Picard iteration - Knowlpedia

For $y'=F(t,y)$ , $y(t_0)=y_0$ , Picard iteration starts from a trial curve, often $y^{(0)}(t)=y_0$ , and sets

y^{(n+1)}(t)=y_0+\int_{t_0}^tF(s,y^{(n)}(s))\,ds.

This is fixed-point iteration for the integral equation.

Convergence mechanism

On a time interval of length at most $h$ from $t_0$ , a state Lipschitz bound $L$ gives $\|Ty-Tz\|_{\sup}\le hL\|y-z\|_{\sup}$ . One must also keep the iterates in a ball where $F$ is defined and bounded. The local existence theorem makes both requirements precise. Repeated time integration also gives factorial bounds, which can prove convergence without a one-step contraction on a longer interval.