标题： 具有正向-反向-正向微分方程及其离散版本的渐近性质用于解决拟单调变分不等式 显示英文标题

标题： Asymptotic Properties of a Forward-Backward-Forward Differential Equation and Its Discrete Version for Solving Quasimonotone Variational Inequalities

摘要： 本文研究了一种前向-后向-前向（FBF）型微分方程及其离散对应形式在求解准单调变分不等式（VIs）中的渐近行为。 在最近针对VIs的连续时间动力系统框架基础上，我们将这些方法扩展以适应准单调算子。 我们在显著放松的条件下建立了弱收敛和强收敛，而无需要求强伪单调性或序列弱到弱连续性。 此外，我们证明了连续轨迹的遍历收敛性，进一步揭示了系统的长期稳定性。 在离散情况下，我们提出了一种新颖的Bregman型算法，该算法结合了基于黄金比例技术的非单调自适应步长规则。 本工作的关键贡献是证明了所提出的算法在算子一致连续的假设下确保强收敛，从而放松了现有方法中普遍存在的标准Lipschitz连续性要求。 数值实验包括无限维和非Lipschitz情形，以说明所提出方法的改进收敛性和更广泛的应用性。 显示英文摘要

摘要： This paper investigates the asymptotic behavior of a forward-backward-forward (FBF) type differential equation and its discrete counterpart for solving quasimonotone variational inequalities (VIs). Building on recent continuous-time dynamical system frameworks for VIs, we extend these methods to accommodate quasimonotone operators. We establish weak and strong convergence under significantly relaxed conditions, without requiring strong pseudomonotonicity or sequential weak-to-weak continuity. Additionally, we prove ergodic convergence of the continuous trajectories, offering further insight into the long-term stability of the system. In the discrete setting, we propose a novel Bregman-type algorithm that incorporates a nonmonotone adaptive step-size rule based on the golden ratio technique. A key contribution of this work is demonstrating that the proposed method ensures strong convergence under the assumption of uniform continuity of the operator, thereby relaxing the standard Lipschitz continuity requirement prevalent in existing methods. Numerical experiments, including infinite-dimensional and non-Lipschitz cases, are presented to illustrate the improved convergence and broader applicability of the proposed approach.