Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex.
Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, $$x_ {k+h} - x_k = \frac{1}{k+h} \left( f(x_{k+h-1})+ ...+ f(x_k) - h x_k \right). $$
Is the sequence $\{x_k\}_k$ Cauchy? If not, under what additional conditions on $f$ does it?