In mathematics, Gronwall's inequality (also called Grönwall's lemma, Gronwall's lemma or Gronwall–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. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants.
Grönwall's lemma is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it is used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem.
It is named for Thomas Hakon Grönwall (1877–1932). Grönwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publications after emigrating to the United States.
The differential form was proven by Grönwall in 1919. The integral form was proven by Richard Bellman in 1943.
A nonlinear generalization of the Gronwall–Bellman inequality is known as Bihari's inequality.
Read more about Gronwall's Inequality: Differential Form, Integral Form For Continuous Functions, Integral Form With Locally Finite Measures
Famous quotes containing the word inequality:
“Nature is unfair? So much the better, inequality is the only bearable thing, the monotony of equality can only lead us to boredom.”
—Francis Picabia (1878–1953)