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International Journal of Differential Equations
Volume 2011 (2011), Article ID 871574, 11 pages
http://dx.doi.org/10.1155/2011/871574
Research Article

Topological Conjugacy between Two Kinds of Nonlinear Differential Equations via Generalized Exponential Dichotomy

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 3 July 2011; Accepted 15 August 2011

Academic Editor: Yuri V. Rogovchenko

Copyright © 2011 Xiaodan Chen and Yonghui Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Based on the notion of generalized exponential dichotomy, this paper considers the topological decoupling problem between two kinds of nonlinear differential equations. The topological equivalent function is given.

1. Introduction and Motivation

Well-known Hartman’s linearization theorem for differential equations states that a 1 : 1 correspondence exists between solutions of a linear autonomous system ̇𝑥=𝐴𝑥 and those of the perturbed system ̇𝑥=𝐴𝑥+𝑓(𝑥), as long as 𝑓 fulfills some goodness conditions, like smallness, continuity, or being Hartman [1]. Based on the exponential dichotomy, Palmer [2] extended this result to the nonautonomous system. Some other improvements of Palmer’s linearization theorem are reported in the literature. For examples, one can refer to Shi [3], Jiang [4], and Reinfelds [5, 6]. Recently, Xia et al. [7] generalized Palmer’s linearization theorem to the dynamic systems on time scales. Consider the linear systeṁ𝑥=𝐴(𝑡)𝑥,(1.1) where 𝑥𝑛 and 𝐴(𝑡) is a 𝑛×𝑛 matrix function.

Definition 1.1. System (1.1) is said to possess an exponential dichotomy [8] if there exists a projection 𝑃 and constants 𝐾>0,𝛼>0 such that 𝑈(𝑡)𝑃𝑈1(𝑠)𝐾𝑒𝛼(𝑡𝑠)𝑈[]𝑈,for𝑠𝑡,𝑠,𝑡,(𝑡)𝐼𝑃1(𝑠)𝐾𝑒𝛼(𝑡𝑠),for𝑡𝑠,𝑠,𝑡(1.2) hold, where 𝑈(𝑡) is a fundamental matrix of linear system ̇𝑥=𝐴(𝑡)𝑥.

However, Lin [9] argued that the notion of exponential dichotomy considerably restricts the dynamics. It is thus important to look for more general types of hyperbolic behavior. Lin [9] proposed the notion of generalized exponential dichotomy which is more general than the classical notion of exponential dichotomy.

Definition 1.2. System (1.1) is said to have a generalized exponential dichotomy if there exists a projection 𝑃 and 𝐾0 such that ||𝑈(𝑡)𝑃𝑈1(||𝑠)𝐾exp𝑡𝑠||𝛼(𝜏)𝑑𝜏,(𝑡𝑠),𝑈(𝑡)(𝐼𝑃)𝑈1(||𝑠)𝐾exp𝑡𝑠𝛼(𝜏)𝑑𝜏,(𝑡𝑠),(1.3) where 𝛼(𝑡) is a continuous function with 𝛼(𝑡)0, satisfying lim𝑡+𝑡0𝛼(𝜉)𝑑𝜉=+,lim𝑡0𝑡𝛼(𝜉)𝑑𝜉=+.

Example 1.3. Consider the system ̇𝑥1̇𝑥2=13001|𝑡|+13|𝑥𝑡|+11𝑥2.(1.4) Then, system (1.4) has a generalized exponential dichotomy, but the classical exponential dichotomy cannot be satisfied.
For this reason, basing on generalized exponential dichotomy, we consider the topological decoupling problem between two kinds of nonlinear differential equations. We prove that there is a 1 : 1 correspondence existing between solutions of topological decoupling systems, namely, ̇𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥) and ̇𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑔(𝑡,𝑥).

2. Existence of Equivalent Function

Consider the following two nonlinear nonautonomous systems:̇𝑥=𝐴(𝑡)𝑥+𝑓(𝑡,𝑥),(2.1)̇𝑥=𝐴(𝑡)𝑥+𝑔(𝑡,𝑥),(2.2) where 𝑥𝑛, 𝐴(𝑡),  𝐵(𝑡) are 𝑛×𝑛 matrices.

Definition 2.1. Suppose that there exists a function 𝐻×𝑛𝑛 such that (i)for each fixed 𝑡, 𝐻(𝑡,) is a homeomorphism of 𝑛 into 𝑛;(ii)𝐻(𝑡,𝑥) as |𝑥|, uniformly with respect to 𝑡;(iii)assume that 𝐺(𝑡,)=𝐻1(𝑡,) has property (ii) too;(iv)if 𝑥(𝑡) is a solution of system (2.1), then 𝐻(𝑡,𝑥(𝑡)) is a solution of system (2.2).If such a map 𝐻 exists, then (2.1) is topologically conjugated to (2.2). 𝐻 is called an equivalent function.

