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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 : 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 .〔, 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 : A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration. Set : and : 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
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