Nonlinear resolvents in the unit disk: geometry and dynamics

TL;DR

This paper introduces a unified approach using a distortion theorem to analyze the geometric and dynamic properties of nonlinear resolvents in the unit disk, including their spirallikeness, starlikeness, and quasiconformal extensions.

Contribution

It develops a method to determine geometric properties of resolvents without restrictions and establishes uniform convergence and characteristics of associated semigroups.

Findings

Determined order of spirallikeness and strong starlikeness of resolvents.

Proved resolvents admit quasiconformal extension to the complex plane.

Established uniform convergence of the resolvent family on the unit disk.

Abstract

In this paper we present a unified approach to the study of geometric and dynamic properties of nonlinear resolvents of holomorphic generators. The idea is to apply the distortion theorem we have established. This method allows us to find order of spirallikeness and of strong starlikeness of resolvents and remove all the restrictions for resolvents to admit quasiconformal extension to the complex plane $\C$. In addition, we use this method to establish the uniform convergence of the resolvent family on the whole unit disk and obtain some characteristics of semigroups generated by these resolvents.