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 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥), 𝑡≠𝑡_𝑘, Δ𝑥(𝑡_𝑘)=𝐴(𝑡_𝑘)𝑥(𝑡_𝑘)+𝑓(𝑡_𝑘,𝑥), 𝑘∈ℤ, is topologically conjugated to 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡), 𝑡≠𝑡_𝑘, Δ𝑥(𝑡_𝑘)=𝐴(𝑡_𝑘)𝑥(𝑡_𝑘), 𝑘∈ℤ, 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. Finally, two examples are given to show the feasibility of our results.

1. Introduction

A basic linearization theorem 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. Since they did not discuss the Hölder regularity of the topologically equivalent function 𝐻(𝑡,𝑥), for this reason, recently, Xia et al. [5] gave a rigorous proof of the Hölder regularity. Xia et al. [6, 7] proved a version of generalized Hartman-Grobman theorem for dynamic systems on time scales. It should be noted that the abovementioned works are based on the linear differential equations with uniform exponential dichotomy. Therefore, motivated by [8], in this paper, we have a version of generalized Hartman-Grobman theorem for the impulsive differential equations when the linear system has ordinary dichotomy.

Our main objective in this paper is to prove that, when the impulsive linear system has an ordinary dichotomy, the nonlinear system 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥),𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥,𝑘∈ℤ, (1) is topologically conjugated to its linear part 𝑥̇(𝑡)=𝐴(𝑡)𝑥(𝑡),𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘,𝑘∈ℤ, (2) 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. Finally, two examples are given to show the feasibility of our results.

2. Definitions

Consider the linear nonautonomous system with impulses at times {𝑡_𝑘}_𝑘∈ℤ: 𝑥̇(𝑡)=𝐴(𝑡)𝑥,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘,𝑘∈ℤ, (3) 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, 𝑥∈ℝ^𝑛, and 𝐴(𝑡) and 𝐴(𝑡) are 𝑛×𝑛 matrixes.

Definition 1. System (3) is said to be an ordinary dichotomy, if there exists a projection 𝑃(𝑃²=𝑃) and a constant 𝐾>0 such that 𝐔(𝑡)𝑃𝐔⁻¹(𝑠(),𝐔(𝑡)𝐔)≤𝐾𝑡≥𝑠)(𝐼−𝑃⁻¹(𝑠)≤𝐾(𝑡≤𝑠), (4) where 𝐔(𝑡) is a fundamental matrix of linear system (3) and is given by 𝐔(𝑡)=Φ(𝑡)_{𝑡𝑘∈[𝑡0,𝑡)}Φ⁻¹𝑡_𝑘𝐼+𝐴𝑡_𝑘Φ𝑡_𝑘Φ⁻¹𝑡₀,𝑡≥𝑡₀, (5) where Φ(𝑡) is a fundamental matrix of the system 𝑥̇=𝐴(𝑡)𝑥, provided that Φ(𝑡_𝑘) is invertible, for all 𝑡_𝑘≥𝑡₀. In what follows, we will assume that 𝐔(𝑡) is invertible for all 𝑡∈ℝ.

Definition 2. In Definition 1, if 𝐔(𝑡)𝑃→0 as 𝑡→+∞, then system (3) is said to possess an ordinary dichotomy with a positive asymptotically stable manifold; if 𝐔(𝑡)(𝐼−𝑃)→0 as 𝑡→−∞, then system (3) is said to possess an ordinary dichotomy with a negatively asymptotically stable manifold; if both of them hold, then system (3) is said to possess an ordinary dichotomy with asymptotically stable manifolds.

3. Main Result and Proof

Consider the following nonautonomous impulse systems: 𝑥̇(𝑡)=𝐴(𝑡)𝑥,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘𝑥̇(𝑡)(𝑡)(),𝑘∈ℤ,=𝐴𝑥+𝑓𝑡,𝑥,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ, (6)(7) 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, 𝑥∈ℝ^𝑛, and 𝐴(𝑡) and 𝐴(𝑡) are 𝑛×𝑛 matrixes.

Definition 3. 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)assume that 𝐺(𝑡,⋅)=𝐻⁻¹(𝑡,⋅) has property (ii) also;(iv)if 𝑥(𝑡) is a solution of system (7), then 𝐻(𝑡,𝑥(𝑡)) is a solution of system (6).
If such a map 𝐻 exists, then system (7) is topologically conjugated to (6). 𝐻 is an equivalent function.

Theorem 4. Suppose that the impulsive linear system (6) has an ordinary dichotomy and for any 𝑥,𝑥₁,𝑥₂∈ℝ^𝑛 and 𝑡∈ℝ one has(𝐻₁)‖𝑓(𝑡,𝑥)‖≤𝜓(𝑡),(𝐻₂)‖𝑓(𝑡,𝑥)‖≤𝜓(𝑡),(𝐻₃)‖𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)‖≤𝑟(𝑡)‖𝑥₁−𝑥₂‖,(𝐻₄)‖𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)‖≤𝑟(̃𝑡)‖𝑥₁−𝑥₂‖,(𝐻₅)∫^+∞_−∞𝜓(𝑡)d𝑡+∑_{𝑡𝑘∈(−∞,+∞)}𝜓(𝑡_𝑘)<𝑁,(𝐻₆)∫^+∞_−∞𝑟(𝑡)d𝑡+∑_{𝑡𝑘∈(−∞,+∞)}𝑟(̃𝑡_𝑘)<𝐶, 𝑘∈ℤ,where 𝜓(𝑡), 𝑟(𝑡) are integrable functions and 𝜓(𝑡_𝑘), 𝑟(̃𝑡_𝑘) are summable functions in ℝ, and 𝐶 and 𝑁 are positive constants. Then system (7) is topologically conjugated to system (6).

