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

TL;DR

This paper investigates the long-term behavior of trajectories in a nonautonomous evolution equation driven by a quasi-nonexpansive operator in Hilbert spaces, establishing weak convergence and exponential rates under certain conditions.

Contribution

It provides new convergence results and rate estimates for nonautonomous evolution equations involving quasi-nonexpansive operators, including applications to adaptive Douglas-Rachford systems.

Findings

Weak convergence of trajectories to fixed points.

Exponential-type convergence rates under metric subregularity.

Application to adaptive Douglas-Rachford dynamical system.

Abstract

We study the asymptotic behavior 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 analyze the asymptotic behavior 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.