A posteriori contraction ball
= A posteriori contraction ball
Let $A$ be differentiable on a neighbourhood of a closed radius-$r$ ball in a <Banach space>, with $\|DA\|\leq\kappa<1$ on that ball. If $\|A(p_0)-p_0\|\leq(1-\kappa)r$, the mean-value estimate gives $A(B)\subseteq B$ and $\|A(p)-A(q)\|\leq\kappa\|p-q\|$. The <contraction mapping theorem> supplies exactly one <fixed point> in $B$, and its distance from $p_0$ is at most $\|A(p_0)-p_0\|/(1-\kappa)$.
// Target: dynamical-systems.bigb