Peano existence theorem
E128381
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Peano existence theorem canonical | 2 |
| 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]
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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.
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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.
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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.
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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.
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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.
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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 |
local existence and uniqueness theorem
ⓘ
surface form:
Picard–Lindelöf 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 |
local existence and uniqueness theorem
ⓘ
surface form:
Picard–Lindelöf theorem
|
| hasWeakerRegularityAssumptionThan |
local existence and uniqueness theorem
ⓘ
surface form:
Picard–Lindelöf 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 ⓘ local existence and uniqueness theorem ⓘ
surface form:
Lipschitz condition
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
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Instruction
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Input
Subject: Peano existence theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Peano existence theorem in terms of uniqueness