Remark 5 (pure continuous case). If impulsive jump operators are absent, then system (6) and (7) reduces to pure continuous systems. That is, ̇(𝑡)̇(𝑡)()𝑥=𝐴𝑥,𝑡∈ℝ,𝑥=𝐴𝑥+𝑓𝑡,𝑥,𝑡∈ℝ. (8) Then Theorem 4 reduces to main results in [8].

Remark 6 (Pure discrete cases). A difference system, or a pure discrete-time system, is a special case of systems with impulses. Thus, instead of (6), we have only Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘,𝑘∈ℤ, (9) or for the perturbed case (7), Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ. (10) Now, Δ𝑥(𝑡_𝑘) means 𝑥(𝑡_𝑘+1)−𝑥(𝑡_𝑘), so that we may write the linear system in the canonical way as 𝑥𝑡_𝑘+1=𝐴𝑡_𝑘𝑥𝑡_𝑘,𝑘∈ℤ, (11) or similarly for the perturbed system, 𝑥𝑡_𝑘+1=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ, (12) where 𝐴=(𝐼+𝐴). Then we have the following.

Corollary 7. Suppose that Δ𝑥(𝑡_𝑘)=𝐴(𝑡_k)𝑥(𝑡_𝑘) has an ordinary dichotomy, and for any 𝑥,𝑥₁,𝑥₂∈ℝ^𝑛. If 𝑓(𝑡,𝑥) satisfies 𝑓(≤𝜓(𝑡),𝑓𝑡,𝑥)𝑡,𝑥₁−𝑓𝑡,𝑥₂≤𝑟̃(𝑡)𝑥₁−𝑥₂,_{𝑡𝑘∈(−∞,+∞)}𝜓𝑡_𝑘<𝑁,_{𝑡𝑘∈(−∞,+∞)}𝑟̃𝑡_𝑘<𝐶, (13) then system (10) is topologically equivalent to system (9).

Remark 8. We point out that the conditions in Theorem 4 can be approached. For example, taking 𝜓(𝑡)=𝑟(̃𝑡)=1/𝑡², if we assume that the interval [𝑛,𝑛+1] contains finite number of sequences 𝑡_𝑘, then _{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘=_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘<+∞. (14) In particular, if 𝑡=𝑡_𝑘=𝑘, 𝑘=1,2,…, then ^+∞_𝑡𝑘=1𝜓𝑡_𝑘=^+∞_𝑡𝑘=1𝑟̃𝑡_𝑘=𝜋²6. (15)

Before the proof of Theorem 4, let us make some discussions about ordinary dichotomy and introduce some lemmas. Note first that if system (6) has an ordinary dichotomy then a fundamental matrix can be chosen such that the projection 𝑃=^𝐼𝑟₀ in (4). In fact, for any projection 𝑃, there exists an invertible matrix 𝑇 such that 𝑇⁻¹𝑃𝑇=^𝐼𝑟₀. If (4) holds for 𝐔(𝑡) and 𝑃; then (4) holds for 𝐔(𝑡) and 𝑇⁻¹𝑃𝑇, this implies that 𝑇⁻¹𝑃𝑇=^𝐼𝑟₀ is the required projection if 𝐔(𝑡)𝑇 is chosen as a fundamental matrix. Furthermore, in (4), we can assume that 𝐔(𝑡)=(𝜙₁(𝑡),…,𝜙_𝑛(𝑡)) with 𝜙_𝑖(𝑡) being unbounded on ℝ for 𝑖=1,…,𝑟−𝑠,𝑟+𝑡,…,𝑛 and 𝜙_𝑗(𝑡) bounded on ℝ for 𝑗=𝑟−𝑠+1,…,𝑟+𝑡−1 and 𝑃=^𝐼𝑟₀=𝑃₁+𝑃₂ with 𝑃₁=^{𝐼𝑟−𝑠}₀, 𝐼−𝑃=𝑃₃+𝑃₄ with 𝑃₄=⁰_{𝐼𝑛+1−𝑟−𝑡}. Then |𝐔(𝑡)(𝑃₂+𝑃₃)𝐔⁻¹(𝑠)| is bounded on ℝ, |𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)| is bounded on 𝑡≥𝑠 (and unbounded on 𝑡≤𝑠), and |𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)| is bounded on 𝑡≤𝑠 (and unbounded on 𝑡≥𝑠).

In what follows, we assume that the assumptions in Theorem 4 always hold. Let 𝑋(𝑡,𝑡₀,𝑥₀) be a solution of (7) satisfying the initial condition 𝑋(𝑡₀)=𝑥₀ and 𝑌(𝑡,𝑡₀,𝑦₀) a solution of (6) satisfying the initial condition 𝑌(𝑡₀)=𝑦₀.

Lemma 9. For each (𝜏,𝜉), the system 𝑍̇=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, (16) has a unique bounded solution ℎ(𝑡,(𝜏,𝜉)) with (𝑃₂+𝑃₃)𝐔⁻¹(0)ℎ(0,(𝜏,𝜉))=0.

