The <contraction mapping theorem> states that a <contraction mapping> $T:S\to S$ on a nonempty <complete metric space> has a unique <fixed point>. Explicitly, if $d(Tu,Tv)\leq qd(u,v)$ for all $u,v\in S$ and one fixed $0\leq q<1$, then the iterates $u_{n+1}=Tu_n$ converge to that fixed point from every $u_0\in S$. Both self-mapping and completeness are needed; the set need not be a <vector space>.

Solution

Past exam of the differential equations 1 course of the University of Oxford / 2017 / 2 / a / i / Solution

 Solution (source code)