Picard–Lindelöf theorem

From Example Problems
(Redirected from Picard Iteration)
Jump to: navigation, search

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.

See also