Proof. For each (𝜏,𝜉), the solution of system (16) satisfying ℎ(0,(𝜏,𝜉))=𝑥₀ is ℎ(𝑡,(𝜏,𝜉))=𝐔(𝑡)𝐔⁻¹(0)𝑥₀−^𝑡₀𝐔(𝑡)𝐔⁻¹(𝑠)×𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠−_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,()(𝑡)𝐔𝜏,𝜉=𝐔⁻¹(0)𝑥₀−⁴_𝑖=1^𝑡₀𝐔(𝑡)𝑃_𝑖𝐔⁻¹(𝑠)(((−×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_{4𝑖=1𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃_𝑖𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,()(𝑡)𝐔𝜏,𝜉=𝐔⁻¹(0)𝑥₀−^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)(((+×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_0−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)(((−×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_0+∞−_𝑡^+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)(((−×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)+_{𝑡𝑘∈()−∞,0}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)−_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(−𝜏,𝜉)_{𝑡𝑘∈[)0,+∞}−_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉). (17) Noting that ⁰_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓((((𝑡)𝑃𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠=𝐔₁𝐔⁻¹(0)⁰_−∞𝐔(0)𝑃₁𝐔⁻¹(𝑠)((((𝑡)𝑃×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠≜𝐔₁𝐔⁻¹(0)𝑥₁,_0+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓((((𝑡)𝑃𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠=𝐔₄𝐔⁻¹(0)₀^+∞𝐔(0)𝑃₄𝐔⁻¹(𝑠)((((𝑡)𝑃×𝑓𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠≜𝐔₄𝐔⁻¹(0)𝑥₂. (18) On the other hand, _{𝑡𝑘∈(−∞,0)}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,()(𝑡)𝑃𝜏,𝜉=𝐔₁𝐔⁻¹(0)_{𝑡𝑘∈()−∞,0}𝐔(0)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,()(𝑡)𝑃𝜏,𝜉≜𝐔₁𝐔⁻¹(0)𝑥₃,_{𝑡𝑘∈[0,+∞)}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,((𝑡)𝑃𝜏,𝜉)=𝐔₄𝐔⁻¹(0)_{𝑡𝑘∈[)0,+∞}𝐔(0)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,()(𝑡)𝑃𝜏,𝜉≜𝐔₄𝐔⁻¹(0)𝑥₄. (19) It follows from (17) that ℎ(𝑡,(𝜏,𝜉))=𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(0)𝑥₀+𝐔(𝑡)𝑃₁𝐔⁻¹(0)×𝑥₀+𝑥₁+𝑥₃+𝐔(𝑡)𝑃₄𝐔⁻¹(0)𝑥₀−𝑥₂−𝑥₄−^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠+_𝑡^+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠−_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)+_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)−_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉). (20) We can assert that 𝑃₁𝐔⁻¹(0)(𝑥₀+𝑥₁+𝑥₃)=0 and 𝑃₄𝐔⁻¹(0)(𝑥₀−𝑥₂−𝑥₄)=0 hold. Otherwise, ℎ(𝑡,(𝜏,𝜉)) will be unbounded since it concludes unbounded part 𝐔(𝑡)𝑃₁𝐔⁻¹(0)(𝑥₀+𝑥₁+𝑥₃) or/and 𝐔(𝑡)𝑃₄𝐔⁻¹(0)(𝑥₀−𝑥₂−𝑥₄). Thus ℎ(𝑡,(𝜏,𝜉))=𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(0)𝑥₀−^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(((+𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠−^𝑡₀𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠−_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)+_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,()−𝜏,𝜉_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉). (21) Moreover, if ℎ(𝑡,(𝜏,𝜉)) satisfies the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)ℎ(0,(𝜏,𝜉))=0, then