Linearization of Nonautonomous Impulsive System with Nonuniform Exponential Dichotomy

Abstract

This paper gives a version of Hartman-Grobman theorem for the impulsive differential equations. We assume that the linear impulsive system has a nonuniform exponential dichotomy. Under some suitable conditions, we proved that the nonlinear impulsive system is topologically conjugated to its linear system. Indeed, we do construct the topologically equivalent function (the transformation). Moreover, the method to prove the topological conjugacy is quite different from those in previous works (e.g., see Barreira and Valls, 2006).

1. Introduction

A basic contribution to the linearization problem for autonomous differential equations is the famous Hartman-Grobman theorem (see [1, 2]). Then Palmer successfully generalized the standard Hartman-Grobman theorem to nonautonomous differential equations (see [3]). Then Fenner and Pinto [4] generalized Hartman-Grobman theorem to impulsive differential equations. However, they did not discuss the Hölder regularity of the topologically equivalent function 𝐻(𝑡,𝑥). Then Xia et al. [5] gave a rigorous proof of the Hölder regularity. Xia et al. [6, 7] gave a version of the generalized Hartman-Grobman theorem for dynamic systems on measure chains. It should be noted that the above mentioned works are based on the linear differential equations with uniform exponential dichotomy. Recently, Barreira and Valls have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way [8–11]. So, a version of the Hartman-Grobman theorem is also given for differential equations with nonuniform hyperbolicity (see [12]). However, they did not discuss the impulsive systems with nonuniform hyperbolicity. For this reason, in this paper, we considered the linearization of impulsive differential equations with nonuniform hyperbolicity. Moreover, our method to prove the topological conjugacy used in this paper is completely different from that in [12]. We divided the proof into several lemmas and constructed a concrete topologically equivalent function.

2. Definitions

Consider the linear nonautonomous system with impulses at times {𝑡_𝑘}_𝑘∈ℤ as follows: 𝑥̇(𝑡)=𝐴(𝑡)𝑥,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘,𝑘∈ℤ, (1) where Δ𝑥(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.

A perturbed nonautonomous system with impulsive is therefore described by 𝑥̇(𝑡)=𝐴(𝑡)𝑥+𝑓(𝑡,𝑥),𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ, (2) where, in systems (1) and (2), 𝑥∈ℝ^𝑛, 𝐴(𝑡) and 𝐴(𝑡) are 𝑛×𝑛 matrixes.

Definition 1 (see [11, 12]). The impulsive system (1) is said to be a nonuniform exponential dichotomy in ℝ, if there exist a projection 𝑃(𝑡) and positive constants 𝛼,𝑘, and 𝜀≥0, such that ‖𝑇()𝑃(𝑠{()|𝑠‖𝑇()𝑄(𝑠{()|𝑠𝑡,𝑠)‖≤𝑘exp−𝛼𝑡−𝑠+𝜀|},𝑡≥𝑠,𝑡,𝑠)‖≤𝑘exp−𝛼𝑠−𝑡+𝜀|},𝑡≤𝑠, (3) where 𝑄(𝑡)=Id−𝑃(𝑡) is the complementary projection and 𝑇(𝑡,𝑠) is the evolution operator of the impulsive system (1), which satisfies 𝑇(𝑡,𝑠)𝑃(𝑠)=𝑃(𝑡)𝑇(𝑡,𝑠),𝑡,𝑠∈ℝ.

Definition 2 (see [5, 7]). 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)−𝑥‖ uniformly bounded with respect to 𝑡;(iii)𝐺(𝑡,⋅)=𝐻⁻¹(𝑡,⋅) has property (ii) also;(iv)if 𝑥(𝑡) is a solution of system (2), then 𝐻(𝑡,𝑥(𝑡)) is a solution of system (1).If such a map 𝐻_𝑡 exists, then system (2) is topologically conjugated to (1). 𝐻 is an equivalent function.

