具有无界非线性项的广义逐段常变量微分方程拓扑等价函数的正则性  

Holder regularity of topological equivalence functions of DEPCAGs with unbounded nonlinear terms

作　　者：邹长武[1] 夏永辉 Manuel Pinto Changwu Zou;Yonghui Xia;Manuel Pinto

机构地区：[1]福州大学数学与计算机科学学院,福州350116 [2]浙江师范大学数学与计算机科学学院,金华321004 [3]Departamento de Matematica,Universidad de Chile,Casilla 653,Chile

出　　处：《中国科学：数学》2020年第6期847-872,共26页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11671176,11601085和11871251);福建省自然科学基金(批准号:2018J01001);福建省教育厅基金(批准号:JAT160082);华侨大学启动基金(批准号:Z16J0039);FONDECYT基金(批准号:1120709和1170466)资助项目。

摘　　要：关于微分方程的拓扑等价以及拓扑等价函数的正则性一直是微分方程研究的焦点之一.当非线性项有界时,已经有很多学者证明了微分方程的拓扑等价函数是H?lder正则的.然而,当非线性项无界时,拓扑等价的正则性尚无突破性结果.在无界情形下, Zou和Shi(2017)得到了拓扑线性化的条件.除Zou和Shi(2017)给出的条件外,在附加的一个前提下,本文证明拓扑等价函数的H?lder正则性.事实上,当系统是有界时,系统线性化的条件足以保证其H?lder正则性.但是,当系统是无界时,这个附加条件是必不可少的.本文举例说明这个事实.这是首篇考虑无界系统拓扑等价H?lder正则的文章.The study on the H?lder regularity of topological equivalence functions has been the focus of significant attention. When the nonlinear term is bounded, many scholars have concluded that topological equivalence functions are Holder continuous. However, studies of the Holder regularity of topological equivalence functions when the nonlinear term is unbounded have not been conducted. Under the assumption that the nonlinear term is unbounded, sufficient conditions were obtained to guarantee that a DEPCAG(differential equation with piecewise constant arguments of a generalized type) is topologically linearized to its linear system in Zou and Shi(2017).In addition to conditions in Zou and Shi(2017), we prove that the topological equivalence functions are Holder continuous under an additional assumption in this paper. We mentioned that the conditions to guarantee the existence of topological conjugacy implies the Holder regularity for bounded system. However, in this paper,we show that this is not valid for unbounded system. That is, the additional condition cannot be removed for unbounded case. Then, we give an example that shows that the additional condition is necessary for the H?lder regularity of topological equivalence functions while the system is unbounded. As far as we know, this is the first paper considering the H?lder regularity of topological equivalence functions for unbounded system. Most of the literature focused on the bounded system.

关 键 词：微分方程 逐段常变量微分方程 正则性 

分 类 号：O175[理学—数学]