= Picard operator on a closed ball
{c}
{title2=$(\mathcal Pg)(t)=b+\int_a^t f(s,g(s))\,ds$}
= Picard operator
{c}
{synonym}
Suppose $f$ is continuous and Lipschitz in its state variable on a closed cylinder $|t-a|\leq h$, $\|y-b\|\leq k$, with bounds $\|f\|\leq M$ and Lipschitz constant $L_f$. On a smaller interval of radius $\eta$, the displayed <Picard operator> maps continuous curves in that <closed ball> into itself when $\eta M\leq k$, and it is a <contraction mapping> in the <supremum norm> when $\eta L_f<1$. This function space is complete because it is a closed subset of a <Banach space> of continuous vector-valued <functions>. The <contraction mapping theorem> gives a unique <fixed point>, and the <fundamental theorem of calculus> makes it a solution of the <ordinary differential equation>. Choosing $\eta M<k$ also prevents any solution with the same initial data from first exiting the cylinder during that interval.
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