The Douglas–Rachford splitting method is a classical and powerful method that is widely used in engineering fields for finding a zero of the sum of two operators. In this paper, we begin by proposing an abstract second-order dynamic system involving a generalized cocoercive operator to find a zero of the operator in a real Hilbert space. Then we develop a second-order adaptive Douglas–Rachford dynamic system for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With proper tuning of the parameters such that the adaptive Douglas–Rachford operator is quasi-nonexpansive, we demonstrate that the trajectory of the proposed adaptive system converges weakly to a fixed point of the adaptive operator. When the strong monotonicity strictly outweighs the weak one, we further derive the strong convergence of shadow trajectories to the solution of the original problem. Finally, two simulation examples are reported to corroborate the effectiveness of the proposed adaptive system.

Douglas–Rachford分裂方法是一种经典而强大的方法，广泛用于工程领域中，以寻找两个算子之和为零。在本文中，我们从提出一个抽象的二阶动态系统开始，该系统涉及一个广义的矫顽算子，以在真实的希尔伯特空间中找到该算子的零。然后，我们开发了一个二阶自适应Douglas-Rachford动力学系统，用于查找两个算子之和的零，其中一个是强单调的，而另一个是弱单调的。通过适当调整参数，使自适应Douglas–Rachford算子是拟非扩张的，我们证明了所提出的自适应系统的轨迹微弱地收敛到了自适应算子的一个固定点。当强单调性严格胜过弱单调性时，我们进一步推导了阴影轨迹的强收敛性，以解决原始问题。最后，报告了两个仿真示例，以证实所提出的自适应系统的有效性。