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There is at most one continuous invariant mean

Paweł Pasteczka

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.

There is at most one continuous invariant mean

Abstract

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping (where is an interval and ), the functional equation has at most one solution in the family of continuous means . Some general approach to the latter equation is also given.

Paper Structure

This paper contains 5 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Applications
  4. Weakly contractive mean-type mappings
  5. An applications in solving a functional equation

Key Result

Lemma 1

Let $D$ be a Hausdorff, $\sigma$-compact topological space and $F \colon D \to [0,\infty)$ be a continuous function. If $T \colon [0,\infty) \to 2^D$ is nondecreasing, right-continuous (we consider topological limit on $2^D$) and such that Then $\vec{F} \circ T$ is constant.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Proposition 1: MatPas21, Theorem 2
  • Corollary 1
  • proof
  • Theorem 2
  • proof