ℎ(𝑡,(𝜏,𝜉))=−^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(((+𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)+_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)−_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉). (22) Now, we prove that ℎ(𝑡,(𝜏,𝜉)) is bounded. Due to the boundedness of 𝐔(𝑡)𝑃₂=(0,…,0,𝜙_{𝑟−𝑠+1},…,𝜙_𝑟,0,…,0) and 𝐔(𝑡)𝑃₃=(0,…,0,𝜙_𝑟+1,…,𝜙_{𝑟+𝑡−1},0,…,0), we assume that |𝐔(𝑡)𝑃₂𝐔⁻¹(0)|≤𝐶₁, |𝐔(𝑡)𝑃₃𝐔⁻¹(0)|≤𝐶₂, where 𝐶₁, 𝐶₂ are some positive constants. Together with (4) and (𝐻₁), it follows that, if 𝑡≥0, then we have ^𝑡₀𝐔(𝑡)𝑃₂𝐔⁻¹(𝑠)𝑓(((≤𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡0𝐔(𝑡)𝑃₂𝐔⁻¹(𝑠)⋅𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠≤𝐾^𝑡₀𝜓(𝑠)d𝑠; (23) if 𝑡<0, then we have ^𝑡₀𝐔(𝑡)𝑃₂𝐔⁻¹(𝑠)𝑓(((≤𝐔(𝑡)𝑃𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠₂𝐔⁻¹(0)^𝑡₀𝐔(0)𝑃₂𝐔⁻¹(𝑠⋅𝑓((()𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠≤𝐾𝐶₁⁰_𝑡𝜓(𝑠)d𝑠. (24) Similarly, if 𝑡≥0, then we have ^𝑡₀𝐔(𝑡)𝑃₃𝐔⁻¹(𝑠)𝑓(((≤𝐔(𝑡)𝑃𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠₃𝐔⁻¹(0)^𝑡₀𝐔(0)𝑃₃𝐔⁻¹(𝑠⋅𝑓((()𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠≤𝐾𝐶₂^𝑡₀𝜓(𝑠)d𝑠; (25) if 𝑡<0, then we have ^𝑡₀𝐔(𝑡)𝑃₃𝐔⁻¹(𝑠)𝑓(((≤𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠_𝑡0𝐔(𝑡)𝑃₃𝐔⁻¹(𝑠)⋅𝑓(𝑠,𝑋(𝑠,(𝜏,𝜉)))d𝑠≤𝐾⁰_𝑡𝜓(𝑠)d𝑠. (26) On the other hand, it follows from (4) and (𝐻₂) that, if 𝑡≥0, then we have _{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂𝐔⁻¹𝑡_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)≤𝐾_{𝑡𝑘∈[0,𝑡)}𝜓𝑡_𝑘. (27) If 𝑡<0, then we have _{𝑡𝑘∈(]𝑡,0}𝐔(𝑡)𝑃₂𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_k,𝑋𝑡_𝑘,()≤𝐔(𝑡)𝑃𝜏,𝜉₂𝐔⁻¹(0)_{𝑡𝑘∈(]𝑡,0}𝐔(0)𝑃₂𝐔⁻¹×𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)≤𝐾𝐶_{1𝑡𝑘∈(𝑡,0]}𝜓𝑡_𝑘. (28) Similarly, if 𝑡≥0, then we have _{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,()≤𝐔(𝑡)𝑃𝜏,𝜉₃𝐔⁻¹(0)_{𝑡𝑘∈[)0,𝑡}𝐔(0)𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)≤𝐾𝐶_{2𝑡𝑘∈[0,𝑡)}𝜓𝑡_𝑘. (29) If 𝑡<0, then we have _{𝑡𝑘∈(]𝑡,0}𝐔(𝑡)𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝜏,𝜉)≤𝐾_{𝑡𝑘∈(𝑡,0]}𝜓𝑡_𝑘. (30) Therefore, we get ^𝑡₀𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑓(((𝐶𝑠,𝑋𝑠,𝜏,𝜉)))d𝑠≤max₁+1,𝐶₂+1𝐾^+∞_−∞𝜓(𝑡)𝐶d𝑡≤𝐾₁+𝐶₂+1^+∞_−∞𝜓(𝑡)d𝑡,_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,()𝐶𝜏,𝜉≤max₁+1,𝐶₂+1𝐾_{𝑡𝑘∈(−∞,+∞)}𝜓𝑡_𝑘≤𝐾𝐶₁+𝐶₂+1_{𝑡𝑘∈(−∞,+∞)}𝜓𝑡_𝑘. (31) Similarly, we get ^𝑡₀𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑟(𝑠)𝐶d𝑠≤𝐾₁+𝐶₂+1^+∞_−∞𝑟(𝑡)d𝑡,_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑟̃𝑡_𝑘≤𝐾𝐶₁+𝐶₂+1_{𝑡𝑘∈(−∞,+∞)}𝑟̃𝑡_𝑘. (32) It follows from (4), (𝐻₅), (22), and (31) that ℎ(𝑡,(𝜏,𝜉))≤𝐾^𝑡_−∞𝜓(𝑠)d𝑠+𝐾_𝑡^+∞𝜓(𝑠)d𝑠+𝐾_{𝑡𝑘∈()−∞,𝑡}𝜓𝑡_𝑘+𝐾_{𝑡𝑘∈[)𝑡,+∞}𝜓𝑡_𝑘+𝐾𝐶₁+𝐶₂+1^+∞_−∞𝜓(𝑡)𝐶d𝑡+𝐾₁+𝐶₂+1_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘=𝐾^+∞_−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘+𝐾𝐶₁+𝐶₂+1^+∞_−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘<𝐾𝑁+𝐾𝑁𝐶₁+𝐶₂𝐶+1=𝐾𝑁₁+𝐶₂+2. (33) So ℎ(𝑡,(𝜏,𝜉)) is a bounded solution, and the bounded solution is unique with the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)ℎ(0,(𝜏,𝜉))=0. The proof of Lemma 9 is complete.