Theorem 2.2. Suppose that ̇𝑥=𝐴(𝑡)𝑥 has a generalized exponential dichotomy. If 𝑓(𝑡,𝑥),𝑔(𝑡,𝑥) fulfill ||||||𝑓𝑓(𝑡,𝑥)𝐹(𝑡),𝑡,𝑥1𝑓𝑡,𝑥2||||𝑥𝑟(𝑡)1𝑥2||,||||||𝑔𝑔(𝑡,𝑥)𝐺(𝑡),𝑡,𝑥1𝑔𝑡,𝑥2||||𝑥𝑟(𝑡)1𝑥2||,𝑁𝑁(𝑡,𝐹,𝐺)𝐵,(𝑡,𝑟)𝐿<1,(2.3) where 𝑁(𝑡,𝐹,𝐺)=𝑡𝐾exp𝑡𝑠(+𝛼(𝜑)𝑑𝜑𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,𝑡+𝐾exp𝑡𝑠(𝛼(𝜑)𝑑𝜑𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,𝑁(𝑡,𝑟)=𝑡𝐾exp𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠+𝑡+𝐾exp𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠,(2.4) where 𝐹(𝑡),𝐺(𝑡),𝑟(𝑡)0 are integrable functions and 𝐵, 𝐿 are positive constants, then the nonlinear nonautonomous system (2.1) is topologically equivalent to the nonlinear nonautonomous system (2.2). Moreover, the equivalent functions 𝐻(𝑡,𝑥),𝐺(𝑡,𝑦) fulfill ||||||||𝐻(𝑡,𝑥)𝑥𝐵,𝐺(𝑡,𝑦)𝑦𝐵.(2.5)

In what follows, we always suppose that the conditions of Theorem 2.2 are satisfied. Denote that 𝑋(𝑡,𝑡0,𝑥0) is a solution of (2.2) satisfying the initial condition 𝑋(𝑡0)=𝑥0 and that 𝑌(𝑡,𝑡0,𝑦0) is a solution of (2.1) satisfying the initial condition 𝑌(𝑡0)=𝑦0. To prove the main results, we first prove some lemmas.

Lemma 2.3. For each (𝜏,𝜉), system 𝑧=𝐴(𝑡)𝑧𝑓(𝑡,𝑋(𝑡,𝜏,𝜉))+𝑔(𝑡,𝑋(𝑡,𝜏,𝜉)+𝑧)(2.6) has a unique bounded solution (𝑡,(𝜏,𝜉)) with |(𝑡,(𝜏,𝜉))|𝐵.

