Invariant means and iterates of mean-type mappings

Abstract

A classical result states that for two continuous, strict means M, N :

There is at most one continuous invariant mean

“…In this section, similarly to [2], we show a simple application of our results in solving functional equations Theorem 2. Let M : I p → I p be a weakly-contractive mean-type mapping such that there exists a continuous M-invariant mean K : I p → I.…”

“…Let us emphasize that it is sufficient to verify that the inequality above is valid for n = n 0 (v). Moreover in a special case p = 2 it was proved [2] that M is weakly contractive if and only if M 2 is contractive. However, due to [3], for every p > 2 we can construct a weakly contractive mean-type mapping on I p such that the function…”

“…In some recent studies by Pasteczka [4] and Matkowski-Pasteczka [2,3] it was proved that if M is not continuous then there also exist an M-invariant mean, but it is no longer uniquely determined. On the other hand, it was proved that in a narrow case p = 2 every contractive mean-type mapping M : I 2 → I has at most one continuous M-invariant mean (see [4]).…”

There is at most one continuous invariant mean

“…Let us emphasize that it is sufficient to verify if the inequality above is valid for n = n 0 (v). Moreover in a special case p = 2 it was proved [2] that M is weakly contractive if and only if M 2 is contractive. However, due to [3], for every p > 2 we can construct weakly contractive mean-type mapping on I p such that the function…”

“…In few recent studies by Pasteczka [4] and Matkowski-Pasteczka [2,3] it was proved that if M is not continuous then there also exist an M-invariant mean, but it is no longer uniquely determined. On the other hand, it was proved that in a narrow case p = 2 every contractive meantype mapping M : I 2 → I has at most one continuous M-invariant mean (see [4]).…”