Gronwall's Inequality bounds a function satisfying a certain integral or differential inequality by the solution of the corresponding equation obtained by replacing the inequality with equality, allowing control over how large a quantity can grow once a bound on its own rate of growth is known. Named for Thomas Hakon Gronwall, it is a standard technical tool for proving uniqueness and continuous dependence results for ordinary and partial differential equations.
Facts
StatementFor u'(t) <= beta(t) u(t) on the interior of interval I, u(t) <= u(a) exp(integral from a to t of beta(s) ds). 2 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Wikipedia: Grönwall's inequality
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall-Bellman inequality) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation.
2. Gronwall's inequality, Wikipedia
Differential form section
u(t) less than or equal to u(a) exp(integral a to t of beta(s) ds)
Publication history section
Gronwall published his original result in 1919, in a paper titled Note on the derivatives with respect to a parameter of the solutions of a system of differential equations in the Annals of Mathematics.
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