Proof. Let 𝔹 be the set of all the continuous bounded functions 𝑥(𝑡) with |𝑥(𝑡)|𝐵. For each (𝜏,𝜉) and any 𝑧(𝑡)𝔹, define the mapping 𝑇 as follows: 𝑇𝑧(𝑡)=𝑡𝑈(𝑡)𝑃𝑈1([]𝑠)𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1[](𝑠)𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.7)
Simple computation leads to ||||𝑇𝑧(𝑡)𝑡||𝑈(𝑡)𝑃𝑈1(||(+𝑠)𝐹(𝑠)+𝐺(𝑠))𝑑𝑠𝑡+||𝑈(𝑡)(𝐼𝑃)𝑈1||(𝑠)(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠𝑡𝐾exp𝑡𝑠+𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠𝑡+𝐾exp𝑡𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠𝐵,(2.8) which implies that 𝑇 is a self-map of a sphere with radius 𝐵. For any 𝑧1(𝑡),𝑧2(𝑡)𝔹, ||𝑇𝑧1(𝑡)𝑇𝑧2(||𝑡)𝑡||𝑈(𝑡)𝑃𝑈1(||𝑧𝑠)𝑟(𝑠)1(𝑠)𝑧2(+𝑠)𝑑𝑠𝑡+||𝑈(𝑡)(𝐼𝑃)𝑈1||𝑧(𝑠)𝑟(𝑠)1(𝑠)𝑧2𝑧(𝑠)𝑑𝑠1𝑧2𝑡𝐾exp𝑡𝑠+𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠𝑡+𝐾exp𝑡𝑠𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠𝐿1𝑧2.(2.9)
Due to the fact that 𝐿<1, 𝑇 has a unique fixed point, namely, 𝑧0(𝑡), and 𝑧0(𝑡)=𝑡𝑈(𝑡)𝑃𝑈1(𝑔𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧0(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧0(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠,(2.10) it is easy to show that 𝑧0(𝑡) is a bounded solution of (2.6). Now, we are going to show that the bounded solution is unique. For this purpose, we assume that there is another bounded solution 𝑧1(𝑡) of (2.6). Thus, 𝑧1(𝑡) can be written as follows: 𝑧1(𝑡)=𝑈(𝑡)𝑈1(0)𝑥0+𝑡0𝑈(𝑡)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈1(0)𝑥0+𝑡0[]𝑈𝑈(𝑡)𝑃+(𝐼𝑃)1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈1(0)𝑥0+𝑡𝑈(𝑡)𝑃𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠0𝑈(𝑡)𝑃𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1+(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠0+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.11)
Note that 0𝑈(𝑡)𝑃𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈1(0)0𝑈(0)𝑃𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑈(𝑡)𝑈1(||||0)0𝑈(0)𝑃𝑈1(𝑔𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(||||𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑈(𝑡)𝑈1(0)0𝐾exp0𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,(2.12) which implies that 0𝑈(0)𝑃𝑈1(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠))𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠 is convergent; denote it by 𝑥1. That is, 0𝑈(𝑡)𝑃𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈1(0)𝑥1.(2.13) Similarly, 0+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈1(0)𝑥2.(2.14) Therefore, it follows from the expression of 𝑧1(𝑡) that 𝑧1(𝑡)=𝑈(𝑡)𝑈1𝑥(0)0𝑥1+𝑥2+𝑡𝑔𝑈(𝑡)𝑃𝑈(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.15)
Noticing that 𝑧1(𝑡) is bounded, 𝑡𝑈(𝑡)𝑃𝑈(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠))𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠))𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠 is also bounded. So, 𝑈(𝑡)𝑈1(0)(𝑥0𝑥1+𝑥2) is bounded. But we see that 𝑧=𝐴(𝑡)𝑧 does not have a nontrivial bounded solution. Thus, 𝑈(𝑡)𝑈1(0)(𝑥0𝑥1+𝑥2)=0; it follows that 𝑧1(𝑡)=𝑡𝑈(𝑡)𝑃𝑈1(𝑔𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.16) Simple calculating shows ||𝑧1(𝑡)𝑧0(||𝑡)𝑡||𝑈(𝑡)𝑃𝑈1(||||𝑧𝑠)𝑟(𝑠)1(𝑠)𝑧0(||+𝑠)𝑑𝑠𝑡+||𝑈(𝑡)(𝐼𝑃)𝑈1||||𝑧(𝑠)𝑟(𝑠)1(𝑠)𝑧0||𝑧(𝑠)𝑑𝑠1𝑧0𝑡𝐾exp𝑡𝑠+𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑡+𝐾exp𝑡𝑠𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝐿1𝑧0.(2.17)

Therefore, 𝑧1𝑧0𝐿𝑧1𝑧0, consequently 𝑧1(𝑡)𝑧0(𝑡). This implies that the bounded solution of (2.6) is unique. We may call it (𝑡(𝜏,𝜉)). From the above proof, it is easy to see that |(𝑡,(𝜏,𝜉))|𝐵.

Lemma 2.4. For each (𝜏,𝜉), the system 𝑧=𝐴(𝑡)𝑧+𝑓(𝑡,𝑋(𝑡,𝜏,𝜉)+𝑧)𝑔(𝑡,𝑋(𝑡,𝜏,𝜉))(2.18) has a unique bounded solution ̃𝑔(𝑡,(𝜏,𝜉)) and |̃𝑔(𝑡,(𝜏,𝜉))|𝐵.

Proof. The proof is similar to that of Lemma 2.3.

Lemma 2.5. Let 𝑥(𝑡) be any solution of the system (2.1), then 𝑧(𝑡)=0 is the unique bounded solution of system 𝑧=𝐴(𝑡)𝑧+𝑓(𝑡,𝑥(𝑡)+𝑧)𝑓(𝑡,𝑥(𝑡)).(2.19)

