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We study the asymptotic behavior of the trajectory of a nonautonomousevolution equation governed by a quasi-nonexpansive operator in Hilbert spaces.We prove the weak convergence of the trajectory to a fixed point of theoperator by relying on Lyapunov analysis. Under a metric subregularitycondition, we further derive a flexible global exponential-type rate for thedistance of the trajectory to the set of fixed points. The results obtained areapplied to analyze the asymptotic behavior of the trajectory of an adaptiveDouglas-Rachford dynamical system, which is applied for finding a zero of thesum of two operators, one of which is strongly monotone while the other one isweakly monotone.

Asymptotic behavior of a nonautonomous evolution equation governed by a quasi-nonexpansive operator