广义ode的线性化和Hölder连续性及其在测量微分方程中的应用

IF 2.3 2区 数学 Q1 MATHEMATICS
Weijie Lu , Yonghui Xia
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引用次数: 0

摘要

线性化是由poincar(1890)提出的,它是连接线性系统和非线性系统的桥梁。本文建立了Banach空间中一类新的由Kurzweil积分定义的微分方程的线性化定理,即广义ode(简称GODEs)。在Banach空间x上,用dxdτ=D[A(t)x+F(x,t)]表示非线性偏微分方程,其中A:R→B(x)是x上的有界线性算子,F:X×R→x是Kurzweil可积的。字母D表示gdes是通过它的解来定义的,dxdτ只是一个符号。因此,微分方程本质上是一类积分方程的符号表示。由于偏微分方程与偏微分方程的理论差异,偏微分方程的Hartman-Grobman定理很难推广到偏微分方程。为了克服这一困难,我们首先在库兹韦尔积分意义上构造非线性方程有界解的公式。然后,我们建立了一个Hartman-Grobman型线性化定理,它是连接线性偏微分方程及其非线性扰动的桥梁。进一步,我们利用gronwall型不等式(Perron-Stieltjes积分意义下)证明了共轭都是Hölder连续的。最后,应用于测度微分方程和脉冲微分方程,结果是有效的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Linearization and Hölder continuity of generalized ODEs with application to measure differential equations
Linearization, initiated by Poincaré (1890) [44], is a bridge connecting the linear system with the nonlinear system. In this paper, we establish the linearization theorem for a new kind of differential equations defined by Kurzweil integral, so-called generalized ODEs (for short, GODEs) in a Banach space. The nonlinear GODEs are formulated asdxdτ=D[A(t)x+F(x,t)] on a Banach space X, where A:RB(X) is a bounded linear operator on X and F:X×RX is Kurzweil integrable. The letter D means that the GODEs are defined via its solution and dxdτ is only a notation. Thus, GODEs are essentially a notational representation of a class of integral equations. Due to the differences between the theory of GODEs and ODEs, it is difficult to extend the Hartman-Grobman theorem of ODEs to the GODEs. To overcome the difficulty, we first construct the formula for bounded solutions of the nonlinear GODEs in the Kurzweil integral sense. Afterwards, we establish a Hartman-Grobman type linearization theorem which is a bridge connecting the linear GODEs with their nonlinear perturbations. Furthermore, we show that the conjugacies are both Hölder continuous by using the Gronwall-type inequality (in the Perron-Stieltjes integral sense). Finally, applications to the measure differential equations and impulsive differential equations, our results are effective.
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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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