非特征端点条件下PM函数的迭代根

数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 557-569.

非特征端点条件下PM函数的迭代根

唐肖¹,李林^2,*()

¹ 重庆师范大学数学科学学院重庆 401331
² 嘉兴学院数学系浙江嘉兴 314001

收稿日期:2021-02-01 出版日期:2022-04-26 发布日期:2022-04-18
通讯作者: 李林 E-mail:mathll@163.com
基金资助:
国家自然科学基金(12001537);国家自然科学基金(11671061);国家自然科学基金(12026207);浙江省自然科学基金(LY18A010017);重庆师范大学博士启动基金(20XLB033)

Characteristic Endpoints Question for Piecewise Monotone Functions

Xiao Tang¹,Lin Li^2,*()

¹ School of Mathematical Science, Chongqing Normal University, Chongqing 401331
² Faculty of Mathematics, Jiaxing University, Zhejiang Jiaxing 314001

Received:2021-02-01 Online:2022-04-26 Published:2022-04-18
Contact: Lin Li E-mail:mathll@163.com
Supported by:
the NSFC(12001537);the NSFC(11671061);the NSFC(12026207);the NSF of Zhejiang Province(LY18A010017);the PhD Start-up Fund of Chongqing Normal University(20XLB033)

摘要/Abstract

摘要：

在特征端点条件下, 高度为1的PM函数的任意阶连续迭代根的存在性已经被证明.这就产生了一个在没有特征端点条件下的公开问题, 称为特征端点问题.当非单调点个数小于等于迭代根阶数时, 此问题在大部分情况下已解决.该文将研究非单调点个数大于迭代根阶数的情形, 给出高度为2且阶数也为2的连续迭代根存在的充分条件, 部分回答了特征端点问题.

关键词: 迭代根, 逐段单调函数, 特征区间, 特征端点条件

Abstract:

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. This raises an open problem about iterative roots without this condition, called characteristic endpoints problem. This problem is solved almost completely when the number of forts is equal to or less than the order. In this paper, we study the case that the number of forts is greater than the order and give a sufficient condition for existence of continuous iterative roots of order 2 with height 2, answering the characteristic endpoints problem partially.

Key words: Iterative root, Piecewise monotone function, Characteristic interval, Characteristic endpoints condition

中图分类号:

O175

唐肖,李林. 非特征端点条件下PM函数的迭代根[J]. 数学物理学报, 2022, 42(2): 557-569.

Xiao Tang,Lin Li. Characteristic Endpoints Question for Piecewise Monotone Functions[J]. Acta mathematica scientia,Series A, 2022, 42(2): 557-569.

图/表 4

表 1

图 1

图 2

图 3

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阶数	$ f $的性质	存在性或不存在性
$ n= 2 $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)=1 $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)=1 $	存在
$ 2<n<N(F) $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)<n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在
$ n=N(F) $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)\le n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)=n $	存在
$ n=N(F)+1 $	$ f $在$ [a', b'] $上$ \uparrow $, $ H(f)\le n $	不存在
	$ f $在$ [a', b'] $上$ \downarrow $, $ H(f)<n-1 $	不存在

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