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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Picard–Lindelöf theorem ： ウィキペディア英語版 Picard–Lindelöf theorem In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Consider the initial value problem :$y\text{'}(t)=f(t,y(t)),\backslash qquad\; y(t\_0)=y\_0.$ Suppose is uniformly Lipschitz continuous in (meaning the Lipschitz constant can be taken independent of ) and continuous in . Then, for some value , there exists a unique solution to the initial value problem on the interval $(t\_0+\backslash varepsilon)$.〔, Theorem I.3.1〕 == Proof sketch == The proof relies on transforming the differential equation, and applying fixed-point theory. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :$y(t)\; -\; y(t\_0)\; =\; \backslash int\_^t\; f(s,y(s))\; \backslash ,\; ds.$ A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration. Set :$\backslash varphi\_0(t)=y\_0$ and :$\backslash varphi\_(t)=y\_0+\backslash int\_^t\; f(s,\backslash varphi\_k(s))\backslash ,ds.$ It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" is convergent and that the limit is a solution to the problem. An application of Grönwall's lemma to , where and are two solutions, shows that , thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point). 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「Picard–Lindelöf theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.