3. Main Results and Proof

Theorem 3. Suppose that the linear impulsive system (1) has a nonuniform exponential dichotomy (i.e., system (1) has an evolution operator 𝑇(𝑡,𝑠) satisfying (3)) and, for any 𝑥,𝑥₁,𝑥₂∈ℝ^𝑛 and 𝑡∈ℝ, one assumes that(H₁)‖𝑓(𝑡,𝑥)‖≤𝜇exp(−𝜀|𝑡|),(H₂)‖𝑓(𝑡,𝑥)‖≤𝜇exp(−𝜀|𝑡|),(H₃)‖𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)‖≤𝑟exp(−𝜀|𝑡|)‖𝑥₁−𝑥₂‖,(H₄)‖𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)‖≤𝑟exp(−𝜀|𝑡|)‖𝑥₁−𝑥₂‖,(H₅)2𝑘𝑟𝛼⁻¹+2𝑘𝑟𝑁[1+(1/(1−exp(−𝛼)))]<1,where 𝜇,𝑟≥0, 𝑘, 𝛼, and 𝜀 are the same constants in (3), and 𝑁 is a positive integer such that the intervals [𝑛,𝑛+1) contain no more than 𝑁 terms of the sequences {𝑡_𝑘}_𝑘∈ℤ, for all 𝑛∈ℤ. Then system (2) is topologically conjugated to system (1).

We divide the proof of Theorem 3 into several lemmas.

In what follows, we always suppose that the conditions of Theorem 3 are satisfied. Denote that 𝑋(𝑡,𝑡₀,𝑥₀) is a solution of the system (2) satisfying the initial condition 𝑋(𝑡₀)=𝑥₀, and that 𝑌(𝑡,𝑡₀,𝑦₀) is a solution of the system (1) satisfying the initial condition 𝑌(𝑡₀)=𝑦₀.

Lemma 4. If system (1) has a nonuniform exponential dichotomy, then 𝑥(𝑡)=0 is the unique bounded solution of system (1).

