Forward-Backward-Forward Dynamical System for Solving Mixed Variational Inequality Problems

TL;DR

This paper introduces a forward-backward-forward dynamical system designed to solve mixed variational inequality problems in Hilbert spaces, demonstrating convergence and stability under certain conditions, supported by numerical examples.

Contribution

The paper proposes a novel dynamical system approach with convergence and stability analysis for mixed variational inequalities, extending existing methods.

Findings

Weak convergence of trajectories under Lipschitz and monotonicity conditions

Global exponential stability with h-strong pseudomonotonicity

Numerical examples confirming convergence behavior

Abstract

We study in this paper a forward-backward-forward dynamical system for solving a mixed variational inequality problem in a real Hilbert space. For the convergence analysis of our proposed system, we apply the Lyapunov analysis to obtain the weak convergence of the generated trajectories when the associated operator is Lipschitz continuous and satisfies the general monotonicity condition. We also assume that the involved real-valued convex function satisfies some mild assumptions. Furthermore, the Lipschitz continuous operator is taken to be $h-$strongly pseudomonotone to establish the global exponential stability of the equilibrium point of the system for all the orbits generated. Finally, we present some numerical examples which illustrate how the trajectories of the proposed system converge to the equilibrium point of the proposed dynamical system.