Proof. Obviously, 𝑧0 is a bounded solution of system (2.19). We show that the bounded solution is unique. If not, then there is another bounded solution 𝑧1(𝑡), which can be written as follows: 𝑧1(𝑡)=𝑈(𝑡)𝑈1(0)𝑧1(0)+𝑡0𝑈(𝑡)𝑈1(𝑓𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.20) By Lemma 2.3, we can get 𝑧1(𝑡)=𝑡𝑈(𝑡)𝑃𝑈1(𝑓𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑓(𝑠)𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.21) It follows that ||𝑧1(||𝑡)𝑡||𝑈(𝑡)𝑃𝑈1(||||𝑧𝑠)𝑟(𝑠)1(||+𝑠)𝑑𝑠𝑡+||𝑈(𝑡)(𝐼𝑃)𝑈1||||𝑧(𝑠)𝑟(𝑠)1||(𝑠)𝑑𝑠𝑡𝐾exp𝑡𝑠||𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)1||+(𝑠)𝑑𝑠𝑡+𝐾exp𝑡𝑠||𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)1||||𝑧(𝑠)𝑑𝑠𝐿1||.(𝑡)(2.22) That is, 𝑧1𝐿|𝑧1|. Consequently, 𝑧1(𝑡)0. This completes the proof of Lemma 2.5.

Lemma 2.6. Let 𝑦(𝑡) be any solution of the system (2.2), then 𝑧(𝑡)=0 is the unique bounded solution of system 𝑧=𝐴(𝑡)𝑧+𝑔(𝑡,𝑦(𝑡)+𝑧)𝑔(𝑡,𝑦(𝑡)).(2.23)

Proof. Obviously, 𝑧0 is a bounded solution of system (2.23). We will show that the bounded solution is unique. If not, then there is another bounded solution 𝑧1(𝑡). Then, 𝑧1(𝑡) can be written as follows: 𝑧1(𝑡)=𝑈(𝑡)𝑈1(0)𝑧1(0)+𝑡0𝑈(𝑡)𝑈1(𝑔𝑠)𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑔(𝑠,𝑌(𝑠,𝜏,𝜉))𝑑𝑠.(2.24)
By Lemma 2.3, we can get 𝑧1(𝑡)=𝑡𝑈(𝑡)𝑃𝑈1(𝑔𝑠)𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑔(𝑠,𝑌(𝑠,𝜏,𝜉))𝑑𝑠𝑡+𝑈(𝑡)(𝐼𝑃)𝑈1𝑔(𝑠)𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧1(𝑠)𝑔(𝑠,𝑌(𝑠,𝜏,𝜉))𝑑𝑠.(2.25) Then, it follows that ||𝑧1(||𝑡)𝑡||𝑈(𝑡)𝑃𝑈1(||||𝑧𝑠)𝑟(𝑠)1(||+𝑠)𝑑𝑠𝑡+||𝑈(𝑡)(𝐼𝑃)𝑈1||||𝑧(𝑠)𝑟(𝑠)1||(𝑠)𝑑𝑠𝑡𝐾exp𝑡𝑠||𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)1||+(𝑠)𝑑𝑠𝑡+𝐾exp𝑡𝑠||𝑧𝛼(𝜑)𝑑𝜑𝑟(𝑠)1||||𝑧(𝑠)𝑑𝑠𝐿1||.(𝑡)(2.26) That is, 𝑧1𝐿𝑧1. Consequently, 𝑧1(𝑡)0. This completes the proof of Lemma 2.6.

Now, we define two functions as follows:𝐻(𝑡,𝑥)=𝑥+(𝑡,(𝑡,𝑥)),(2.27)𝐺(𝑡,𝑥)=𝑦+̃𝑔(𝑡,(𝑡,𝑦)).(2.28)

Lemma 2.7. For any fixed (𝑡0,𝑥0),𝐻(𝑡,𝑋(𝑡,𝑡0,𝑥0)) is a solution of the system (2.2).

Proof. Replace (𝜏,𝜉) by (𝑡,𝑋(𝑡,𝜏,𝜉)) in (2.6); system (2.6) is not changed. Due to the uniqueness of the bounded solution of (2.6), we can get that (𝑡,(𝑡,𝑋(𝑡,𝑡0,𝑥0)))=(𝑡,(𝑡0,𝑥0)). Thus, 𝐻𝑡,𝑋𝑡,𝑡0,𝑥0=𝑋𝑡,𝑡0,𝑥0𝑡+𝑡,0,𝑥0.(2.29) Differentiating it, noticing that 𝑋(𝑡,𝑡0,𝑥0), (𝑡,(𝑡0,𝑥0)) are the solutions of the (2.1), and (2.6), respectively; therefore, we can obtain 𝐻𝑡,𝑋𝑡,𝑡0,𝑥0=𝐴(𝑡)𝑋𝑡,𝑡0,𝑥0+𝑓𝑡,𝑋𝑡,𝑡0,𝑥0𝑡+𝐴(𝑡)𝑡,0,𝑥0𝑓𝑡,𝑋𝑡,𝑡0,𝑥0+𝑔𝑡,𝑋𝑡,𝑡0,𝑥0𝑡+𝑡,0,𝑥0=𝐴(𝑡)𝐻𝑡,𝑋𝑡,𝑡0,𝑥0+𝑔𝑡,𝐻𝑡,𝑡0,𝑥0.(2.30) It indicates that 𝐻(𝑡,𝑋(𝑡,𝑡0,𝑥0)) is the solution of the system (2.2).