Lemma 10. For each (𝜏,𝜉), the system 𝑍̇=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, (34) has a unique bounded solution 𝑔(𝑡,(𝜏,𝜉)) with (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑔(0,(𝜏,𝜉))=0.

Proof. For a bounded continuous function 𝑧(𝑡) of ℝ whose norm ‖𝑧‖=sup_𝑡∈ℝ|𝑧(𝑡)|, we define a map 𝑇 as follows: 𝑇𝑧(𝑡)=^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(()(𝑠−𝑠,𝑌𝑠,𝜏,𝜉+𝑧))d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓(()(𝑠+𝑠,𝑌𝑠,𝜏,𝜉+𝑧))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)(()(𝑠+×𝑓𝑠,𝑌𝑠,𝜏,𝜉+𝑧))d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧𝑡_𝑘−_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧𝑡_𝑘+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧𝑡_𝑘. (35) It follows from (4), (𝐻₅), and (31) that we can also obtain that |𝑇𝑧(𝑡)|≤𝐾^𝑡_−∞𝜓(𝑠)d𝑠+𝐾_𝑡^+∞𝜓(𝑠)d𝑠+𝐾_{𝑡𝑘∈()−∞,𝑡}𝜓𝑡_𝑘+𝐾_{𝑡𝑘∈[)𝑡,+∞}𝜓𝑡_𝑘+𝐾𝐶₁+𝐶₂+1^+∞_−∞𝜓(𝑡)𝐶d𝑡+𝐾₁+𝐶₂+1_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘=𝐾^+∞_−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘+𝐾𝐶₁+𝐶₂×+1_+∞−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝜓𝑡_𝑘<𝐾𝑁+𝐾𝑁𝐶₁+𝐶₂𝐶+1=𝐾𝑁₁+𝐶₂+2≜𝐵. (36) So, we have |𝑇𝑧(𝑡)|≤𝐵. Therefore, 𝑇 is a self-map of a sphere with radius 𝐵.
Moreover, it follows from (4), (𝐻₃), (𝐻₄), (𝐻₆), and (32) that 𝑇𝑧₁(𝑡)−𝑇𝑧₂(𝑡)≤𝐾^𝑡_−∞𝑟(𝑠)𝑧₁(𝑠)−𝑧₂(𝑠)d𝑠+𝐾_𝑡^+∞𝑟(𝑠)𝑧₁(𝑠)−𝑧₂(𝑠+)d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠(𝑠𝑧)×𝑟)₁(𝑠)−𝑧₂(𝑠)d𝑠+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑟̃𝑡_𝑘×𝑧₁𝑡_𝑘−𝑧₂𝑡_𝑘+𝐾_{𝑡𝑘∈()−∞,𝑡}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₂𝑡_𝑘+𝐾_{𝑡𝑘∈[)𝑡,+∞}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₂𝑡_𝑘≤𝐾𝐶₁+𝐶₂+1𝑧₁−𝑧₂×^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘+𝐾𝑧₁−𝑧₂×^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘≤𝐾𝐶𝐶₁+𝐶₂+2𝑧₁−𝑧₂. (37) Let 𝐶 be a positive constant such that 𝐶<1/𝐾(𝐶₁+𝐶₂+2); then 𝐿≜𝐾𝐶(𝐶₁+𝐶₂+2)<1. By the contraction principle map 𝑇 has a unique fixed point 𝑧₀(𝑡); that is, 𝑧₀(𝑡) satisfies 𝑧₀(𝑡)=^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₀(𝑠)−d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₀(𝑠+)d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)×𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₀(𝑠)d𝑠+_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₀𝑡_𝑘−_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₀𝑡_𝑘+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₀𝑡_𝑘. (38) By direct differentiation, we can verify that 𝑧₀(𝑡) is a solution of (34). Furthermore, the solution is bounded with |𝑧₀(𝑡)|≤𝐵 and 𝑃₂+𝑃₃𝐔⁻¹(0)𝑧(0)=0. (39) Now, we are going to prove that the bounded solution with initial condition (39) is unique. For this purpose, we assume that 𝑧₁(𝑡) is another solution of (34). Following steps similar to (17)–(22), it is not hard to show that any bounded solution of (34) with initial value condition (39) can be written as follows: 𝑧₁(𝑡)=^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)×𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)−d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)×𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)+d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)×𝑓𝑠,𝑌(𝑠,𝜏,𝜉)+𝑧₁(𝑠)+d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘−_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑌𝑡_𝑘,𝜏,𝜉+𝑧₁𝑡_𝑘. (40) Calculate 𝑧₁(𝑡)−𝑧₀(𝑡), by (4), (𝐻₃), (𝐻₄), (𝐻₆), and (32) that 𝑧₁(𝑡)−𝑧₀(𝑡)≤𝐾^𝑡_−∞𝑟(𝑠)𝑧₁(𝑠)−𝑧₀(𝑠)d𝑠+𝐾_𝑡^+∞𝑟(𝑠)𝑧₁(𝑠)−𝑧₀(𝑠+)d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠𝑟(𝑠)×𝑧₁)(𝑠)−𝑧₀(𝑠)d𝑠+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑟̃𝑡_𝑘×𝑧₁𝑡_𝑘−𝑧₀𝑡_𝑘+𝐾_{𝑡𝑘∈()−∞,𝑡}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₀𝑡_𝑘+𝐾_{𝑡𝑘∈[)𝑡,+∞}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘−𝑧₀𝑡_𝑘≤𝐾𝐶₁+𝐶₂+1𝑧₁−𝑧₀×^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘+𝐾𝑧₁−𝑧₀×^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘≤𝐾𝐶𝐶₁+𝐶₂+2𝑧₁−𝑧₀=𝐿𝑧₁−𝑧₀. (41) Hence ‖𝑧₁−𝑧₀‖≤𝐿‖𝑧₁−𝑧₀‖, and 𝐿<1, so we have 𝑧₁(𝑡)=𝑧₀(𝑡). This implies that the bounded solution of (34) with initial condition (39) is unique. The proof of Lemma 10 is complete.

Lemma 11. Let 𝑥(𝑡) be any solution of the system (7); then the system 𝑍̇=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, (42) has a unique bounded solution 𝑧(𝑡)=0 with (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑧(0)=0.

Proof. Obviously, 𝑧≡0 is a bounded solution of system (42) with the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑧(0)=0. Now we show that the bounded solution is unique. If not, there is another bounded solution 𝑧₁(𝑡), by Lemma 10, which can be written as follows: 𝑧₁(𝑡)=^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)×𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)((𝑠−−𝑓𝑠,𝑥))d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)×𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)((𝑠+−𝑓𝑠,𝑥))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)×𝑓𝑠,𝑥(𝑠)+𝑧₁(𝑠)((𝑠+−𝑓𝑠,𝑥))d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘−_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝑧₁𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘. (43) Then it follows from (4), (𝐻₃), (𝐻₄), (𝐻₆), and (32) that 𝑧₁(𝑡)≤𝐾^𝑡_−∞𝑟(𝑠)𝑧₁(𝑠)d𝑠+𝐾_𝑡^+∞𝑟(𝑠)𝑧₁(𝑠+)d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑟(𝑠)𝑧₁(𝑠)d𝑠+_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘+𝐾_{𝑡𝑘∈()−∞,𝑡}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘+𝐾_{𝑡𝑘∈[)𝑡,+∞}𝑟̃𝑡_𝑘𝑧₁𝑡_𝑘≤𝐾𝐶₁+𝐶₂+1𝑧₁×^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘+𝐾𝑧₁^+∞_−∞𝑟(𝑡)d𝑡+_{𝑡𝑘∈()−∞,+∞}𝑟̃𝑡_𝑘≤𝐾𝐶𝐶₁+𝐶₂+2𝑧₁=𝐿𝑧₁. (44) Since 𝐿<1, so 𝑧₁(𝑡)≡0 with the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑧(0)=0. The proof of Lemma 11 is complete.

