# Monotone vector fields and generation of nonexpansive semigroups in complete CAT(0) spaces

## TL;DR

This paper introduces a new approach to monotone vector fields in complete CAT(0) spaces, extending the theory beyond Hilbert spaces and Hadamard manifolds, and proves a generation theorem for nonexpansive semigroups.

## Contribution

It develops a novel concept of monotonicity in CAT(0) spaces using tangent space structure, and establishes convergence of exponential formulas for generating nonexpansive semigroups.

## Key findings

- Extended monotone vector field theory to CAT(0) spaces
- Proved convergence of exponential formulas for resolvents
- Improved existing generation theorems in the literature

## Abstract

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the monotonicity defined earlier in CAT(0) spaces by Khatibzadeh and Ranjbar [14] and Chaipunya and Kumam [8]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [13, Theorem 1.3.13], Mayer [19, Theorem 1.13], Stojkovic [22, Theorem 2.18], and Bac\'ak [4, Theorem 1.5] for proper, convex, lower semicontinuous functions in the context of complete CAT(0) spaces, and also by Iwamiya and Okochi [11, Theorem 4.1] for monotone vector fields in the context of Hadamard manifolds.