Carathéodory existence theorem

E547409

mathematical theorem

The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.

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All labels observed (2)

Label	Occurrences
Carathéodory existence theorem canonical	1
Carathéodory solution	1

How this entity was disambiguated

This entity first appeared as the object of triple T5790598 — 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: Carathéodory existence theorem
Context triple: [Peano existence theorem, relatedConcept, Carathéodory existence theorem]

A. Peano existence theorem
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.
B. 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.
C. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
D. Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
E. Darboux theorem
The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
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: Carathéodory existence theorem
Target entity description: The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.

A. Peano existence theorem
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.
B. 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.
C. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
D. Carathéodory’s extension theorem
Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
E. Darboux theorem
The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ
allows	discontinuous right-hand side in time variable ⓘ
appearsIn	advanced textbooks on ordinary differential equations ⓘ monographs on differential equations with discontinuous right-hand sides ⓘ
appliesTo	initial value problems for ordinary differential equations ⓘ
assumes	Carathéodory conditions on the right-hand side ⓘ continuity of f in the state variable for almost every time ⓘ local integrable bound on f ⓘ measurability of f in the time variable ⓘ
comparedTo	Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem
concerns	Carathéodory-type right-hand side f(t,x) ⓘ differential equation x'(t)=f(t,x(t)) ⓘ
concludes	existence of an absolutely continuous solution ⓘ solution satisfies the differential equation almost everywhere ⓘ
context	measurable dependence on time ⓘ non-Lipschitz right-hand sides ⓘ
ensures	local solvability under Carathéodory conditions ⓘ
field	mathematical analysis ⓘ ordinary differential equations ⓘ
generalizes	Picard–Lindelöf existence theorem ⓘ linked to: local existence and uniqueness theorem classical existence theorems for ODEs ⓘ
guarantees	existence of solutions to ordinary differential equations ⓘ
hasVersion	global existence version under growth conditions ⓘ local existence version ⓘ
implies	existence of Carathéodory solutions ⓘ
mayGuarantee	local existence of solutions ⓘ maximal interval of existence for solutions ⓘ
namedAfter	Constantin Carathéodory ⓘ
relatedTo	Filippov theory of differential equations with discontinuous right-hand sides ⓘ Peano existence theorem ⓘ
reliesOn	Lebesgue integration ⓘ absolute continuity of functions ⓘ
requires	for each compact set in state space, an integrable majorant of f ⓘ local boundedness of f by an integrable function ⓘ
sometimesGuarantees	uniqueness of solutions under additional conditions ⓘ
statedFor	systems of ordinary differential equations ⓘ vector-valued unknown functions ⓘ
strengthens	pure measurability assumptions by continuity in state variable ⓘ
topicIn	theory of initial value problems ⓘ
typeOfSolution	Carathéodory solution ⓘ linked to: Carathéodory existence theorem
usedIn	control theory ⓘ differential inclusions and related generalizations ⓘ measure differential equations ⓘ nonlinear dynamical systems ⓘ
usesWeakerConditionsThan	Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness theorem

How these facts were elicited

Referenced by (2)

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

Peano existence theorem → relatedConcept → Carathéodory existence theorem ⓘ

Carathéodory existence theorem → typeOfSolution → Carathéodory solution ⓘ

linked to: Carathéodory existence theorem

All labels observed (2)

How this entity was disambiguated Show

Statements (45)

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Referenced by (2)

How this entity was disambiguated

How these facts were elicited