Projective splitting with backward, half-forward and proximal-Newton steps

We propose and study the weak convergence of a projective splitting algorithm for solving multi-term composite monotone inclusion problems involving the finite sum of $n$ maximal monotone operators, each of which having an inner four-block structure: sum of maximal monotone, Lipschitz continuous, cocoercive and smooth differentiable operators. We show how to perform backward and half-forward steps with respect to the maximal monotone and Lipschitz$+$cocoercive components, respectively, while performing proximal-Newton steps with respect to smooth differentiable blocks.