Convergence rates for boundedly regular systems

Abstract

In this work, we consider a continuous dynamical system associated with the fixed point set of a nonexpansive operator which was originally studied by Boţ and Csetnek (J. Dyn. Diff. Equat. 29(1), pp. 155–168, 2017). Our main results establish convergence rates for the system’s trajectories when the nonexpansive operator satisfies an additional regularity property. This setting is the natural continuous-time analogue to discrete-time results obtained in Bauschke, Noll and Phan (J. Math. Anal. Appl. 421(1), pp. 1–20, 2015) and Borwein, Li and Tam (SIAM J. Optim. 27(1), pp. 1–33, 2017) by using the same regularity properties. Closure properties of the class of Hölder regular operators under taking convex combinations and compositions are also derived.