|
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. There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants. Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a Comparison theorem that can be 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 Grönwall–Bellman inequality is known as Bihari's inequality. Other variants and generalizations can be found in Pachpatte, B.G. (1998). == Differential form == Let denote an interval of the real line of the form or or with . Let and be real-valued continuous functions defined on . If is differentiable in the interior of (the interval without the end points and possibly ) and satisfies the differential inequality : then is bounded by the solution of the corresponding differential ''equation'' : : for all . Remark: There are no assumptions on the signs of the functions and . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grönwall's inequality」の詳細全文を読む スポンサード リンク
|