On a theorem about Mosco convergence in Hadamard spaces

TL;DR

This paper establishes a converse relationship between the convergence of proximal mappings and Mosco convergence of convex functions in Hadamard spaces, extending previous theoretical results.

Contribution

It proves that convergence of proximal mappings implies Mosco convergence of convex functions in Hadamard spaces, providing a new insight into their relationship.

Findings

Proximal mapping convergence implies Mosco convergence.

Extends Bacak's theorem to a converse statement.

Provides conditions under which the implication holds.

Abstract

Let $(f^n),f$ be a sequence of proper closed convex functions defined on a Hadamard space. We show that the convergence of proximal mappings $J^n_{\lambda}x$ to $J_{\lambda}x$, under certain additional conditions, imply Mosco convergence of $f^n$ to $f$. This result is a converse to a theorem of Bacak about Mosco convergence in Hadamard spaces.