Convergence of nonlinear semigroups under nonpositive curvature

TL;DR

This paper extends classical convergence results of semigroups and resolvents from linear spaces to metric spaces with nonpositive curvature, using Mosco convergence of convex functions.

Contribution

It generalizes the convergence of gradient flow semigroups and resolvents to non-linear, nonpositively curved metric spaces, expanding the scope of classical analysis.

Findings

Mosco convergence implies resolvent convergence

Semigroups converge under Mosco convergence in nonpositive curvature spaces

Addresses convergence on sequences of spaces, solving a previous open problem

Abstract

The present paper is devoted to semigroups of nonexpansive mappings on metric spaces of nonpositive curvature. We show that the Mosco convergence of a sequence of convex lsc functions implies convergence of the corresponding resolvents and convergence of the gradient flow semigroups. This extends the classical results of Attouch, Brezis and Pazy into spaces with no linear structure. The same method can be further used to show the convergence of semigroups on a sequence of spaces, which solves a problem of [Kuwae and Shioya, Trans. Amer. Math. Soc., 2008].