Proof. Let 𝑇(𝑡,𝑠) be the evolution operator satisfying 𝑥(𝑡)=𝑇(𝑡,𝑠)𝑥(𝑠) for every 𝑡,𝑠∈ℝ. Then there exists 𝛼,𝑘>0, 𝜀≥0, and a projection 𝑃(𝑡) satisfying (3). We suppose that 𝑥(𝑡) is any bounded solution of the system (1), and it satisfies the initial condition (𝑠,𝑥(𝑠)). Therefore, 𝑥(𝑡) can be written as 𝑥(𝑡)=𝑇(𝑡,𝑠)𝑃(𝑠)𝑥(𝑠)+𝑇(𝑡,𝑠)[Id−𝑃(𝑠)]𝑥(𝑠). Now we prove 𝑃(𝑠)𝑥(𝑠)=0and[Id−𝑃(𝑠)]𝑥(𝑠)=0.
If 𝑃(𝑠)𝑥(𝑠)≠0, considering 𝑡≤0, ‖𝑥(𝑡=‖𝑇()𝑃(𝑠)𝑥(𝑠)((𝑠𝑥(𝑠≥𝑇()𝑃(𝑠)𝑥(𝑠−‖𝑇((𝑠𝑥(𝑠.)‖𝑡,𝑠+𝑇𝑡,𝑠)[Id−𝑃)])‖|‖𝑡,𝑠)‖𝑡,𝑠)[Id−𝑃)])‖| (4) It follows from the first expression of (3) that ‖𝑃(𝑠)𝑥(𝑠)‖=𝑃²(𝑠)𝑥(𝑠=𝑃(𝑠)𝑇₋₁)()𝑇()𝑃(𝑠)𝑥(𝑠≤𝑃(𝑠)𝑇𝑡,𝑠𝑡,𝑠)₋₁(𝑇()𝑃(𝑠)𝑥(𝑠=‖𝑃(𝑠)𝑇(𝑇()𝑃(𝑠)𝑥(𝑠=‖𝑇()𝑃(𝑡𝑇()𝑃(𝑠)𝑥(𝑠{()|𝑡𝑇()𝑃(𝑠)𝑥(𝑠.𝑡,𝑠)‖𝑡,𝑠)‖𝑠,𝑡)‖‖𝑡,𝑠)‖𝑠,𝑡)‖‖𝑡,𝑠)‖≤𝑘exp−𝛼s−𝑡+𝜀|}‖𝑡,𝑠)‖ (5) Namely, ‖𝑇(𝑡,𝑠)𝑃(𝑠)𝑥(𝑠)‖≥𝑘⁻¹exp{𝛼(𝑠−𝑡)−𝜀|𝑡|}‖𝑃(𝑠)𝑥(𝑠)‖. (6) On the other hand, it follows from the second expression of (3) that ‖𝑇((𝑠𝑥(𝑠=‖𝑇((𝑠×[(𝑠𝑥(𝑠≤‖𝑇((𝑠×(𝑠𝑥(𝑠{()|𝑠×(𝑠𝑥(𝑠.𝑡,𝑠)[Id−𝑃)])‖𝑡,𝑠)[Id−𝑃)]Id−𝑃)])‖𝑡,𝑠)[Id−𝑃)]‖‖[Id−𝑃)])‖≤𝑘exp−𝛼𝑠−𝑡+𝜀|}‖[Id−𝑃)])‖ (7) From the above analysis, which implies that ‖𝑥(𝑡)‖≥𝑘⁻¹{𝛼()|𝑡𝑃(𝑠)𝑥(𝑠{()|𝑠(𝑠𝑥(𝑠.exp𝑠−𝑡+𝜀|}‖)‖−𝑘exp−𝛼𝑠−𝑡+𝜀|}‖[Id−𝑃)])‖ (8) Then we obtain ‖𝑥(𝑡)‖→+∞ as 𝑡→−∞. Similarly, if [Id−𝑃(𝑠)]𝑥(𝑠)≠0, we obtain ‖𝑥(𝑡)‖→+∞ as 𝑡→+∞. Consequently, 𝑃(𝑠)𝑥(s)=0 and [Id−𝑃(𝑠)]𝑥(𝑠)=0. Hence, 𝑥(𝑡)=0.

Lemma 5. For each (𝜏,𝜉), system 𝑧^′=𝐴(𝑡)𝑧−𝑓(𝑡,𝑋(𝑡,𝜏,𝜉)),𝑡≠𝑡_𝑘,Δ𝑧𝑡_𝑘=𝐴𝑡_𝑘𝑧𝑡_𝑘−𝑓𝑡_𝑘,𝑋𝑡_𝑘,𝜏,𝜉,𝑘∈ℤ, (9) has a unique bounded solution ℎ(𝑡,(𝜏,𝜉)) with |ℎ(𝑡,(𝜏,𝜉))|≤2𝑘𝜇𝛼⁻¹+2𝑘𝜇𝑁[1+(1/(1−exp(−𝛼)))].

