There is at most one continuous invariant mean

TL;DR

This paper proves that for weakly contractive mean-type mappings, there is at most one continuous invariant mean satisfying a specific functional equation, providing insights into the uniqueness of such means.

Contribution

It establishes the uniqueness of continuous invariant means under weakly contractive mean-type mappings and offers a general approach to the related functional equation.

Findings

At most one continuous invariant mean exists for the given conditions.

The paper provides a general method to analyze the functional equation involved.

The results apply to non-continuous mean-type mappings as well.

Abstract

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping $\mathbf{M} \colon I^p\to I^p$ (where $I$ is an interval and $p \in \mathbb{N}$), the functional equation $K \circ \mathbf{M}=K$ has at most one solution in the family of continuous means $K \colon I^p \to I$. Some general approach to the latter equation is also given.