Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} \sum_{j=0}^n f(u_j)$ ; or equivalently $u_{n+1} - u_n = \frac{1}{n+1} (f(u_n) -u_n)$.

Is it always true that $(u_n)$ converges to some fixed point of $f$ ?

- When $E$ is one-dimensional, this is an easy exercise, but all proofs I know use in a crucial way the ordering of the real numbers.
- I don't expect the two-dimensional case to be easier than the general case.