Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators

In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space \(\mathcal {H}\). We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.