Proof. For each (𝜏,𝜉), let 𝑧₀(𝑡)=−^𝑡_−∞𝑇()𝑃(𝑠)𝑓((+𝑡,𝑠𝑠,𝑋𝑠,𝜏,𝜉))𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓(𝑠,𝑋(𝑠,𝜏,𝜉))𝑑𝑠−_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,𝜏,𝜉+_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,𝜏,𝜉. (10) Differentiating it, then 𝑧₀(𝑡) is a solution of system (9) It follows from (3), (H₁), and (H₂) that we can easily deduce 𝑧₀(𝑡)≤^𝑡_−∞{()|𝑠(|𝑠+𝑘exp−𝛼𝑡−𝑠+𝜀|}×𝜇exp−𝜀|)𝑑𝑠_𝑡+∞{()|𝑠(|𝑠𝑘exp−𝛼𝑠−𝑡+𝜀|}×𝜇exp−𝜀|)𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑘exp−𝛼𝑡−𝑡_𝑘+𝜀𝑡_𝑘×𝜇exp−𝜀𝑡_𝑘+_{𝑡𝑘∈[𝑡,+∞)}𝑘exp−𝛼𝑡_𝑘−𝑡+𝜀𝑡_𝑘×𝜇exp−𝜀𝑡_𝑘≤2𝑘𝜇𝛼⁻¹1+2𝑘𝜇𝑁1+().1−exp−𝛼 (11) It is easy to show that 𝑧₀(𝑡) is a bounded solution of (9). On the other hand, for each (𝜏,𝜉), the linear part of system (9) has a nonuniform exponential dichotomy, by Lemma 4, then system (9) has a unique bounded solution 𝑧₀(𝑡), we denote ℎ(𝑡,(𝜏,𝜉)) and |ℎ(𝑡,(𝜏,𝜉))|≤2𝑘𝜇𝛼⁻¹+2𝑘𝜇𝑁[1+(1/(1−exp(−𝛼)))].

Lemma 6. For each (𝜏,𝜉), the system 𝑍^′=𝐴(𝑡)𝑍+𝑓(𝑡,𝑌(𝑡,𝜏,𝜉)+𝑍),𝑡≠𝑡_𝑘,Δ𝑍𝑡_𝑘=𝐴𝑡_𝑘𝑍𝑡_𝑘+𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑍𝑡_𝑘,𝑘∈ℤ, (12) has a unique bounded solution 𝑔(𝑡,(𝜏,𝜉)) and |𝑔(𝑡,(𝜏,𝜉))|≤2𝑘𝜇𝛼⁻¹+2𝑘𝜇𝑁[1+(1/(1−exp(−𝛼)))].

