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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 Carathéodory's existence theorem ： ウィキペディア英語版 Carathéodory's existence theorem In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation is continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory. == Introduction == Consider the differential equation :$y\text{'}(t)\; =\; f(t,y(t))\; \backslash ,$ with initial condition :$y(t\_0)\; =\; y\_0,\; \backslash ,$ where the function ƒ is defined on a rectangular domain of the form :$R\; =\; \backslash ^n\; \backslash ,:\backslash ,\; |t-t\_0|\; \backslash le\; a,\; |y-y\_0|\; \backslash le\; b\; \backslash \}.$ Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.〔, Theorem 1.2 of Chapter 1〕 However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation :$y\text{'}(t)\; =\; H(t),\; \backslash quad\; y(0)\; =\; 0,$ where ''H'' denotes the Heaviside function defined by : It makes sense to consider the ramp function : as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at $t=0$, because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition. A function ''y'' is called a ''solution in the extended sense'' of the differential equation $y\text{'}\; =\; f(t,y)$ with initial condition $y(t\_0)=y\_0$ if ''y'' is absolutely continuous, ''y'' satisfies the differential equation almost everywhere and ''y'' satisfies the initial condition.〔, page 42〕 The absolute continuity of ''y'' implies that its derivative exists almost everywhere.〔, Theorem 7.18〕 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「Carathéodory's existence 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.