Peano existence theorem

E128381

existence theorem mathematical theorem theorem in ordinary differential equations

The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.

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All labels observed (2)

Label	Occurrences
Peano existence theorem canonical	3
Peano existence theorem in terms of uniqueness	1

How this entity was disambiguated

This entity first appeared as the object of triple T1057163 — 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: Peano existence theorem
Context triple: [local existence and uniqueness theorem, relatedTo, Peano existence theorem]

A. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
B. 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.
C. Banach fixed-point theorem
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.
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. 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.
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: Peano existence theorem
Target entity description: The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.

A. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
B. 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.
C. Banach fixed-point theorem
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.
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. 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.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	existence theorem ⓘ mathematical theorem ⓘ theorem in ordinary differential equations ⓘ
allows	non-unique solution trajectories ⓘ
appliesTo	autonomous differential equations ⓘ first-order ordinary differential equations ⓘ initial value problems ⓘ non-autonomous differential equations ⓘ
assumes	continuity of the right-hand side function ⓘ local continuity conditions ⓘ
comparedTo	Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem
concernsEquation	y' = f(t,y) ⓘ
concernsProblem	y' = f(t,y), y(t0) = y0 ⓘ
concludes	solution exists on some interval around the initial point ⓘ there exists at least one local solution ⓘ
contrastsWith	uniqueness theorems for ODEs ⓘ
doesNotGuarantee	uniqueness of solutions ⓘ
field	differential equations ⓘ mathematical analysis ⓘ ordinary differential equations ⓘ
formalizes	existence of solutions under continuity alone ⓘ
generalizesTo	n-dimensional state space ⓘ systems of ordinary differential equations ⓘ
guarantees	existence of solutions to initial value problems ⓘ
hasConsequence	initial value problem may have infinitely many solutions ⓘ solution set may form a continuum of solutions ⓘ
hasStrongerExistenceConditionThan	Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem
hasWeakerRegularityAssumptionThan	Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem
historicalPeriod	late 19th century ⓘ
implies	solution curves exist under mild assumptions ⓘ
involves	continuous vector fields on the plane or higher dimensions ⓘ
isPartOf	classical theory of ODEs ⓘ
mathematicianAssociated	Giuseppe Peano ⓘ
namedAfter	Giuseppe Peano ⓘ
relatedConcept	Carathéodory existence theorem ⓘ Cauchy problem ⓘ Lipschitz condition ⓘ linked to: local existence and uniqueness theorem integral curves of vector fields ⓘ
requires	f is continuous in a neighborhood of (t0,y0) ⓘ
topicIn	graduate analysis courses ⓘ undergraduate differential equations courses ⓘ
typeOf	local existence theorem ⓘ
usedIn	mathematical modeling ⓘ qualitative theory of differential equations ⓘ theory of dynamical systems ⓘ
weakerConditionThan	Lipschitz continuity ⓘ

How these facts were elicited

Referenced by (4)

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

local existence and uniqueness theorem → relatedTo → Peano existence theorem ⓘ

local existence and uniqueness theorem → strongerThan → Peano existence theorem in terms of uniqueness ⓘ

linked to: Peano existence theorem

Giuseppe Peano → notableWork → Peano existence theorem ⓘ

Carathéodory existence theorem → relatedTo → Peano existence theorem ⓘ

All labels observed (2)

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Statements (46)

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Referenced by (4)

How this entity was disambiguated

How these facts were elicited