Proof. Let 𝔹 be the set of all the continuous bounded functions 𝑧(𝑡) with |𝑧(𝑡)|≤2𝑘𝜇𝛼⁻¹+2𝑘𝜇𝑁[1+(1/(1−exp(−𝛼)))]. For each (𝜏,𝜉) and any 𝑧(𝑡)∈𝔹, define the mapping 𝑇 as follows: 𝑇𝑧(𝑡)=^𝑡_−∞𝑇()𝑃(𝑠)𝑓(()(𝑠−𝑡,𝑠𝑠,𝑌𝑠,𝜏,𝜉+𝑧))𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓(𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧(𝑠))𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧𝑡_𝑘. (13) It follows from (3), (H₁), and (H₂) that |𝑇𝑧(𝑡)|≤^𝑡_−∞{()|𝑠(|𝑠+𝑘exp−𝛼𝑡−𝑠+𝜀|}×𝜇exp−𝜀|)𝑑𝑠_𝑡+∞{()|𝑠(|𝑠𝑘exp−𝛼𝑠−𝑡+𝜀|}×𝜇exp−𝜀|)𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑘exp−𝛼𝑡−𝑡_𝑘+𝜀𝑡_𝑘×𝜇exp−𝜀𝑡_𝑘+_{𝑡𝑘∈[𝑡,+∞)}𝑘exp−𝛼𝑡_𝑘−𝑡+𝜀𝑡_𝑘×𝜇exp−𝜀𝑡_𝑘≤2𝑘𝜇𝛼⁻¹1+2𝑘𝜇𝑁1+1−exp(−𝛼)≜𝐵, (14) which implies that 𝑇 is a self-map of a sphere with radius 𝐵. For any 𝑧₁(𝑡),𝑧₂(𝑡)∈𝔹, and it follows from (3), (H₄), and (H₅), then we have 𝑇𝑧₁(𝑡)−𝑇𝑧₂(𝑡)≤^𝑡_−∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑡−𝑠+𝜀|}×𝑟exp−𝜀|)₁(𝑠)−𝑧₂(𝑠+)𝑑𝑠_𝑡+∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑠−𝑡+𝜀|}×𝑟exp−𝜀|)₁(𝑠)−𝑧₂(𝑠+)𝑑𝑠_{𝑡𝑘∈(−∞,𝑡)}𝑘exp−𝛼𝑡−𝑡_𝑘+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁(𝑠)−𝑧₂(𝑠)+_{𝑡𝑘∈[𝑡,+∞)}𝑘exp−𝛼𝑡_𝑘−𝑡+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁(𝑠)−𝑧₂(𝑠≤)2𝑘𝑟𝛼⁻¹1+2𝑘𝑟𝑁1+()×𝑧1−exp−𝛼₁−𝑧₂. (15) And together with (H₅), 𝑇 has a unique fixed point, namely, 𝑧₀(𝑡), and 𝑧₀(𝑡)=^𝑡_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₀(𝑠)−𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₀(𝑠)𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₀𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₀𝑡_𝑘. (16) It is easy to show that 𝑧₀(𝑡) is a bounded solution of (12). 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 (12). Thus 𝑧₁(𝑡) can be written as follows: 𝑧₁(𝑡)=𝑇(𝑡,0)𝑥₀+^𝑡₀𝑇(𝑡,𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)+𝑑𝑠_{𝑡𝑘∈[0,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘=𝑇(𝑡,0)𝑥₀+^𝑡₀𝑇(𝑃(𝑠)+((𝑠()𝑡,𝑠)[Id−𝑃))]×𝑓𝑠,Y𝑠,𝜏,𝜉+𝑧₁(𝑠)+𝑑𝑠_{𝑡𝑘∈[0,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘+Id−𝑃𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘=𝑇(𝑡,0)𝑥₀+^𝑡_−∞𝑇()𝑃(𝑠)()𝑡,𝑠×𝑓𝑠,𝑌𝑠,𝜏,𝜉+𝑧₁(𝑠)𝑑𝑠−⁰_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)+𝑑𝑠_0+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠−)𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)+𝑑𝑠_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘−_{𝑡𝑘∈(−∞,0)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘+_{𝑡𝑘∈[0,+∞)}𝑇𝑡,𝑡⁺_k𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘. (17) Note