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

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.

Keywords:

• Nonautonomous evolution equation
• Adaptive Douglas-Rachford dynamical system
• Quasi-nonexpansive operator
• Asymptotic behavior
• Metric subregularity

AMS SUBJECT CLASSIFICATIONS:

• 34G25
• 37N40
• 46N10
• 47H05
• 47H09
• 47H10
• 90C25

Acknowledgments

The authors would like to thank the referees and the editor for their helpful comments and suggestions which have led to the improvement of the earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Science Foundation of China [grant number 11471230] and the Scientific Research Foundation of the Education Department of Sichuan Province [grant number 16ZA0213].