= Fixed point theorem for summably Lipschitz iterates
{title2=$\|T^nx-T^ny\|\leq\alpha_n\|x-y\|,\quad\sum_{n\geq1}\alpha_n<\infty$}
Let $T$ be a self-map of a nonempty closed subset of a <Banach space>, with the displayed bounds. Then $T$ has a unique <fixed point> $p$ and every orbit $x_n=T^nx_0$ converges to it. Indeed, $\|x_{n+1}-x_n\|\leq\alpha_n\|x_1-x_0\|$ makes the orbit Cauchy by summing successive increments. The first bound makes $T$ continuous, so the limit is fixed. Since $\alpha_n\to0$, the bounds also imply uniqueness. The tail sum gives $\|p-x_n\|\leq\sum_{k\geq n}\alpha_k\|x_1-x_0\|$ for $n\geq1$. A <Volterra integral equation> supplies factorially decreasing bounds even when the first iterate is not a <contraction mapping>.
Back to article page
