Existence and Global Attractivity of Positive Periodic Solutions for a Two-Species Competitive System with Stage Structure and Impulse

Abstract

A class of nonautonomous two-species competitive system with stage structure and impulse is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantee the existence of at least a positive periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Finally, an illustrative example is given to demonstrate the correctness of the obtained results.

1. Introduction

In recent years, with the increasing applications of theory of differential equations in mathematical ecology, various mathematical models have been proposed in the study of population [1–25]. But most of the previous results focused on the dynamical behaviors (including the stability, attractiveness, persistence, and periodicity of solution) of the systems which have fixed parameters and there is no impulse. Considering that harvest of many populations are not continuous and the periodic environmental factor, it is reasonable to investigate the systems with periodic coefficients and impulse. Impulsive differential systems display a combination of characteristics of both the continuous-time and discrete-time systems [26–30]. In 2006, Chen [1] studied the following non-autonomous almost periodic competitive two-species model with stage structure in one species:𝑥̇₂𝑥̇₁(𝑡)=−𝑎₁(𝑡)𝑥₁(𝑡)+𝑏₁(𝑡)𝑥₂(𝑡),(𝑡)=𝑎₂(𝑡𝑥)̇₃𝑥₁(𝑡)−𝑏₂(𝑡)𝑥₂(𝑡)−𝑐(𝑡)𝑥²²(𝑡)−𝛽₁(𝑡)𝑥₂(𝑡)𝑥₂(𝑡)𝑥₃(𝑡),(𝑡)=𝑥₃(𝑡)𝑑(𝑡)−𝑒(𝑡)𝑥₃(𝑡)−𝛽₂(𝑡)𝑥₂(𝑡),(1.1) where 𝑥₁(𝑡) and 𝑥₂(𝑡) are immature and mature population densities of one species, respectively; 𝑥₃(𝑡) represents the population density of another species; 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝛽_𝑖(𝑡)(𝑖=1,2),𝑐(𝑡),𝑑(𝑡),𝑒(𝑡) are all continuous, almost periodic functions. The competition is between 𝑥₂(𝑡) and 𝑥₃(𝑡). Chen [1] obtained sufficient conditions for the existence of a unique, globally attractive, strictly positive almost periodic solution for system (1.1).

Considering that the harvest is an annual harvest pulse, to describe a system more accurately, we should consider the impulsive differential equation. Motivated by this point of view, we revised system (1.1) into the following form:𝑥̇₂𝑥̇₁(𝑡)=−𝑎₁(𝑡)𝑥₁(𝑡)+𝑏₁(𝑡)𝑥₂(𝑡),𝑡≠𝑡_𝑘,(𝑡)=𝑎𝑥₂̇₃(𝑡)𝑥₂(𝑡)−𝑏₂(𝑡)𝑥₂(𝑡)−𝑐(𝑡)𝑥²²(𝑡)−𝛽₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡),𝑡≠𝑡_𝑘,Δ𝑥_𝑖(𝑡)=𝑥₃(𝑡)𝑑(𝑡)−𝑒(𝑡)𝑥₃(𝑡)−𝛽₂(𝑡)𝑥₂(𝑡),𝑡≠𝑡_𝑘,𝑡_𝑘=𝑥_𝑖𝑡^+𝑘−𝑥_𝑖𝑡^−𝑘=−𝛾_𝑖𝑘𝑥_𝑖(𝑡𝑘),𝑖=1,2,3,𝑘=1,2,…,𝑞,(1.2) where Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡^+𝑘)−𝑥_𝑖(𝑡^−𝑘) are the impulses at moments 𝑡_𝑘 and 𝑡₁<𝑡₂<⋯ is a strictly increasing sequence such that lim_𝑘→∞𝑡_𝑘=+∞;𝑥₁(𝑡) and 𝑥₂(𝑡) are immature and mature population densities of one species, respectively, and 𝑥₃(𝑡) represents the population density of another species. The competition is between 𝑥₂(𝑡) and 𝑥₃(𝑡).

Throughout the paper, we always assume the following. (H₁)𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝛽_𝑖(𝑡)(𝑖=1,2),𝑐(𝑡),𝑑(𝑡),𝑒(𝑡) are all continuous 𝜔 periodic; that is, 𝑎_𝑖(𝑡+𝜔)=𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡+𝜔)=𝑏_𝑖(𝑡), 𝛽_𝑖(𝑡+𝜔)=𝛽_𝑖(𝑡)(𝑖=1,2), 𝑐(𝑡+𝜔)=𝑐(𝑡), 𝑑(𝑡+𝜔)=𝑑(𝑡), 𝑒(𝑡+𝜔)=𝑒(𝑡) for any 𝑡∈𝑅.(H₂)𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝛽_𝑖(𝑡)(𝑖=1,2),𝑐(𝑡),𝑑(𝑡),𝑒(𝑡) are all positive.(H₃)0<𝛾_𝑖𝑘<1,𝑖=1,2,3 for all 𝑘∈𝑁, and there exists a positive integer 𝑞 such that 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔,𝛾_{𝑖(𝑘+𝑞)}=𝛾_𝑖𝑘,𝑖=1,2,3.