Let 𝐻()((,𝐺,𝑡,𝑥=𝑥+ℎ𝑡,𝑡,𝑥))𝑡,𝑦=𝑦+𝑔𝑡,𝑡,𝑦 (45) where ℎ(𝑡,(𝑡,𝑥)) is given by (22), and ℎ(𝑡,(𝑡,𝑥))=−^𝑡_−∞𝐔(𝑡)𝑃₁𝐔⁻¹(𝑠)𝑓(((+𝑠,𝑋𝑠,𝑡,𝑥)))d𝑠_𝑡+∞𝐔(𝑡)𝑃₄𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝑡,𝑥)))d𝑠_𝑡0𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹(𝑠)𝑓(((−𝑠,𝑋𝑠,𝑡,𝑥)))d𝑠_{𝑡𝑘∈()−∞,𝑡}𝐔(𝑡)𝑃₁𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝑡,𝑥)+_{𝑡𝑘∈[)𝑡,+∞}𝐔(𝑡)𝑃₄𝐔⁻¹𝑡⁺_𝑘𝑓𝑡_𝑘,𝑋𝑡_𝑘,(𝑡,𝑥)−_{𝑡𝑘∈[)0,𝑡}𝐔(𝑡)𝑃₂+𝑃₃𝐔⁻¹𝑡⁺_𝑘×𝑓𝑡_k,𝑋𝑡_𝑘,(𝑡,𝑥). (46)

Lemma 12. Let 𝑥(𝑡) be any solution of system (7); then 𝐻(𝑡,𝑥(𝑡)) is a solution of system (6).

Proof. If 𝑥(𝑡) is any solution of system (7), then 𝐻(𝑡,𝑥(𝑡))=𝑥(𝑡)+ℎ(𝑡,(𝑡,𝑥(𝑡)))=𝑥(𝑡)+ℎ(𝑡,(0,𝑥(0))) since, by (46), ℎ(𝑡,(𝑡,𝑥(𝑡)))=ℎ(𝑡,(0,𝑥(0))).
We assume that 𝐻(𝑡)=𝐻(𝑡,𝑥(𝑡)); then we have 𝐻̇(𝑡)(𝑡)𝑥(𝑡)((𝑡(𝑡)ℎ(((𝑡((𝑡(𝑡𝑥(𝑡)(((𝑡(𝑡)𝐻(𝑡),𝑡=𝐴+𝑓𝑡,𝑥))+𝐴𝑡,𝑡,𝑥)))−𝑓𝑡,𝑥))=𝐴)(+ℎ𝑡,𝑡,𝑥))))=𝐴Δ𝐻_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝐴𝑡_𝑘ℎ𝑡_𝑘,𝑡_𝑘,𝑥𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+ℎ𝑡_𝑘,𝑡_𝑘,𝑥𝑡_𝑘=𝐴𝑡_𝑘𝐻𝑡_𝑘. (47) So, 𝐻(𝑡,𝑥(𝑡)) is the solution of system (6).

Lemma 13. Let 𝑦(𝑡) be any solution of system (6); then 𝐺(𝑡,𝑦(𝑡)) is a solution of system (7).

Proof. The proof is similar to Lemma 12.

Lemma 14. For any 𝑡∈ℝ, 𝑥∈ℝ^𝑛, 𝐺(𝑡,𝐻(𝑡,𝑥))=𝑥. (48)

Proof. According to the above arguments, if 𝑥(𝑡) is a solution of system (7), from Lemma 12, 𝐻(𝑡,𝑥(𝑡)) is a solution of (6). On the other hand, in view of Lemma 13, it is easy to see that 𝑥₁(𝑡)=𝐺(𝑡,𝐻(𝑡,𝑥(𝑡))) is another solution of (7). Let 𝐽(𝑡)=𝑥₁(𝑡)−𝑥(𝑡); we have 𝐽̇(𝑡)=𝐴(𝑡)𝑥₁(𝑡)+𝑓𝑡,𝑥₁(𝑡)(𝑡)𝑥(𝑡)((𝑡(𝑡)𝐽(𝑡)((𝑡)(𝑡((𝑡,𝑡−𝐴+𝑓𝑡,𝑥))=𝐴+𝑓𝑡,𝑥+𝐽))−𝑓𝑡,𝑥))Δ𝐽_𝑘=𝐴𝑡_𝑘𝑥₁𝑡_𝑘+𝑓𝑡_𝑘,𝑥₁𝑡_𝑘−𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘=𝐴𝑡_𝑘𝐽𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘+𝐽𝑡_𝑘−𝑓𝑡_𝑘,𝑥𝑡_𝑘. (49) Thus 𝐽(𝑡) is a solution of the system (42). On the other hand, following the definition of 𝐻, 𝐺, we can obtain |𝐽(𝑡=|𝐺(((𝑡(𝑡≤|𝐺(((𝑡((𝑡+|𝐻((𝑡(𝑡=𝑔((((𝑡+|ℎ(((𝑡.)|𝑡,𝐻𝑡,𝑥)))−𝑥)|𝑡,𝐻𝑡,𝑥)))−𝐻𝑡,𝑥))|𝑡,𝑥))−𝑥)|𝑡,𝑡,𝐻𝑡,𝑥))))𝑡,𝑡,𝑥)))| (50) It follows from Lemmas 9 and 10 that 𝐽(𝑡) is a bounded solution of the system (42); by Lemma 11, system (42) has only one zero bounded solution with the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑧(0)=0. Hence 𝐽(𝑡)=0 and thus 𝑥₁(𝑡)=𝑥(𝑡). That is, 𝐺(𝑡,𝐻(𝑡,𝑥(𝑡)))=𝑥(𝑡). Since 𝑥(𝑡) is arbitrary, we have 𝐺(𝑡,𝐻(𝑡,𝑥))=𝑥. (51)

