On the structure of fixed-point sets of asymptotically regular semigroups

TL;DR

This paper investigates the structure of fixed-point sets of asymptotically regular semigroups, showing they are retracts of their domain and exploring conditions for H"older continuity and characterizations of Bynum's coefficients.

Contribution

It extends recent results by demonstrating fixed-point sets are retracts and characterizes Bynum's coefficients and Opial modulus using nets.

Findings

Fixed-point sets are retracts of the domain.

Under certain conditions, the retraction is H"older continuous.

Bynum's coefficients and Opial modulus are characterized via nets.

Abstract

We extend a few recent results of G\'{o}rnicki (2011) asserting that the set of fixed points of an asymptotically regular mapping is a retract of its domain. In particular, we prove that in some cases the resulting retraction is H\"{o}lder continuous. We also characterise Bynum's coefficients and the Opial modulus in terms of nets.