Second-order dynamical system to solve the sum of monotone operators via the KM algorithm

Dynamical systems serve as powerful tools for studying optimization and fixed-point algorithms. Motivated by this perspective, we take a problem to find solution of the form 0 ∈ A ( y ) + B ( y ) in Hilbert spaces using a dynamical system approach based on the Krasnoselskii–Mann method, where A is a set-valued operator, while B is a single-valued operator. Specifically, we propose a second-order dynamical system that arises as the continuous limit of the Krasnoselskii–Mann algorithm and analyze its stability using a Lyapunov function. We also conduct numerical applications to complement the theoretical analysis and visualize the system’s behavior through graphical representations, which enhance the understanding of the inclusion problem. Furthermore, we extend our study by deriving a novel dynamical system using the heavy ball method.