Asymptotic behaviour of a nonautonomous evolution equation governed by a quasi-nonexpansive operator. (9th December 2022)

• Record Type: Journal Article
• Authors: Zhu, Ming; Hu, Rong; Fang, Ya-Ping
• Is Part Of: Optimization. Volume 71:Number 15(2022)
• Page Start: 4463
• Page End: 4495
• Publication Date: 2022-12-09
• Subjects: Nonautonomous evolution equation -- Adaptive Douglas-Rachford dynamical system -- Quasi-nonexpansive operator -- Asymptotic behavior -- Metric subregularity; 34G25 -- 37N40 -- 46N10 -- 47H05 -- 47H09 -- 47H10 -- 90C25; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2021.1949314
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : 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.