that ⁰_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)()𝑑𝑠=𝑇𝑡,0⁰_−∞𝑇(0,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(s()𝑥)𝑑𝑠≜𝑇𝑡,0₁. (18) And together with (3) and (H₁), we have 𝑥₁=⁰_−∞𝑇(0,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)≤𝑑𝑠_0−∞‖𝑇(0,𝑠)𝑃(𝑠)‖𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠≤)𝑑𝑠_0−∞𝑘exp{−𝛼(0−𝑠)+𝜀|𝑠|}𝜇exp(−𝜀|𝑠|)𝑑𝑠=𝑘𝜇𝛼⁻¹. (19) Similarly, ₀^+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)𝑑𝑠≜𝑇(𝑡,0)𝑥₂,𝑥₂≤𝑘𝜇𝛼⁻¹. (20) On the other hand, _{𝑡𝑘∈(−∞,0)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘=𝑇(𝑡,0)_{𝑡𝑘∈(−∞,0)}𝑇0,𝑡⁺_𝑘𝑃𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘≜𝑇(𝑡,0)𝑥₃. (21) And together with (3) and (H₂), we have 𝑥₃=_{𝑡𝑘∈(−∞,0)}𝑇0,𝑡⁺_𝑘𝑃𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘≤_{𝑡𝑘∈(−∞,0)}𝑇0,𝑡⁺_𝑘𝑃𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘≤_{𝑡𝑘∈(−∞,0)}𝑘exp−𝛼0−𝑡_𝑘+𝜀𝑡_𝑘×𝜇exp−𝜀𝑡_𝑘=𝑘𝜇_{𝑡𝑘∈(−∞,0)}exp𝛼𝑡_𝑘1≤𝑘𝜇𝑁1+().1−exp−𝛼 (22) Similarly, _{𝑡𝑘∈[0,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘≜𝑇(𝑡,0)𝑥₄,𝑥₄1≤𝑘𝜇𝑁1+().1−exp−𝛼 (23) Therefore, it follows from the expression of 𝑧₁(𝑡) and it can be written as follows: 𝑧₁(𝑡)=𝑇(𝑡,0)𝑥₀−𝑥₁+𝑥₂−𝑥₃+𝑥₄+^𝑡_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘. (24) Noticing that 𝑧₁(𝑡) is bounded, hence, 𝑇(𝑡,0)(𝑥₀−𝑥₁+𝑥₂−𝑥₃+𝑥₄) is bounded. And it is a solution of system (1). By Lemma 4, we can obtain that 𝑇(𝑡,0)(𝑥₀−𝑥₁+𝑥₂−𝑥₃+𝑥₄)=0. It follows that 𝑧₁(𝑡)=^𝑡_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)𝑑𝑠+_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘. (25) Simple calculation shows 𝑧₁(𝑡)−𝑧₀(t)≤^𝑡_−∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑡−𝑠+𝜀|}×𝑟exp−𝜀|)₁(𝑠)−𝑧₀(𝑠+)𝑑𝑠_𝑡+∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑠−𝑡+𝜀|}×𝑟exp−𝜀|)₁(𝑠)−𝑧₀(𝑠+)𝑑𝑠_{𝑡𝑘∈(−∞,𝑡)}𝑘exp−𝛼𝑡−𝑡_𝑘+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₀𝑡_𝑘+_{𝑡𝑘∈[𝑡,+∞)}𝑘exp−𝛼𝑡_𝑘−𝑡+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₀𝑡_𝑘≤2𝑘𝑟𝛼⁻¹1+2𝑘𝑟𝑁1+()×𝑧1−exp−𝛼₁−𝑧₀. (26) It follows from (H₅) that we can obtain 𝑧₁(𝑡)≡𝑧₀(𝑡). This implies that the bounded solution of (12) is unique. We denote it as 𝑔(𝑡(𝜏,𝜉)). From the above proof, it is easy to see that |𝑔(𝑡,(𝜏,𝜉))|≤2𝑘𝜇𝛼⁻¹+2𝑘𝜇𝑁[1+(1/(1−exp(−𝛼)))].

