physics - Picard-Lindelöf theorem

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

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

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 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 $| φ(t) - ψ(t) |$ , where $φ$ and $ψ$ are two solutions, shows that $\phi(t)\equiv\psi(t)$ , thus proving the uniqueness.

Picard-Lindelöf theorem

See also