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

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 𝑥̇=𝐴𝑥 and those of the perturbed system 𝑥̇=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 system𝑥̇=𝐴(𝑡)𝑥,(1.1) where 𝑥∈ℝ^𝑛 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‖‖)𝑈[𝐼(−𝑡)𝑃𝑃]𝑈𝑈⁻¹₋₁(𝑠)‖‖≤𝐾𝑒^{−𝛼(𝑡−𝑠)},for𝑠≤𝑡,𝑠,𝑡∈ℝ,(𝑠)‖‖≤𝐾𝑒^{𝛼(𝑡−𝑠)},for𝑡≤𝑠,𝑠,𝑡∈ℝ(1.2) hold, where 𝑈(𝑡) is a fundamental matrix of linear system 𝑥̇=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 𝐾≥0 such that ||𝑈||𝑈(𝑡)((𝑡𝐼)𝑃−𝑈𝑃)⁻¹𝑈(𝑠)||₋₁≤𝐾exp−^𝑡𝑠𝛼(𝜏)𝑑𝜏,(𝑡≥𝑠),(𝑠)||≤𝐾exp^𝑡𝑠𝛼(𝜏)𝑑𝜏,(𝑡≤𝑠),(1.3) where 𝛼(𝑡) is a continuous function with 𝛼(𝑡)≥0, satisfying lim_𝑡→+∞∫^𝑡0𝛼(𝜉)𝑑𝜉=+∞,lim_{𝑡→−∞}∫^0𝑡𝛼(𝜉)𝑑𝜉=+∞.

Example 1.3. Consider the system ⎛⎜⎜⎝𝑥̇₁𝑥̇₂⎞⎟⎟⎠=⎛⎜⎜⎜⎜⎝−1₃√|0𝑡|+1₃01√|𝑡|+1⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝𝑥¹𝑥₂⎞⎟⎟⎠.(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, 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥) and 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑔(𝑡,𝑥).

2. Existence of Equivalent Function

Consider the following two nonlinear nonautonomous systems:𝑥𝑥̇̇==𝐴𝐴((𝑡𝑡))𝑥𝑥++𝑓𝑔((𝑡𝑡,,𝑥𝑥)),,((22..12)) where 𝑥∈ℝ^𝑛, 𝐴(𝑡),  𝐵(𝑡) are 𝑛×𝑛 matrices.

