Banach fixed-point theorem

E126344

fixed-point theorem mathematical theorem result in metric space theory

The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.

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All labels observed (4)

Label	Occurrences
Banach fixed-point theorem canonical	7
Banach fixed-point theorem which assumes contraction	1
Banach–Caccioppoli fixed-point theorem	1
contraction mapping theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1056916 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Banach fixed-point theorem
Context triple: [Brouwer fixed-point theorem, relatedTo, Banach fixed-point theorem]

A. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
B. Kakutani fixed-point 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.
C. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
D. Brouwer fixed-point 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.
E. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Banach fixed-point theorem
Target entity description: The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.

A. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
B. Kakutani fixed-point 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.
C. Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
D. Brouwer fixed-point 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.
E. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	fixed-point theorem ⓘ mathematical theorem ⓘ result in metric space theory ⓘ
alsoKnownAs	Banach–Caccioppoli fixed-point theorem ⓘ linked to: Banach fixed-point theorem contraction mapping theorem ⓘ linked to: Banach fixed-point theorem
appliesTo	complete metric spaces ⓘ
assumes	Lipschitz constant strictly less than 1 ⓘ self-map on a complete metric space ⓘ
coreConcept	complete metric space ⓘ contraction mapping ⓘ fixed point ⓘ iterative approximation ⓘ
field	analysis ⓘ functional analysis ⓘ metric space theory ⓘ
guarantees	convergence of Picard iteration ⓘ existence of a fixed point ⓘ geometric rate of convergence of iterates ⓘ uniqueness of a fixed point ⓘ
hasApplicationDomain	dynamical systems ⓘ nonlinear analysis ⓘ numerical analysis ⓘ ordinary differential equations ⓘ partial differential equations ⓘ
hasConsequence	existence of invariant points for contractions ⓘ stability of iterative schemes under contractions ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	fixed point is unique ⓘ limit of iterates is a fixed point ⓘ sequence of iterates converges in the metric space ⓘ sequence of iterates is Cauchy ⓘ
involves	Cauchy sequence ⓘ Lipschitz constant ⓘ
logicalForm	if a mapping is a contraction on a complete metric space then it has a unique fixed point ⓘ
namedAfter	Stefan Banach ⓘ
provides	error estimate for distance to fixed point ⓘ iterative method to find fixed point ⓘ
relatedTo	Brouwer fixed-point theorem ⓘ Lipschitz continuity ⓘ Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem Schauder fixed-point theorem ⓘ
requires	contraction mapping ⓘ metric space completeness ⓘ
typeOf	existence and uniqueness theorem ⓘ
usedFor	constructive proofs in analysis ⓘ iterative numerical methods ⓘ proving existence and uniqueness of solutions to differential equations ⓘ proving existence and uniqueness of solutions to integral equations ⓘ

How these facts were elicited

Referenced by (10)

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

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

local existence and uniqueness theorem → isProvedBy → Banach fixed-point theorem ⓘ

Stefan Banach → notableWork → Banach fixed-point theorem ⓘ

Stefan Banach → eponymOf → Banach fixed-point theorem ⓘ

Schauder fixed-point theorem → relatedTo → Banach fixed-point theorem ⓘ

Schauder fixed-point theorem → contrastWith → Banach fixed-point theorem which assumes contraction ⓘ

linked to: Banach fixed-point theorem

Picard iteration → basedOn → Banach fixed-point theorem ⓘ

Banach fixed-point theorem → alsoKnownAs → contraction mapping theorem ⓘ

linked to: Banach fixed-point theorem

Banach fixed-point theorem → alsoKnownAs → Banach–Caccioppoli fixed-point theorem ⓘ

linked to: Banach fixed-point theorem

Tarski’s fixed point theorem → relatedTo → Banach fixed-point theorem ⓘ

All labels observed (4)

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Statements (48)

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Referenced by (10)

How this entity was disambiguated

How these facts were elicited