A dynamical system associated with the fixed points set of a nonexpansive operator

TL;DR

This paper analyzes a dynamical system linked to nonexpansive operators, establishing convergence properties and rates, with applications to finding zeros of sums of monotone and cocoercive operators.

Contribution

It introduces a new dynamical system framework for nonexpansive operators, proving convergence and convergence rates, and unifies several existing systems as special cases.

Findings

Weak convergence of trajectories to fixed points

Convergence rate of o(1/√t) for fixed point residuals

Application to zeros of sums of monotone and cocoercive operators

Abstract

We study the existence and uniqueness of (locally) absolutely continuous trajectories of a dynamical system governed by a nonexpansive operator. The weak convergence of the orbits to a fixed point of the operator is investigated by relying on Lyapunov analysis. We show also an order of convergence of $o(\frac{1}{\sqrt{t}})$ for the fixed point residual of the trajectory of the dynamical system. We apply the results to dynamical systems associated with the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive one. Several dynamical systems from the literature turn out to be particular instances of this general approach.