Lemma 15. For any 𝑡∈ℝ, 𝑦∈ℝ^𝑛, 𝐻𝑡,𝐺𝑡,𝑦=𝑦. (52)

Proof. If 𝑦(𝑡) is any solution of system (6), from Lemma 13, 𝐺(𝑡,𝑦(𝑡)) is a solution of (7). On the other hand, in view of Lemma 12, it is easy to see that 𝑦₁(𝑡)=𝐻(𝑡,𝐺(𝑡,𝑦(𝑡))) is another solution of (6). Let 𝐽(𝑡)=𝑦₁(𝑡)−𝑦(𝑡); we have 𝐽̇(𝑡)=𝐴(𝑡)𝑦₁(𝑡)(𝑡)𝑦(𝑡)(𝑡)𝐽(𝑡),𝑡−𝐴=𝐴Δ𝐽_𝑘=𝐴𝑡_𝑘𝑦₁𝑡_𝑘−𝐴𝑡_𝑘𝑦𝑡_𝑘=𝐴𝑡_𝑘𝐽𝑡_𝑘. (53) Thus 𝐽(𝑡) is a solution of the system (6). On the other hand, following the definition of 𝐻, 𝐺, we can obtain |𝐽(𝑡=𝐻(𝑡)(𝑡≤𝐻(𝑡)(𝑡)+𝐺(𝑡)(𝑡=ℎ(𝑡)+𝑔(𝑡).)|𝑡,𝐺𝑡,𝑦−𝑦)𝑡,𝐺𝑡,𝑦−𝐺𝑡,𝑦𝑡,𝑦−𝑦)𝑡,𝑡,𝐺𝑡,𝑦𝑡,𝑡,𝑦 (54) It follows from Lemmas 9 and 10 that 𝐽(𝑡) is a bounded solution of the system (6), and it is easy to see that system (6) has only one zero bounded solution with the initial condition (𝑃₂+𝑃₃)𝐔⁻¹(0)𝑥(0)=0; therefore, 𝐽(𝑡)=0 and thus 𝑦₁(𝑡)=𝑦(𝑡). That is, 𝐻(𝑡,𝐺(𝑡,𝑦(𝑡)))=𝑦(𝑡). Since 𝑦(𝑡) is arbitrary, we have 𝐻𝑡,𝐺𝑡,𝑦=𝑦. (55) So 𝐻 and 𝐺 are inverses of each other for each fixed 𝑡 and they are both homeomorphisms for each fixed 𝑡.

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

Proof of Theorem 4. We are going to show that 𝐻(𝑡,⋅) satisfies the four conditions of Definition 3.
Proof of Condition (i). For any fixed 𝑡, it follows from Lemmas 14 and 15 that 𝐻(𝑡,⋅) is homeomorphism and 𝐺(𝑡,⋅)=𝐻⁻¹(𝑡,⋅).
Proof of Condition (ii). It follows from 𝐻(𝑡,𝑥)=𝑥+ℎ(𝑡,(𝑡,𝑥)) and Lemma 9 that |𝐻(𝑡,𝑥)−𝑥| is bounded, uniformly with respect to 𝑡.
Proof of Condition (iii). It follows from 𝐺(𝑡,𝑦)=𝑦+𝑔(𝑡,(𝑡,𝑦)) and Lemma 10 that |𝐺(𝑡,𝑦)−𝑦| is bounded, uniformly with respect to 𝑡.
Proof of Condition (iv). Following from Lemma 12 and Lemma 13, we easily prove that condition (iv) is true.
Therefore, system (7) is topologically conjugated to system (6). This completes the proof of Theorem 4.

