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In mathematics, the Picard-Lindelöf theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem

<math>y'(t)=f(t,y(t)),\quad y(t_0)=y_0<math>

has exactly one solution if f is Lipschitz continuous in <math>y<math>, continuous in <math>t<math> as long as <math>y(t)<math> stays bounded.

A simple proof of existence of the solution is successive approximation: (also called Picard iteration)

Set

<math>\varphi_0(t)=y_0 \,\!<math>

and

<math>\varphi_i(t)=y_0+\int_{t_0}^{t}f(s,\varphi_{i-1}(s))\,ds.<math>

It can then be shown rather easily that the sequence of the <math>\varphi_i \,\!<math> (called the Picard iterates) is convergent and that the limit is a solution to the problem.

An application of Grönwall's lemma to <math>|\phi(t)-\psi(t)|<math>, where <math>\phi<math> and <math>\psi<math> are two solutions, shows that <math>\phi(t)\equiv\psi(t)<math>, thus proving the uniqueness.

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