# Picard–Lindelöf theorem

(Redirected from Picard Iteration)

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 $\psi$ are two solutions, shows that $\varphi (t)\equiv \psi (t)$, thus proving the uniqueness.