Pexider invariance equation for embeddable mean-type mappings

IF 0.9 3区数学 Q2 MATHEMATICS

Aequationes Mathematicae Pub Date : 2025-01-03 DOI:10.1007/s00010-024-01139-0

Paweł Pasteczka

引用次数: 0

Abstract

We prove that whenever $M_1,\dots ,M_n:I^k \rightarrow I$, ($n,k \in \mathbb {N}$) are symmetric, continuous means on the interval I and $S_1,\dots ,S_m:I^k \rightarrow I$ ($m < n$) satisfy a sort of embeddability assumptions then for every continuous function $\mu :I^n \rightarrow \mathbb {R}$ which is strictly monotone in each coordinate, the functional equation

$$ \mu (S_1(v),\dots ,S_m(v),\underbrace{F(v),\dots ,F(v)}_{(n-m)\text { times}})=\mu (M_1(v),\dots ,M_n(v)) $$

has the unique solution $F=F_\mu :I^k \rightarrow I$ which is a mean. We deliver some sufficient conditions so that $F_\mu $ is well-defined (in particular uniquely determined) and study its properties. The aim of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018).

查看原文本刊更多论文

可嵌入均值型映射的Pexider不变性方程

我们证明了只要$M_1,\dots ,M_n:I^k \rightarrow I$, （$n,k \in \mathbb {N}$）是对称的，区间I和$S_1,\dots ,S_m:I^k \rightarrow I$ （$m < n$）上的连续均值满足一种可嵌入性假设，那么对于每一个在各坐标上严格单调的连续函数$\mu :I^n \rightarrow \mathbb {R}$，函数方程$$ \mu (S_1(v),\dots ,S_m(v),\underbrace{F(v),\dots ,F(v)}_{(n-m)\text { times}})=\mu (M_1(v),\dots ,M_n(v)) $$有唯一解$F=F_\mu :I^k \rightarrow I$，该解是均值。我们提供了一些充分条件，使$F_\mu $定义良好（特别是唯一确定），并研究了它的性质。本研究的目的是对（Himmel和Matkowski， 2018）中引入的beta型手段家族提供一个广泛的概述。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Aequationes Mathematicae MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

1.70

自引率

12.50%

发文量

审稿时长

>12 weeks

期刊介绍： aequationes mathematicae is an international journal of pure and applied mathematics, which emphasizes functional equations, dynamical systems, iteration theory, combinatorics, and geometry. The journal publishes research papers, reports of meetings, and bibliographies. High quality survey articles are an especially welcome feature. In addition, summaries of recent developments and research in the field are published rapidly.