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.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in mathematical analysis ⓘ |
| alsoKnownAs |
Bellman–Gronwall inequality
NERFINISHED
ⓘ
Gronwall–Bellman inequality NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
generalized Gronwall inequality NERFINISHED ⓘ linear Gronwall inequality NERFINISHED ⓘ nonlinear Gronwall inequality NERFINISHED ⓘ |
| implies | growth of solutions is at most exponential under given conditions ⓘ |
| mathematicalSubjectClassification |
26D10
ⓘ
34A40 ⓘ |
| namedAfter | Thomas Hakon Gronwall NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
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