Definition 2.1. Suppose that there exists a function 𝐻∶ℝ×ℝ^𝑛→ℝ^𝑛 such that (i)for each fixed 𝑡, 𝐻(𝑡,⋅) is a homeomorphism of ℝ^𝑛 into ℝ^𝑛;(ii)𝐻(𝑡,𝑥)→∞ as |𝑥|→∞, uniformly with respect to 𝑡;(iii)assume that 𝐺(𝑡,⋅)=𝐻⁻¹(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 𝑥̇=𝐴(𝑡)𝑥 has a generalized exponential dichotomy. If 𝑓(𝑡,𝑥),𝑔(𝑡,𝑥) fulfill ||𝑓𝑡,𝑥₁||𝑓(𝑡,𝑥)||≤𝐹(𝑡),||𝑔𝑡,𝑥₁−||𝑔𝑓(𝑡,𝑡𝑥,)𝑥||₂≤||𝐺≤(𝑟𝑡)(𝑡,)||𝑥₁−𝑥₂||,−𝑁𝑁𝑔((𝑡𝑡,𝑡,,𝑟𝐹𝑥),₂≤𝐺||)𝐿≤≤<𝑟𝐵(1𝑡,),||𝑥₁−𝑥₂||,(2.3) where 𝑁(𝑡,𝐹,𝐺)=+^𝑡−∞𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,_𝑡+∞𝑁(𝑡,𝑟)=_𝑡−∞𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠+^𝑡+∞𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(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)−𝑥||≤𝐵,||𝐺(𝑡,𝑦)−𝑦||≤𝐵.(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)𝑧−𝑓(𝑡,𝑋(𝑡,𝜏,𝜉))+𝑔(𝑡,𝑋(𝑡,𝜏,𝜉)+𝑧)(2.6) has a unique bounded solution ℎ(𝑡,(𝜏,𝜉)) with |ℎ(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)=−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(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)𝑃𝑈⁻¹(𝑠)||(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠_𝑡+∞≤||𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)||(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠_𝑡−∞+𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠_𝑡+∞≤𝐵,𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠(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)𝑃𝑈⁻¹(𝑠)||𝑟(𝑠)𝑧₁(𝑠)−𝑧₂(𝑠)𝑑𝑠_𝑡+∞≤‖‖𝑧||𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)||𝑟(𝑠)𝑧₁(𝑠)−𝑧₂(𝑠)𝑑𝑠₁+−𝑧₂‖‖^𝑡−∞𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠_𝑡+∞≤𝐿‖‖𝑧₁𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)𝑑𝑠−𝑧₂‖‖.(2.9)
Due to the fact that 𝐿<1, 𝑇 has a unique fixed point, namely, 𝑧₀(𝑡), and 𝑧₀(𝑡)=−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₀(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔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)𝑈⁻¹(0)𝑥₀=+𝑈(^𝑡0𝑡)𝑈𝑈(𝑡)₋₁𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠(0)𝑥₀=+𝑈(^𝑡0𝑡)𝑈𝑈(𝑡)₋₁[𝑃+(𝐼−𝑃)]𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠(0)𝑥₀+−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_0−∞+𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_0+∞−𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.11)
Note that ^0−∞=𝑈𝑈(𝑡)(𝑃𝑡)𝑈𝑈⁻¹₋₁(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠≤𝑈(𝑡)𝑈₋₁(0)^0−∞𝑈(0)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠≤𝑈(𝑡)𝑈₋₁(0)||||^0−∞𝑈(0)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠||||(0)^0−∞𝐾exp−^0𝑠𝛼(𝜑)𝑑𝜑(𝐹(𝑠)+𝐺(𝑠))𝑑𝑠,(2.12) which implies that ∫^0−∞𝑈(0)𝑃𝑈⁻¹(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠))−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠 is convergent; denote it by 𝑥₁. That is, ^0−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈⁻¹(0)𝑥₁.(2.13) Similarly, ^0+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠=𝑈(𝑡)𝑈⁻¹(0)𝑥₂.(2.14) Therefore, it follows from the expression of 𝑧₁(𝑡) that 𝑧₁(𝑡)=+𝑈(𝑡_𝑡−∞)𝑈⁻¹(0)𝑥₀−𝑥₁+𝑥₂−𝑈(𝑡)𝑃𝑈(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.15)
Noticing that 𝑧₁(𝑡) is bounded, ∫𝑓(^𝑡−∞𝑠,𝑈𝑋((𝑡𝑠)𝑃,𝜏𝑈,(𝜉𝑠)))[]𝑔𝑑(𝑠𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠))−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))]𝑑𝑠−∫^𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)[𝑔(𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠))− is also bounded. So, 𝑈(𝑡)𝑈⁻¹(0)(𝑥₀−𝑥₁+𝑥₂) is bounded. But we see that 𝑧^=𝐴(𝑡)𝑧 does not have a nontrivial bounded solution. Thus, 𝑈(𝑡)𝑈⁻¹(0)(𝑥₀−𝑥₁+𝑥₂)=0; it follows that 𝑧₁(𝑡)=−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔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)𝑃𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)−𝑧₀(𝑠)||𝑑𝑠_𝑡+∞≤‖‖𝑧||𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)−𝑧₀(𝑠)||𝑑𝑠₁+−𝑧₀‖‖^𝑡−∞𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)_𝑡+∞≤𝐿‖‖𝑧₁𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)−𝑧₀‖‖.(2.17)