4. Examples

Now we present two examples to show the feasibility of our results. Consider the following impulsive systems: 𝑥^′(𝑡)=𝐴(𝑡)𝑥,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘𝑥,𝑘∈ℤ,_′(𝑡)=𝐴(𝑡)𝑥+𝑓(𝑡,𝑥),𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐴𝑡_𝑘𝑥𝑡_𝑘+𝑓𝑡_𝑘,𝑥𝑡_𝑘,𝑘∈ℤ, (56)(57) where 𝐴(𝑡)−1=diag|𝑡|11+,−1+𝑡²,11+|𝑡|,𝑓()=1,arctan𝑡𝑡,𝑥3𝜋1+𝑡²(sin𝑥,1,cos𝑥,1)^𝑇,𝑓()=1𝑡,𝑥(1+𝑡)²(cos𝑥,sin𝑥,1,1)^𝑇. (58) From system (57), we can easily see that 𝑓(𝑡,𝑥) satisfies 𝑓(<1𝑡,𝑥)1+𝑡²(𝑡),𝑓(<1≜𝜓𝑡,𝑥)(1+𝑡)²≜𝜓(𝑡),𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂<13𝜋1+𝑡²𝑥₁−𝑥₂≜𝑟(𝑡)𝑥₁−𝑥₂,𝑓𝑡,𝑥₁−𝑓𝑡,𝑥₂<1(1+𝑡)²𝑥₁−𝑥₂≜𝑟̃(𝑡)𝑥₁−𝑥₂. (59) It is easy to see that the fundamental matrix of 𝑥^′=𝐴(𝑡)𝑥 is Φ(𝑡)=diag𝑒^{∫𝑡0−(1/(1+|𝑢|))d𝑢},𝑒^{−arctan𝑡},𝑒^{∫𝑡0(1/(1+|𝑢|))d𝑢},𝑒^{∫𝑡0arctan𝑢d𝑢}. (60) Then the fundamental matrix of the impulsive linear system (56) is 𝐔(𝑡)=Φ(𝑡)_{𝑡𝑘∈[)0,𝑡}Φ⁻¹𝑡_𝑘𝐼+𝐴𝑡_𝑘Φ𝑡_𝑘Φ⁻¹(0). (61) In what follows, we give two examples of 𝐴(𝑡_𝑘).

Example 1. Taking 𝐴𝑡_𝑘−1=diag21,−21,−21,−2, (62) denote that 𝑁 is a positive constant such that the interval [0,𝑡) contains no more than 𝑁{𝑡_𝑘}_𝑘∈ℤ. Then we have 𝐔(𝑡)1=diag2^𝑁𝑒^{∫𝑡0−(1/(1+|𝑢|))d𝑢},12^𝑁𝑒^{−arctan𝑡},12^𝑁𝑒^{∫𝑡0(1/(1+|𝑢|))d𝑢},12^𝑁𝑒^{∫𝑡0arctan𝑢d𝑢}. (63) Let 𝑃=diag{1,1,0,0}. (64) Consequently, 𝐔(𝑡)𝑃𝐔⁻¹(𝑠)1=diag4^𝑁𝑒^{∫𝑡𝑠−(1/(1+|𝑢|))d𝑢},14^𝑁𝑒^{−arctan𝑡+arctan𝑠},𝐔(𝑡)𝐔,0,0)(𝐼−𝑃⁻¹(𝑠)1=diag0,0,4^𝑁𝑒^{∫𝑡𝑠(1/(1+|𝑢|))d𝑢},14^𝑁𝑒^{∫𝑡𝑠arctan𝑢d𝑢}. (65) Obviously, 𝐔(𝑡)𝑃𝐔⁻¹(𝑠≤1)4^𝑁𝐔(𝑡)𝐔<1,)(𝐼−𝑃⁻¹(𝑠≤1)4^𝑁<1. (66) That is, system (56) possesses an ordinary dichotomy. On the other hand, we see that ^+∞_−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈(−∞,+∞)}𝜓𝑡_𝑘=^+∞_−∞11+𝑡²d𝑡+_{𝑡𝑘∈()−∞,+∞}11+𝑡_𝑘²𝜋=𝑁𝜋+²6_+∞−∞,𝑟(𝑡)d𝑡+_{𝑡𝑘∈(−∞,+∞)}𝑟̃𝑡_𝑘=^+∞_−∞13𝜋1+𝑡²d𝑡+_{𝑡𝑘∈()−∞,+∞}11+𝑡_𝑘²1=𝑁3+𝜋²6. (67) From Theorem 4, we conclude that (56) and (57) are topologically conjugated.

Example 2. Taking 𝐴𝑡_𝑘−1=diag𝑡_𝑘1+2,−𝑡_𝑘1+2,−𝑡_𝑘1+2,−𝑡_𝑘,{}.+2𝑃=diag1,1,0,0 (68) Taking 𝑡_𝑘=𝑘=0,1,2,…, we have 𝐔(𝑡)1=diag𝑒𝑘+2^{∫𝑡0−(1/(1+|𝑢|))d𝑢},1𝑒𝑘+2^{−arctan𝑡},1𝑒𝑘+2^{∫𝑡0(1/(1+|𝑢|))d𝑢},1𝑒𝑘+2^{∫𝑡0arctan𝑢d𝑢}. (69) That is, 𝐔(𝑡)𝑃𝐔⁻¹(𝑠)1=diag(𝑘+2)²𝑒^{∫𝑡𝑠−(1/(1+|𝑢|))d𝑢},1(𝑘+2)²𝑒^{−arctan𝑡+arctan𝑠},𝐔(𝑡)𝐔,0,0)(𝐼−𝑃⁻¹(𝑠)1=diag0,0,(𝑘+2)²𝑒^{∫𝑡𝑠(1/(1+|𝑢|))d𝑢},1(𝑘+2)²𝑒^{∫𝑡𝑠arctan𝑢d𝑢}. (70) Obviously, 𝐔(𝑡)𝑃𝐔⁻¹(𝑠≤1)(𝑘+2)²𝐔(𝑡)𝐔<1,)(𝐼−𝑃⁻¹(𝑠≤1)(𝑘+2)²<1. (71) In this case, system (56) also possesses an ordinary dichotomy. On the other hand, we have ^+∞_−∞𝜓(𝑡)d𝑡+_{𝑡𝑘∈(−∞,+∞)}𝜓𝑡_𝑘𝜋=𝜋+²6_+∞−∞,𝑟(𝑡)d𝑡+_{𝑡𝑘∈(−∞,+∞)}𝑟̃𝑡_𝑘=13+𝜋²6. (72) From Theorem 4, we conclude that (56) and (57) are topologically conjugated.

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.