The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

Abstract

We study a nonlinear semigroup associated with a nonexpansive mapping on an Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature—both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete as well as continuous times.