Lemma 7. Let 𝑥(𝑡) be any solution of the system (2); then 𝑧(𝑡)=0 is the unique bounded solution of system 𝑧^′=𝐴(𝑡)𝑍+𝑓(𝑡,𝑥(𝑡)+𝑍)−𝑓(𝑡,𝑥(𝑡)),𝑡≠𝑡_𝑘,Δ𝑍𝑡_𝑘=𝐴𝑡_𝑘𝑍𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑍𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ. (27)

Proof. Obviously, 𝑧≡0 is a bounded solution of system (27). We show that the bounded solution is unique; if not, there is another bounded solution 𝑧₁(𝑡), which can be written as follows: 𝑧₁(𝑡)=𝑇(𝑡,0)𝑧₁(0)+^𝑡₀𝑇(𝑡,𝑠)𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)−𝑓(𝑠,𝑥(𝑠))𝑑𝑠+_{𝑡𝑘∈[0,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘. (28) By Lemma 6, we can get 𝑧₁(𝑡)=^𝑡_−∞𝑇(𝑡,𝑠)𝑃(𝑠)𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)((𝑠−−𝑓𝑠,𝑥))𝑑𝑠_𝑡+∞𝑇(𝑡,𝑠)𝑄(𝑠)𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)((𝑠+−𝑓𝑠,𝑥))𝑑𝑠_{𝑡𝑘∈(−∞,𝑡)}𝑇𝑡,𝑡⁺_𝑘𝑃𝑡⁺_𝑘𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘−_{𝑡𝑘∈[𝑡,+∞)}𝑇𝑡,𝑡⁺_𝑘𝑄𝑡⁺_𝑘𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘. (29) Then it follows from (3), (H₃), and (H₄) that 𝑧₁(𝑡)≤^𝑡_−∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑡−𝑠+𝜀|}×𝑟exp−𝜀|)₁(𝑠+)𝑑𝑠_𝑡+∞{()|𝑠(|𝑠𝑧𝑘exp−𝛼𝑠−𝑡+𝜀|}×𝑟exp−𝜀|)₁(𝑠+)𝑑𝑠_{𝑡𝑘∈(−∞,𝑡)}𝑘exp−𝛼𝑡−𝑡_𝑘+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁𝑡_𝑘+_{𝑡𝑘∈[𝑡,+∞)}𝑘exp−𝛼𝑡_𝑘−𝑡+𝜀𝑡_𝑘×𝑟exp−𝜀𝑡_𝑘𝑧₁𝑡_𝑘≤2𝑘𝑟𝛼⁻¹1+2𝑘𝑟𝑁1+()𝑧1−exp−𝛼¹, (30) And, together with (H₅), consequently, 𝑧₁(𝑡)≡0. This completes the proof of Lemma 7.

Now we define two functions as follows: 𝐻()((,𝐺().𝑡,𝑥=𝑥+ℎ𝑡,𝑡,𝑥))𝑡,𝑥=𝑦+𝑔𝑡,𝑡,𝑦 (31)(32)

Lemma 8. For any fixed (𝑡₀,𝑥₀), 𝐻(𝑡,𝑋(𝑡,𝑡₀,𝑥₀)) is a solution of system (1).

Proof. Replace (𝜏,𝜉) by (𝑡,𝑋(𝑡,𝜏,𝜉)) in (9); system (9) is not changed. Due to the uniqueness of the bounded solution of (9), we can get that ℎ(𝑡,(𝑡,𝑋(𝑡,𝑡₀,𝑥₀)))=ℎ(𝑡,(𝑡₀,𝑥₀)). Thus 𝐻𝑡,𝑋𝑡,𝑡₀,𝑥₀=𝑋𝑡,𝑡₀,𝑥₀+ℎ𝑡,𝑡₀,𝑥₀. (33) Differentiating it and noticing that 𝑋(𝑡,𝑡₀,𝑥₀) and ℎ(𝑡,(𝑡₀,𝑥₀)) are the solutions of (2) and (9), respectively, we can obtain 𝐻𝑡,𝑋𝑡,𝑡₀,𝑥₀^′=𝐴(𝑡)𝑋𝑡,𝑡₀,𝑥₀+𝑓𝑡,𝑋𝑡,𝑡₀,𝑥₀+𝐴(𝑡)ℎ𝑡,𝑡₀,𝑥₀−𝑓𝑡,𝑋𝑡,𝑡₀,𝑥₀=𝐴(𝑡)𝑋𝑡,𝑡₀,𝑥₀+ℎ𝑡,𝑡₀,𝑥₀=𝐴(𝑡)𝐻𝑡,𝑋𝑡,𝑡₀,𝑥₀Δ𝐻𝑡_𝑘,𝑋𝑡_𝑘,𝑡₀,𝑥₀=𝐴𝑡_𝑘𝑋𝑡_𝑘,𝑡₀,𝑥₀+𝑓𝑡_𝑘,𝑋𝑡_𝑘,𝑡₀,𝑥₀+𝐴𝑡_𝑘ℎ𝑡_𝑘,𝑡_𝑘,𝑋𝑡_𝑘,𝑡₀,𝑥₀−𝑓𝑡_𝑘,𝑋𝑡_𝑘,𝑡₀,𝑥₀=𝐴𝑡_𝑘𝑋𝑡_𝑘,𝑡₀,𝑥₀+ℎ𝑡_𝑘,𝑡₀,𝑥₀=𝐴𝑡_𝑘𝐻𝑡_𝑘,𝑋𝑡_𝑘,𝑡₀,𝑥₀. (34) It indicates that 𝐻(𝑡,𝑋(𝑡,𝑡₀,𝑥₀)) is the solution of system (1).

