A continuous-time inertial Newton method for structured monotone inclusions

In a real Hilbert space $ {{\rm{H}}} $, we investigate the convergence properties of a second-order inertial dynamic with Newton-type correction, designed to solve structured monotone inclusions of the form $ Ax + Bx \ni 0 $, where $ A : {{\rm{H}}} \rightrightarrows {{\rm{H}}} $ is maximally monotone and $ B : {{\rm{H}}} \to {{\rm{H}}} $ is cocoercive. The system combines viscous damping, inertial effects, and a correction term based on the time derivative of a regularized forward-backward operator. Under suitable assumptions, we establish well-posedness and prove weak convergence of the trajectories. Strong convergence is obtained under additional monotonicity conditions. The analysis relies on a Lyapunov functional tailored to the structure of the dynamics and opens the way for further investigations on discretization schemes and numerical algorithms. To illustrate the theoretical findings, we provide numerical simulations that highlight the impact of the Newton-type correction term on convergence behavior. The experiments explore different damping regimes and show how the proposed dynamic effectively balances stability and acceleration in structured monotone inclusion problems.