Therefore, ‖𝑧₁−𝑧₀‖≤𝐿‖𝑧₁−𝑧₀‖, consequently 𝑧₁(𝑡)≡𝑧₀(𝑡). This implies that the bounded solution of (2.6) is unique. We may call it ℎ(𝑡(𝜏,𝜉)). From the above proof, it is easy to see that |ℎ(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 𝑔̃(𝑡,(𝜏,𝜉)) and |𝑔̃(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, 𝑧≡0 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)𝑈⁻¹(0)𝑧₁(0)+^𝑡0𝑈(𝑡)𝑈⁻¹(𝑠)𝑓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)=−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑓𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑓𝑠,𝑋(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠.(2.21) It follows that ||𝑧₁(𝑡)||≤+^𝑡−∞||𝑈(𝑡)𝑃𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡+∞≤||𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡−∞+𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡+∞≤𝐿||𝑧₁𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠(𝑡)||.(2.22) That is, ‖𝑧₁‖≤𝐿|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, 𝑧≡0 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)𝑈⁻¹(0)𝑧₁(0)+^𝑡0𝑈(𝑡)𝑈⁻¹(𝑠)𝑔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)=−^𝑡−∞𝑈(𝑡)𝑃𝑈⁻¹(𝑠)𝑔𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑔(𝑠,𝑌(𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)𝑔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)𝑃𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡+∞≤||𝑈(𝑡)(𝐼−𝑃)𝑈⁻¹(𝑠)||𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡−∞+𝐾exp−^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠_𝑡+∞≤𝐿||𝑧₁𝐾exp^𝑡𝑠𝛼(𝜑)𝑑𝜑𝑟(𝑠)||𝑧₁(𝑠)||𝑑𝑠(𝑡)||.(2.26) That is, ‖𝑧₁‖≤𝐿‖𝑧₁‖. Consequently, 𝑧₁(𝑡)≡0. This completes the proof of Lemma 2.6.

Now, we define two functions as follows:𝐻𝐺((𝑡𝑡,,𝑥𝑥))==𝑥𝑦++ℎ𝑔̃((𝑡𝑡,,((𝑡𝑡,,𝑦𝑥)))).,((22..2278))

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 ℎ(𝑡,(𝑡,𝑋(𝑡,𝑡₀,𝑥₀)))=ℎ(𝑡,(𝑡₀,𝑥₀)). Thus, 𝐻𝑡,𝑋𝑡,𝑡₀,𝑥₀=𝑋𝑡,𝑡₀,𝑥₀+ℎ𝑡,𝑡₀,𝑥₀.(2.29) Differentiating it, noticing that 𝑋(𝑡,𝑡₀,𝑥₀), ℎ(𝑡,(𝑡₀,𝑥₀)) are the solutions of the (2.1), and (2.6), respectively; therefore, we can obtain 𝐻𝑡,𝑋𝑡,𝑡₀,𝑥₀^=−𝐴𝑓(𝑡)𝑡𝑋,𝑋𝑡,𝑡𝑡,₀𝑡,₀𝑥₀+𝑓𝑡,𝑋𝑡,𝑡₀,𝑥₀+𝐴(𝑡)ℎ𝑡,𝑡₀,𝑥₀=𝐴(𝑡)𝐻𝑡,𝑋,𝑡𝑥,𝑡₀₀+𝑔𝑡,𝑋𝑡,𝑡₀,𝑥₀+ℎ𝑡,𝑡₀,𝑥₀,𝑥₀+𝑔𝑡,𝐻𝑡,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∈ℝ,𝑦∈ℝ^𝑛,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)==𝑦𝐴^1((𝑡𝑡))𝑦−₁𝑦^(𝑡)=𝐴(𝑡)𝐽((𝑡𝑡))++𝑔𝑔(𝑡,𝑡,𝑦𝑦(₁𝑡)(𝑡+)𝐽−(𝑡𝐴))(−𝑡)𝑦𝑔((𝑡𝑡),𝑦−(𝑔𝑡))(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)||=≤=≤2||||||𝐻ℎ𝐻𝐵(.𝑡((,𝑡𝑡,(,𝑡𝐺𝐺,𝐺((𝑡𝑡,(,𝑡𝑦𝑦,(𝑦(𝑡𝑡)())𝑡)))))−)−)||𝑦𝐺+(𝑡(||)𝑔𝑡||,̃(𝑦𝑡,(𝑡()𝑡),||𝑦+))||||𝐺(𝑡,𝑦(𝑡))−𝑦(𝑡)||(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∈ℝ,𝑥∈ℝ^𝑛,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,⋅)=𝐻⁻¹(𝑡,⋅).Proof of Condition (ii). From (2.27) and Lemma 2.3, we derive |𝐻(𝑡,𝑥)−𝑥|=|ℎ(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|=|𝑔̃(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).