Lemma 2.8. For any fixed (𝑡0,𝑦0), 𝐺(𝑡,𝑌(𝑡,𝑡0,𝑦0)) is a solution of the system (2.1).

Proof. The proof is similar to Lemma 2.7.

Lemma 2.9. For any 𝑡,𝑦𝑛,𝐻(𝑡,𝐺(𝑡,𝑦))𝑦.

Proof. Let 𝑦(𝑡) be any solution of the system (2.2). From Lemma 2.8, 𝐺(𝑡,𝑦(𝑡)) is a solution of system (2.1). Then, by Lemma 2.7, we see that 𝐻(𝑡,𝐺(𝑡,𝑦(𝑡))) is a solution of (2.2), written as 𝑦1(𝑡). Denote 𝐽(𝑡)=𝑦1(𝑡)𝑦(𝑡). Differentiating, we have 𝐽(𝑡)=𝑦1(𝑡)𝑦(𝑡)=𝐴(𝑡)𝑦1(𝑡)+𝑔𝑡,𝑦1(𝑡)𝐴(𝑡)𝑦(𝑡)𝑔(𝑡,𝑦(𝑡))=𝐴(𝑡)𝐽(𝑡)+𝑔(𝑡,𝑦(𝑡)+𝐽(𝑡))𝑔(𝑡,𝑦(𝑡)),(2.31) which implies that 𝐽(𝑡) is a solution of system (2.23). On the other hand, following the definition of 𝐻 and 𝐺 and Lemmas 2.3 and 2.4, we can obtain ||||=||||||||+||||=||||+||||𝐽(𝑡)𝐻(𝑡,𝐺(𝑡,𝑦(𝑡)))𝑦(𝑡)𝐻(𝑡,𝐺(𝑡,𝑦(𝑡)))𝐺(𝑡,𝑦(𝑡))𝐺(𝑡,𝑦(𝑡))𝑦(𝑡)(𝑡,(𝑡,𝐺(𝑡,𝑦(𝑡))))̃𝑔(𝑡,(𝑡,𝑦))2𝐵.(2.32)
This implies that 𝐽(𝑡) is a bounded solution of system (2.23). However, by Lemma 2.6, system (2.23) has only one zero solution. Hence, 𝐽(𝑡)0, consequently 𝑦1(𝑡)𝑦(𝑡), that is, 𝐻(𝑡,𝐺(𝑡,𝑦))=𝑦(𝑡). Since 𝑦(𝑡) is any solution of the system (2.2), Lemma 2.9 follows.

Lemma 2.10. For any 𝑡,𝑥𝑛,𝐺(𝑡,𝐻(𝑡,𝑥))𝑥.

Proof. The proof is similar to Lemma 2.10.

Now, we are in a position to prove the main results.

Proof of Theorem 2.2. We are going to show that 𝐻(𝑡,) satisfies the four conditions of Definition 2.1.Proof of Condition (i). For any fixed 𝑡, it follows from Lemmas 2.9 and 2.10 that 𝐻(𝑡,) is homeomorphism and 𝐺(𝑡,)=𝐻1(𝑡,).Proof of Condition (ii). From (2.27) and Lemma 2.3, we derive |𝐻(𝑡,𝑥)𝑥|=|(𝑡,(𝑡,𝑥))|𝐵. So, |𝐻(𝑡,𝑥)| as |𝑥|, uniformly with respect to 𝑡.Proof of Condition (iii). From (2.23) and Lemma 2.4, we derive |𝐺(𝑡,𝑦)𝑦|=|̃𝑔(𝑡,(𝑡,𝑦))|𝐵. So, |𝐺(𝑡,𝑦)| as |𝑥|, uniformly with respect to 𝑡.Proof of Condition (iv). Using Lemmas 2.7 and 2.8, we easily prove that Condition (iv) is true.

Hence, systems (2.1) and (2.2) are topologically conjugated. This completes the proof of Theorem 2.2.

Acknowledgments

The authors would like to express their gratitude to the editor and anonymous reviewers for their careful reading which improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grant (no. 10901140) and ZJNSFC under Grant (no. Y6100029).

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