Third-Order Dynamical Systems for Generalized Inverse Mixed Variational Inequality Problems

In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the system under suitable continuity assumptions, and prove their exponential convergence to the unique solution under strong monotonicity and Lipschitz continuity conditions. Furthermore, we derive an explicit discretization of the proposed dynamical system, leading to a forward -backward algorithm with double inertial effects. We then establish the linear convergence of the generated iterates to the unique solution.

💡 Research Summary

The manuscript addresses the problem of solving a generalized inverse mixed variational inequality (GIMVI) in a real Hilbert space H. The GIMVI unifies several classical models: given a closed convex set Ω⊂H, continuous operators F,g:H→H, and a proper convex lower‑semicontinuous function h:Ω→ℝ∪{+∞}, one seeks w*∈H such that F(w*)∈Ω and
⟨g(w*), v − F(w*)⟩ + h(v) − h(F(w*)) ≥ 0 for all v∈Ω.

Main contributions

1. Third‑order continuous‑time dynamics. The authors propose the ODE
   w‴(t)+a₂ w″(t)+a₁ w′(t)+a₀