In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of three monotone operators are analyzed. The framework modifies the forward–backward–forward dynamical system by adding a cocoercive operator to the inclusion. The existence, uniqueness, and weak asymptotic convergence of the generated trajectories are discussed. A variable metric forward–backward–half forward dynamical system with the essence of non-self-adjoint linear operators is introduced. The proposed notion, in turn, extends the forward–backward–forward dynamical system and forward–backward dynamical system in the framework of variable metric by relaxing some conditions on the metrics. The distributed dynamical system is further explored to compute a generalized Nash equilibrium in a monotone game as an application. A numerical example is provided to illustrate the convergence of trajectories.

本文分析了与包含三个单调算子的包含问题相关的一阶前向-后向-半前向动力系统。该框架通过向包含中添加强制算子来修改向前-向后-向前动态系统。讨论了生成轨迹的存在性、唯一性和弱渐近收敛性。介绍了一种具有非自伴随线性算子本质的变度量前向-后向-半前向动力系统。反过来，所提出的概念通过放宽度量的一些条件，在可变度量的框架中扩展了前向-后向-前向动力系统和前向-后向动力系统。进一步探索分布式动态系统以计算单调博弈中的广义纳什均衡作为应用程序。提供了一个数值例子来说明轨迹的收敛。