Second-Order Projection Dynamical Model for Solving Inverse Mixed Variational Inequalities and Application

In this paper, we introduce a new projection-based second-order dynamical system for solving inverse mixed variational inequality problems (IMVIs) with two time-dependent parameters. Such problems arise naturally in optimization and equilibrium theory and are often difficult to handle using standard first-order methods. The proposed dynamical model is designed by combining inertial effects with projection operators, which improves both stability and convergence behavior.

Under the assumptions that the underlying operator is strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solutions to the proposed system. Moreover, we prove that the equilibrium point of the dynamical system is globally stable. The stability analysis is carried out using an appropriate Lyapunov function, which provides a clear theoretical justification for the convergence of the trajectories.

Additionally, a discrete-time version of the continuous model is derived, resulting in an inertial projection-type iterative algorithm. It is demonstrated that, under suitable parameter choices, the generated sequence converges linearly to the unique solution of the IMVI. This discrete scheme is simple to implement and can be viewed as an efficient numerical realization of the continuous dynamics.

Finally, several numerical experiments are presented to illustrate the effectiveness of the proposed approach. The results confirm the theoretical findings and demonstrate the method's accuracy, fast convergence, and robustness when applied to inverse mixed variational inequalities and related optimization problems.