Third-Order Resolvent Dynamical Systems for Mixed Variational Inequalities: Convergence and Stability Analysis

Abstract

This paper investigates a class of resolvent dynamical systems for developing inertial proximal methods to solve mixed variational inequalities. Third-order dynamics introduce additional inertia and damping effects, enabling the modeling of complex real-world systems while balancing stability and convergence speed. By exploiting the equivalence between the stationary points of the proposed dynamical system and the solutions of mixed variational inequalities, we show that the system trajectories converge to the unique solution of the problem. The convergence properties of the proposed inertial proximal methods are analyzed under mild conditions, assuming only monotonicity of the underlying operators. Furthermore, using a Lyapunov function framework, this study establishes the global asymptotic and exponential stability of the equilibrium points without requiring explicit solutions of the system. The proposed implicit and explicit discretization schemes provide a continuous-time perspective for designing efficient algorithms for solving mixed variational inequalities. To the best of our knowledge, this is the first work to apply third-order dynamical systems to mixed variational inequalities, offering a foundation for future extensions to stochastic, nonmonotone, and nonconvex equilibrium problems.