Lemma 9. For any fixed (𝑡₀,𝑦₀), 𝐺(𝑡,𝑌(𝑡,𝑡₀,𝑦₀)) is a solution of the system (2).

Proof. The proof is similar to Lemma 8.

Lemma 10. For any 𝑡∈ℝ,𝑦∈ℝ^𝑛, 𝐻(𝑡,𝐺(𝑡,𝑦))≡𝑦.

Proof. Let 𝑦(𝑡) be any solution of system (1). By Lemma 9, 𝐺(𝑡,𝑦(𝑡)) is a solution of system (2). Then by Lemma 8, we see that 𝐻(𝑡,𝐺(𝑡,𝑦(𝑡))) is a solution of system (1) written as 𝑦₁(𝑡). Denote 𝐽(𝑡)=𝑦₁(𝑡)−𝑦(𝑡). Differentiating it, we have 𝐽^′(𝑡)=𝑦^′₁(𝑡)−𝑦^′(𝑡)=𝐴(𝑡)𝑦₁(𝑡)(𝑡)𝑦(t)(𝑡)𝐽(𝑡),(𝑡)−𝐴=𝐴Δ𝐽=Δ𝑦₁(𝑡)−Δ𝑦(𝑡)=𝐴𝑡_𝑘𝑦₁𝑡_𝑘−𝐴𝑡_𝑘𝑦𝑡_𝑘=𝐴𝑡_𝑘𝐽𝑡_𝑘, (35) which implies that 𝐽(𝑡) is a solution of system (1). On the other hand, following the definition of 𝐻, 𝐺, and Lemmas 5 and 6, we can obtain |𝐽(𝑡=𝐻(𝑡)(𝑡≤𝐻(𝑡)(𝑡)+𝐺(𝑡)(𝑡=ℎ(𝑡)+𝑔)|𝑡,𝐺𝑡,𝑦−𝑦)𝑡,𝐺𝑡,𝑦−𝐺𝑡,𝑦𝑡,𝑦−𝑦)𝑡,𝑡,𝐺𝑡,𝑦𝑡,𝑡,𝑦≤4𝑘𝜇𝛼⁻¹1+4𝑘𝜇𝑁1+().1−exp−𝛼 (36) This implies that 𝐽(𝑡) is a bounded solution of system (1). However, by Lemma 4, system (1) 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 (1), then Lemma 10 follows.

Lemma 11. For any 𝑡∈ℝ,𝑥∈ℝ^𝑛, 𝐺(𝑡,𝐻(𝑡,𝑥))≡𝑥.

Proof. The proof is similar to Lemma 10.

Now we are in a position to prove the main result.

Proof of Theorem 3. We are going to show that 𝐻(𝑡,⋅) satisfies the four conditions of Definition 2 in the following. Proof of condition (i) for any fixed 𝑡, it follows from Lemmas 10 and 11 that 𝐻(𝑡,⋅) is homeomorphism and 𝐺(𝑡,⋅)=𝐻⁻¹(𝑡,⋅). Proof of condition (ii) it follows from (31) and Lemma 5 that |𝐻(𝑡,𝑥)−𝑥| is bounded uniformly with respect to 𝑡. Proof of condition (iii) it follows from (32) and Lemma 6 that |𝐺(𝑡,𝑦)−𝑦| is bounded uniformly with respect to 𝑡. Proof of condition (iv) it follows from Lemma 8 and Lemma 9 that we easily prove that condition (iv) is true.
Hence, system (2) is topologically conjugated to system (1). This completes the proof of Theorem 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant (nos. 11271333 and 11171090) and ZJNSFC.