Carathéodory existence theorem
E547409
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Carathéodory existence theorem canonical | 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]
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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.
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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.
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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.
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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.
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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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| relatedTo |
Filippov theory of differential equations with discontinuous right-hand sides
NERFINISHED
ⓘ
Peano existence theorem NERFINISHED ⓘ |
| reliesOn |
Lebesgue integration
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| usedIn |
control theory
ⓘ
differential inclusions and related generalizations ⓘ measure differential equations ⓘ nonlinear dynamical systems ⓘ |
| usesWeakerConditionsThan | Picard–Lindelöf theorem NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Carathéodory existence theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.