Suppose $u_{i+1}=f(u_i)$, for $i=0, 1, 2,...$, and $z_i=u_i-\alpha\cdot(u_{i+1}-u_i)$. Furthermore, let $\Delta u_i=f(u_i)-u_i$, and let $v_i=f(f(u_i))- 2f(u_i)+u_i.$ Can it be shown that $\Delta z_i= \Delta u_i - \alpha\cdot v_i$
The question was raised from (p. 26): http://biostats.bepress.com/cgi/viewcontent.cgi?article=1063&context=jhubiostat