Kakutani fixed-point theorem

E3648

fixed-point theorem mathematical theorem

The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.

Aliases (3)

Statements (50)

Predicate	Object
instanceOf	fixed-point theorem → mathematical theorem →
appliesTo	correspondences in finite-dimensional normed spaces → set-valued maps on Euclidean spaces →
assumption	correspondence is upper hemicontinuous → domain is a nonempty compact convex subset of a Euclidean space → graph of the correspondence is closed → set-valued map has nonempty values → values of the correspondence are convex sets →
conclusion	there exists a fixed point → there exists x such that x is in F(x) →
field	functional analysis → game theory → mathematical analysis → topology →
generalizes	Brouwer fixed-point theorem →
hasCondition	domain is compact → domain is convex → graph is closed or correspondence is upper hemicontinuous → values are convex → values are nonempty →
hasProofMethod	topological arguments → use of Brouwer fixed-point theorem →
impliesExistenceOf	fixed point of a correspondence →
involvesStructure	Euclidean space → topological vector space →
isToolFor	nonconstructive existence proofs → proof of Nash's theorem on equilibria in finite games →
namedAfter	Shizuo Kakutani →
namedAfterOccupation	Japanese-American mathematician →
relatedTo	Brouwer fixed-point theorem → Glicksberg fixed-point theorem → Schauder fixed-point theorem →
timePeriod	20th century →
usedFor	existence of Nash equilibrium → existence of competitive equilibria → existence of equilibria in finite games → existence of general equilibrium in economics → existence of solutions to differential inclusions → existence of solutions to variational inequalities → existence proofs in game theory → existence proofs in mathematical economics →
usesConcept	compactness → convexity → correspondence (set-valued map) → fixed point → multivalued function → nonempty convex compact subset → set-valued function → upper hemicontinuity →

Referenced by (10)

Subject (surface form when different)	Predicate
Hugo Steinhaus ("Kakutani fixed-point theorem (contributions)") → Shizuo Kakutani → Shizuo Kakutani ("Kakutani’s theorem on random ergodic theorems") →	notableWork
Brouwer fixed-point theorem → Glicksberg fixed-point theorem →	relatedTo
Glicksberg fixed-point theorem →	extends
Glicksberg fixed-point theorem ("Kakutani fixed-point theorem to infinite-dimensional settings") →	generalizes
Shizuo Kakutani →	hasTheoremNamedAfter
Shizuo Kakutani →	knownFor
Non-cooperative Games →	usesTool