Non-linear resolvents of holomorphically accretive mappings

TL;DR

This paper investigates the properties of non-linear resolvents of holomorphically accretive mappings in complex Banach spaces, providing new theoretical insights and estimates using a unified approach and a refined inverse function theorem.

Contribution

It introduces a unified method to analyze non-linear resolvents, proving their accretivity, covering properties, and starlikeness under mild conditions.

Findings

Established a covering result for resolvents

Proved accretivity and squeezing ratio estimates

Showed resolvents are starlike mappings under certain conditions

Abstract

In this paper, we present new results on holomorphically accretive mappings and their resolvents defined on the open unit ball of a complex Banach space. We employ a unified approach to examine various properties of non-linear resolvents by applying a distortion theorem we have established. This method enables us to prove a covering result and to establish the accretivity of resolvents along with estimates of the squeezing ratio. Furthermore, we prove that under certain mild conditions, a non-linear resolvent is a starlike mapping of a specified order. As a key tool, we first introduce a refined version of the inverse function theorem for mappings satisfying so-called one-sided estimates.