Invariant means and iterates of mean-type mappings

Janusz Matkowski,Paweł Pasteczka

doi:10.1007/s00010-019-00668-3

Abstract

A classical result states that for two continuous, strict means M,,N :I^2 rightarrow I (I is an interval) there exists a unique (M, N)-invariant mean K :I^2 rightarrow I, i.e. such a mean that K circ (M,N)=K and, moreover, the sequence of iterates ((M,N)^n)_{n=1}^infty converge to (K, K) pointwise. Recently it was proved that continuity assumption cannot be omitted in general. We show that if K is a unique (M, N)-invariant mean then, without continuity assumption, (M,N)^n rightarrow (K,K).

Highlights

It is known that if M and N are continuous bivariate means in an interval I, and the mean-type mapping (M, N ) is diagonally contractive, that is,|M (x, y) − N (x, y)| < |x − y|, x, y ∈ I, x = y, (1) there is a unique mean K : I2 → I that is (M, N )-invariant and, the sequence of iterates ((M, N )n)n∈N of the mean-type mapping (M, N ) converges to (K, K) ([5, Theorem 4.5 (i),(ii)]).For the results of this type, with more restrictive assumptions, see [1]
We show that if K is a unique (M, N )-invariant mean without continuity assumption, (M, N )n → (K, K)
Instead of (1) it was assumed that both means are strict; in [2] it was assumed that at most one mean is not strict, and condition (1) appeared first in [4]

Summary

Introduction

It is known that if M and N are continuous bivariate means in an interval I, and the mean-type mapping (M, N ) is diagonally contractive, that is,. There is a unique mean K : I2 → I that is (M, N )-invariant and, the sequence of iterates ((M, N )n)n∈N of the mean-type mapping (M, N ) converges to (K, K) ([5, Theorem 4.5 (i),(ii)]). Instead of (1) it was assumed that both means are strict; in [2] it was assumed that at most one mean is not strict, and condition (1) appeared first in [4] (see [6]) In all these papers the uniqueness of the invariant mean was obtained under the condition that it is continuous. 3 we give conditions under which the uniqueness of the invariant mean guarantee the relevant convergence of the sequence of iterates of the mean-type mapping (see Theorem 3.3).

Preliminaries

Weakly contractive mean-type mappings

Mean-type mappings with diagonally-contractive iterates and invariant means