Finding the forward-Douglas-Rachford-forward method

We consider the monotone inclusion problem with a sum of 3 operators, in which 2 are monotone and 1 is monotone-Lipschitz. The classical Douglas–Rachford and Forward-backward-forward methods respectively solve the monotone inclusion problem with a sum of 2 monotone operators and a sum of 1 monotone and 1 monotone-Lipschitz operators. We first present a method that naturally combines Douglas–Rachford and Forward-backward-forward and show that it solves the 3 operator problem under further assumptions, but fails in general. We then present a method that naturally combines Douglas–Rachford and forward-reflected-backward, a recently proposed alternative to Forward-backward-forward by Malitsky and Tam [arXiv:1808.04162, 2018]. We show that this second method solves the 3 operator problem generally, without further assumptions.