Invariant property for discontinuous mean-type mappings

TL;DR

This paper investigates invariant means for pairs of discontinuous two-variable means satisfying a specific inequality, establishing the existence of minimal and maximal invariant means and analyzing the uniqueness of continuous invariant means.

Contribution

It extends the theory of invariant means to discontinuous cases, proving existence of extremal invariant means and the uniqueness of continuous invariant means.

Findings

Existence of smallest and largest invariant means for discontinuous pairs.

At most one continuous invariant mean exists per pair.

Continuity of invariant means is not guaranteed in the discontinuous setting.

Abstract

It is known that if $M,\,N$ are continuous two-variable means such that $|M(x,y)-N(x,y)| < |x-y|$ for every $x,\ y$ with $x\ne y$, then there exists a unique invariant mean (which is continuous too). We are looking for invariant means for pairs satisfying the inequality above, but continuity of means is not assumed. In this setting the invariant mean is no longer uniquely defined, but we prove that there exist the smallest and the biggest one. Furthermore it is shown that there exists at most one continuous invariant mean related to each pair.