local existence and uniqueness theorem
E22820
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.
All labels observed (7)
How this entity was disambiguated
This entity first appeared as the object of triple T179425 — 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: local existence and uniqueness theorem Context triple: [implicit function theorem, logicalForm, local existence and uniqueness theorem]
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A.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
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B.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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C.
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.
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D.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- 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: local existence and uniqueness theorem Target entity description: 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.
-
A.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
-
B.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
C.
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.
-
D.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in differential equations ⓘ |
| alsoKnownAs |
local existence and uniqueness theorem
ⓘ
surface form:
Cauchy–Lipschitz theorem
local existence and uniqueness theorem ⓘ
surface form:
Picard existence theorem
local existence and uniqueness theorem ⓘ
surface form:
Picard–Lindelöf theorem
|
| appliesTo |
equations of the form y' = f(t,y)
ⓘ
initial conditions of the form y(t0) = y0 ⓘ systems of ordinary differential equations ⓘ |
| assumes |
f is continuous in t
ⓘ
f is locally Lipschitz in y ⓘ |
| category |
local existence and uniqueness theorem
self-linksurface differs
ⓘ
surface form:
existence and uniqueness theorem
|
| concerns |
first-order ordinary differential equations
ⓘ
initial value problems ⓘ |
| concludes |
existence of a unique solution on some neighborhood of t0
ⓘ
existence of an interval around t0 where a solution exists ⓘ |
| ensures | solution is defined on some open interval containing t0 ⓘ |
| field |
analysis
ⓘ
ordinary differential equations ⓘ |
| guarantees |
existence of a solution through a given initial point
ⓘ
local existence of solutions ⓘ local uniqueness of solutions ⓘ uniqueness of a solution through a given initial point ⓘ |
| hasConsequence |
deterministic behavior of solutions near initial data
ⓘ
well-posedness of local initial value problems ⓘ |
| hasScope | local in time ⓘ |
| historicallyAssociatedWith |
Augustin-Louis Cauchy
ⓘ
Ernst Lindelöf ⓘ Rudolf Lipschitz ⓘ Émile Picard ⓘ |
| implies | solutions depend continuously on initial data (locally) ⓘ |
| isPartOf | theory of initial value problems ⓘ |
| isProvedBy |
Banach fixed-point theorem
ⓘ
Picard iteration ⓘ |
| isTaughtIn |
introductory analysis courses
ⓘ
undergraduate differential equations courses ⓘ |
| motivates |
study of Lipschitz continuity
ⓘ
use of contraction mappings in analysis ⓘ |
| relatedTo |
Peano existence theorem
ⓘ
global existence theorems ⓘ |
| requiresConditionOn |
Lipschitz continuity of f in y
ⓘ
continuity of f in a neighborhood of (t0,y0) ⓘ local Lipschitz condition in y ⓘ |
| strongerThan |
Peano existence theorem
ⓘ
surface form:
Peano existence theorem in terms of uniqueness
|
| usesConcept |
Lipschitz condition
ⓘ
complete metric space ⓘ fixed-point theorem ⓘ |
| weakerThan | global existence and uniqueness theorems ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: local existence and uniqueness theorem Description of subject: 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.
Referenced by (14)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Picard–Lindelöf theorem
this entity surface form:
Cauchy–Lipschitz theorem
this entity surface form:
Picard existence theorem
local existence and uniqueness theorem
→
category
→
local existence and uniqueness theorem
self-linksurface differs
ⓘ
this entity surface form:
existence and uniqueness theorem
this entity surface form:
Picard–Lindelöf theorem
this entity surface form:
Picard–Lindelöf theorem
this entity surface form:
Picard–Lindelöf theorem
Peano existence theorem
→
hasStrongerExistenceConditionThan
→
local existence and uniqueness theorem
ⓘ
this entity surface form:
Picard–Lindelöf theorem
Peano existence theorem
→
hasWeakerRegularityAssumptionThan
→
local existence and uniqueness theorem
ⓘ
this entity surface form:
Picard–Lindelöf theorem
this entity surface form:
Lipschitz condition
this entity surface form:
Picard–Lindelöf theorem for ordinary differential equations
this entity surface form:
Cauchy–Lipschitz theorem