  In a Hilbert framework, we introduce continuous and discrete dynamical
systems which aim at solving inclusions governed by structured monotone
operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a
convex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoercive
operator. We first consider the extension to this setting of the regularized
Newton dynamic with two potentials. Then, we revisit some related dynamical
systems, namely the semigroup of contractions generated by $A$, and the
continuous gradient projection dynamic. By a Lyapunov analysis, we show the
convergence properties of the orbits of these systems.
  The time discretization of these dynamics gives various forward-backward
splitting methods (some new) for solving structured monotone inclusions
involving non-potential terms. The convergence of these algorithms is obtained
under classical step size limitation. Perspectives are given in the field of
numerical splitting methods for optimization, and multi-criteria decision
processes.
