&printsec=frontcover#v=onepage&q=multivalued&f=false Topological Fixed Point Principles for Boundary Value Problems]'', Kluwer Academic Publishers, 2003

* J.-P. Aubin and A. Cellina, ''Differential Inclusions, Set-Valued Maps And Viability Theory'', Grundl. der Math. Wiss. 264, Springer - Verlag, Berlin, 1984

* J.-P. Aubin and [[Hélène Frankowska|H. Frankowska]], ''Set-Valued Analysis'', Birkhäuser, Basel, 1990

* [[Dušan Repovš|D. Repovš]] and P.V. Semenov, [https://www.springer.com/gp/book/9780792352778?cm_mmc=sgw-_-ps-_-book-_-0-7923-5277-7 ''Continuous Selections of Multivalued Mappings''], Kluwer Academic Publishers, Dordrecht 1998

* E. U. Tarafdar and M. S. R. Chowdhury, [https://books.google.com/books?id=Cir88lF64xIC ''Topological methods for set-valued nonlinear analysis''], World Scientific, Singapore, 2008

* {{cite journal |lastlast1=Mitroi |firstfirst1=F.-C. |last2=Nikodem |first2=K. |last3=Wąsowicz |first3=S. |year=2013 |title=Hermite-Hadamard inequalities for convex set-valued functions |journal=Demonstratio Mathematica |volume=46 |issue=4 |pages=655–662 |doi=10.1515/dema-2013-0483 |doi-access=free}}

== See also ==

* [[Selection theorem]]

* [[Ursescu theorem]]

* [[Binary relation]]

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