Brouwer fixed-point theorem

E22815

fixed-point theorem topological theorem

The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.

Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Schauder fixed-point theorem	1

Statements (49)

Predicate	Object
instanceOf	fixed-point theorem ⓘ topological theorem ⓘ
appliesTo	closed disks ⓘ closed intervals ⓘ closed n-dimensional balls ⓘ compact convex subsets of Euclidean space ⓘ continuous functions ⓘ
assumes	compactness of the domain ⓘ continuity of the map ⓘ convexity of the domain ⓘ
category	existence theorem ⓘ
conclusion	existence of a fixed point ⓘ
dimension	does not generally extend to infinite-dimensional spaces ⓘ holds in all finite dimensions ⓘ
domainCondition	compact set ⓘ convex set ⓘ subset of Euclidean space ⓘ
equivalentTo	Sperner's lemma under suitable conditions ⓘ
failsIf	domain is not compact ⓘ domain is not convex ⓘ map is not continuous ⓘ
field	economics ⓘ functional analysis ⓘ game theory ⓘ nonlinear analysis ⓘ topology ⓘ
hasCombinatorialVersion	Sperner's lemma ⓘ
implies	Borsuk–Ulam theorem in certain formulations ⓘ
importance	fundamental result in topology ⓘ
mapCondition	continuous self-map ⓘ
namedAfter	Luitzen Egbertus Jan Brouwer ⓘ
nonConstructive	true ⓘ
proofMethod	combinatorial arguments ⓘ degree theory ⓘ homology theory ⓘ topological methods ⓘ
proposedBy	Luitzen Egbertus Jan Brouwer ⓘ surface form: L. E. J. Brouwer
relatedTo	Banach fixed-point theorem ⓘ Kakutani fixed-point theorem ⓘ Schauder fixed-point theorem ⓘ
statement	Every continuous function from a closed n-dimensional ball to itself has at least one fixed point. ⓘ Every continuous function from a compact convex subset of R^n to itself has at least one fixed point. ⓘ
usedIn	combinatorial topology ⓘ differential equations ⓘ general equilibrium theory in economics ⓘ nonlinear boundary value problems ⓘ proof of Nash equilibrium existence ⓘ topological degree theory ⓘ
yearProved	1911 ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kakutani fixed-point theorem → generalizes → Brouwer fixed-point theorem ⓘ

Glicksberg fixed-point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

Kakutani fixed-point theorem → relatedTo → Brouwer fixed-point theorem ⓘ

this entity surface form: Schauder fixed-point theorem