We study the asymptotic behaviour of the trajectory of a nonautonomous evolution equation governed by a quasi-nonexpansive operator in Hilbert spaces. We prove the weak convergence of the trajectory to a fixed point of the operator by relying on Lyapunov analysis. Under a metric subregularity condition, we further derive a flexible global exponential-type rate for the distance of the trajectory to the set of fixed points. The results obtained are applied to analyse the asymptotic behaviour of the trajectory of an adaptive Douglas–Rachford dynamical system, which is applied for finding a zero of the sum of two operators, one of which is strongly monotone while the other one is weakly monotone.

我们研究了希尔伯特空间中由准非膨胀算子控制的非自治演化方程轨迹的渐近行为。我们依靠 Lyapunov 分析证明了轨迹到算子固定点的弱收敛性。在度量子正则条件下，我们进一步推导出轨迹到固定点集的距离的灵活全局指数型速率。所得结果用于分析自适应道格拉斯-拉赫福德动力系统轨迹的渐近行为，该系统用于寻找两个算子之和的零，其中一个算子是强单调的，另一个是弱单调的。