Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 3 / b / Solution

Write $P_f={\mathrm{prox}}_f$ and $P_h={\mathrm{prox}}_h$. Since $w_k=x_k+z_{k-1}$,and thereforeWith reflected proximal maps $R_f=2P_f-I$ and $R_h=2P_h-I$,Firm nonexpansiveness of each proximal map is equivalent to nonexpansiveness of its reflection. Thus $R_f{R}_h$ is nonexpansive, and its average with the identity is firmly nonexpansive. This is the Douglas–Rachford method.