## Abstract and Applied Analysis

### Linearization of Impulsive Differential Equations with Ordinary Dichotomy

#### Abstract

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system $\stackrel{˙}{x}(t)=A(t)x(t)+f(t,x)$, $t\ne {t}_{k}$, $\mathrm{\Delta }x({t}_{k})=\stackrel{~}{A}({t}_{k})x({t}_{k})+\stackrel{~}{f}({t}_{k},x)$, $k\in \Bbb Z$, is topologically conjugated to $\stackrel{˙}{x}(t)=A(t)x(t)$, $t\ne {t}_{k}$, $\mathrm{\Delta }x({t}_{k})=\stackrel{~}{A}({t}_{k})x({t}_{k})$, $k\in \Bbb Z$, where $\mathrm{\Delta }x({t}_{k})=x({t}_{k}^{+})-x({t}_{k}^{-})$, $x({t}_{k}^{-})=x({t}_{k})$, represents the jump of the solution $x(t)$ at $t={t}_{k}$. Finally, two examples are given to show the feasibility of our results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 632109, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273208

Digital Object Identifier
doi:10.1155/2014/632109

Mathematical Reviews number (MathSciNet)
MR3178879

Zentralblatt MATH identifier
07022783

#### Citation

Gao, Yongfei; Yuan, Xiaoqing; Xia, Yonghui; Wong, P. J. Y. Linearization of Impulsive Differential Equations with Ordinary Dichotomy. Abstr. Appl. Anal. 2014 (2014), Article ID 632109, 11 pages. doi:10.1155/2014/632109. https://projecteuclid.org/euclid.aaa/1412273208

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