local existence and uniqueness theorem

E22820

mathematical theorem theorem in differential equations

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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All labels observed (8)

Label	Occurrences
Picard–Lindelöf theorem	9
Cauchy–Lipschitz theorem	2
local existence and uniqueness theorem canonical	2
Lipschitz condition	1
Picard existence theorem	1
Picard–Lindelöf existence theorem	1
Picard–Lindelöf theorem for ordinary differential equations	1
existence and uniqueness theorem	1

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]

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
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	Cauchy–Lipschitz theorem ⓘ linked to: local existence and uniqueness theorem Picard existence theorem ⓘ linked to: local existence and uniqueness theorem Picard–Lindelöf theorem ⓘ linked to: local existence and uniqueness 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	existence and uniqueness theorem ⓘ linked to: local 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 in terms of uniqueness ⓘ linked to: Peano existence theorem
usesConcept	Lipschitz condition ⓘ complete metric space ⓘ fixed-point theorem ⓘ
weakerThan	global existence and uniqueness theorems ⓘ

How these facts were elicited