Picard–Lindelöf theorem

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

$y'(t)=f(t,y(t)),\quad y(t_0)=y_0$

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

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

Set

$\varphi_0(t)=y_0 \,\!$

and

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

It can then be shown rather easily, by using the Banach fixed point theorem, that the sequence of the $\varphi_i \,\!$ (called the Picard iterates) is convergent and that the limit is a solution to the problem.

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

Picard–Lindelöf theorem

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