Firm non-expansive mappings in weak metric spaces

TL;DR

This paper introduces firm non-expansive mappings in weak metric spaces, generalizing existing concepts from Banach and geodesic spaces, and establishes key properties linking minimal displacement and asymptotic behavior.

Contribution

It extends the concept of non-expansive mappings to weak metric spaces and proves that several asymptotic measures are equal for these mappings, generalizing prior theorems.

Findings

Minimal displacement equals linear rate of escape for firm non-expansive mappings.

Asymptotic step size is equal to minimal displacement.

Generalizes a theorem by Reich and Shafrir to weak metric spaces.

Abstract

We introduce the notion of firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.