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In a Hilbert framework, we introduce continuous and discrete dynamicalsystems which aim at solving inclusions governed by structured monotoneoperators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of aconvex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoerciveoperator. We first consider the extension to this setting of the regularizedNewton dynamic with two potentials. Then, we revisit some related dynamicalsystems, namely the semigroup of contractions generated by $A$, and thecontinuous gradient projection dynamic. By a Lyapunov analysis, we show theconvergence properties of the orbits of these systems.  The time discretization of these dynamics gives various forward-backwardsplitting methods (some new) for solving structured monotone inclusionsinvolving non-potential terms. The convergence of these algorithms is obtainedunder classical step size limitation. Perspectives are given in the field ofnumerical splitting methods for optimization, and multi-criteria decisionprocesses.

Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator