Gronwall inequality
E695946
Gronwall inequality is a fundamental result in analysis that provides bounds on functions satisfying certain integral or differential inequalities, widely used to prove uniqueness and stability of solutions to differential equations.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Bellman–Gronwall inequality | 1 |
| Gronwall inequality canonical | 1 |
| Gronwall–Bellman inequality | 1 |
| discrete Gronwall inequality | 1 |
| generalized Gronwall inequality | 1 |
| linear Gronwall inequality | 1 |
| nonlinear Gronwall inequality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7833350 — 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: Gronwall inequality Context triple: [Lyapunov inequality, relatedTo, Gronwall inequality]
-
A.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
-
B.
Young's inequality
Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
-
C.
Korn inequality
Korn inequality is a fundamental result in functional analysis and the mathematical theory of elasticity that provides bounds relating the full gradient of a vector field to its symmetric part, ensuring control of deformations by their strains.
-
D.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
E.
Sobolev inequality
The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
- 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: Gronwall inequality Target entity description: Gronwall inequality is a fundamental result in analysis that provides bounds on functions satisfying certain integral or differential inequalities, widely used to prove uniqueness and stability of solutions to differential equations.
-
A.
Lyapunov inequality
The Lyapunov inequality is a fundamental result in stability theory and analysis that provides bounds relating norms or moments of functions or solutions to differential equations, widely used in studying the stability of dynamical systems.
-
B.
Young's inequality
Young's inequality is a fundamental result in mathematical analysis that provides an upper bound for the product of two nonnegative numbers in terms of their powers, playing a key role in convex analysis and functional inequalities.
-
C.
Korn inequality
Korn inequality is a fundamental result in functional analysis and the mathematical theory of elasticity that provides bounds relating the full gradient of a vector field to its symmetric part, ensuring control of deformations by their strains.
-
D.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
E.
Sobolev inequality
The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in mathematical analysis ⓘ |
| alsoKnownAs |
Bellman–Gronwall inequality
ⓘ
linked to:
Gronwall inequality
Gronwall–Bellman inequality ⓘ
linked to:
Gronwall inequality
|
| appearsIn |
textbooks on differential equations
ⓘ
textbooks on functional analysis ⓘ textbooks on real analysis ⓘ |
| appliesTo |
locally integrable functions
ⓘ
nonnegative functions ⓘ |
| consequence |
continuous dependence of solutions on parameters
ⓘ
uniqueness of solutions to initial value problems ⓘ |
| field |
analysis
ⓘ
differential equations ⓘ integral equations ⓘ |
| hasForm |
differential inequality
ⓘ
integral inequality ⓘ |
| hasVariant |
discrete Gronwall inequality
ⓘ
linked to:
Gronwall inequality
generalized Gronwall inequality ⓘ
linked to:
Gronwall inequality
linear Gronwall inequality ⓘ
linked to:
Gronwall inequality
nonlinear Gronwall inequality ⓘ
linked to:
Gronwall inequality
|
| implies | growth of solutions is at most exponential under given conditions ⓘ |
| mathematicalSubjectClassification |
26D10
ⓘ
34A40 ⓘ |
| namedAfter | Thomas Hakon Gronwall ⓘ |
| relatedTo |
Lyapunov stability theory
ⓘ
a priori bounds ⓘ comparison principle ⓘ |
| timeDomain | usually stated on an interval of the real line ⓘ |
| typicalAssumption | integrand is bounded by linear term in the unknown function ⓘ |
| typicalConclusion | exponential bound on the unknown function ⓘ |
| typicalCondition |
integrating factor is nonnegative
ⓘ
kernel function is integrable ⓘ |
| typicalFormulation |
differential form involving derivative of the function
ⓘ
integral form with upper limit t and integration variable s ⓘ |
| usedFor |
a priori estimates
ⓘ
bounding solutions of differential inequalities ⓘ comparison of solutions of differential equations ⓘ continuous dependence on initial data ⓘ proving stability of solutions to differential equations ⓘ proving uniqueness of solutions to differential equations ⓘ |
| usedIn |
control theory
ⓘ
dynamical systems ⓘ numerical analysis of differential equations ⓘ ordinary differential equations ⓘ partial differential equations ⓘ stochastic differential equations ⓘ |
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: Gronwall inequality Description of subject: Gronwall inequality is a fundamental result in analysis that provides bounds on functions satisfying certain integral or differential inequalities, widely used to prove uniqueness and stability of solutions to differential equations.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gronwall inequality
linked to: Gronwall inequality
linked to: Gronwall inequality
linked to: Gronwall inequality
linked to: Gronwall inequality
linked to: Gronwall inequality