On the stability of a generalized additive functional equation

TL;DR

This paper investigates the stability of a generalized additive functional equation on homogeneous spaces under group actions, establishing conditions for stability related to fiber bundle structures in topological groups.

Contribution

It extends the Hyers-Ulam stability concept to a broader class of functional equations on homogeneous spaces with new conditions involving fiber bundles.

Findings

Stability holds for continuous functions on homogeneous spaces of strongly amenable groups.

The stability depends on the canonical projection being a fiber bundle.

Provides conditions under which the generalized additive functional equation is stable.

Abstract

We consider Hyers-Ulam stability of a functional equation for continuous functions on a space on which a topological group acts, analogous to the additive functional equation on a group. We show, among other things, that our generalized additive equation, for continuous functions on a homogenous space of a strongly amenable topological group, is stable provided that the canonical projection from that group to its homogenous space is a fiber bundle.