The principle object of this paper is by using Mawhin’s continuation theorem of coincidence degree theory and by constructing the Lyapunov functions to investigate the stability and existence of periodic solutions of (1.2). To the best of my knowledge, it is the first time to deal with the existence and stability of periodic solutions of (1.2).

The organization of the paper is as follows. In Section 2, we introduce some notations and definitions and state some preliminary results needed in later sections. We then establish, in Section 3, some simple criteria for the existence of positive periodic solutions of system (1.2) by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin [31]. The uniqueness and global attractivity of the positive periodic solution are presented in Section 4. In Section 5, an illustrative example is given to demonstrate the correctness of the obtained results.

2. Preliminaries

We will introduce some notations and definitions and state some preliminary results. Consider the impulsive system𝑥̇(𝑡)=Δ𝑥𝑓((𝑡𝑡),∣𝑥),_{𝑡=𝑡𝑘}𝑡≠𝑡_𝑘,𝑘=1,2,…,=𝐼_𝑘𝑥𝑡^−𝑘,(2.1) where 𝑥∈𝑅^𝑛,𝑓∶𝑅×𝑅^𝑛→𝑅^𝑛 is continuous and 𝑓(𝑡+𝜔,𝑥)=𝑓(𝑡,𝑥);𝐼_𝑘∶𝑅^𝑛→𝑅^𝑛 are continuous, and there exists a positive integer 𝑞 such that 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔,𝐼_𝑘+𝑞(𝑥)=𝐼_𝑘(𝑥) with 𝑡_𝑘∈𝑅,𝑡_𝑘+1>𝑡_𝑘,lim_𝑘→∞=∞,Δ𝑥(𝑡)∣_{𝑡=𝑡𝑘}=𝑥(𝑡^+𝑘)−𝑥(𝑡^−𝑘). For 𝑡_𝑘≠0(𝑘=1,2,…),[0,𝜔]∩{𝑡_𝑘}={𝑡₁,𝑡₂,…,𝑡_𝑞}. As we know, {𝑡_𝑘} are called points of jump.

Let us recall some definitions. For the Canchy problem,Δ𝑥𝑥̇((𝑡𝑡))∣=𝑓_{𝑡=𝑡𝑘}(𝑡,𝑥),𝑡∈[0,𝜔],𝑡≠𝑡_𝑘,=𝐼_𝑘𝑥𝑡^−𝑘,𝑥(0)=𝑥₀.(2.2)

Definition 2.1. A map 𝑥∶[0,𝜔]→𝑅^𝑛 is said to be a solution of (2.2), if it satisfied the following conditions:(i)𝑥(𝑡) is a piecewise continuous map with first-class discontinuity points in 𝑡_𝑘∩[0,𝜔], and at each discontinuity point it is continuous on the left;(ii)𝑥(𝑡) satisfies (2.2).

Definition 2.2. A map 𝑥∶[0,𝜔]→𝑅^𝑛 is said to be an 𝜔 periodic solution of (2.1), if (i)𝑥(𝑡) satisfies (i) and (ii) of Definition 2.1 in the interval [0,𝜔] and(ii)𝑥(𝑡) satisfies 𝑥(𝑡+𝜔−0)=𝑥(𝑡−0),𝑡∈𝑅.

Obviously, if 𝑥(𝑡) is a solution of (2.2) defined on [0,𝜔], such that 𝑥(0)=𝑥(𝜔), then, by the periodicity of (2.2) in 𝑡, the function 𝑥^∗(𝑡) defined by 𝑥^∗(𝑡)=𝑥𝑥_∗(𝑡−𝑗𝜔),𝑡∈[𝑗𝜔,(𝑗+1)𝜔]⧵𝑡^𝑘,(𝑡)isleftcontinuousat𝑡=𝑡_𝑘(2.3) is a 𝜔 periodic solution of (2.1).

For system (1.2), seeking the periodic solutions is equivalent to seeking solutions of the following boundary value problem:𝑥̇₂𝑥̇₁(𝑡)=−𝑎₁(𝑡)𝑥₁(𝑡)+𝑏₁(𝑡)𝑥₂(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,(𝑡)=𝑎𝑥₂̇₃(𝑡)𝑥₂(𝑡)−𝑏₂(𝑡)𝑥₂(𝑡)−𝑐(𝑡)𝑥²²(𝑡)−𝛽₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,Δ𝑥_𝑖(𝑡)=𝑥₃(𝑡)𝑑(𝑡)−𝑒(𝑡)𝑥₃(𝑡)−𝛽₂(𝑡)𝑥₂(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,𝑡_𝑘=𝑥_𝑖𝑡^+𝑘−𝑥_𝑖𝑡^−𝑘=−𝛾_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,3,𝑥_𝑖(0)=𝑥_𝑖(𝜔),𝑘=1,2,…,𝑞.(2.4)

3. Existence of Positive Periodic Solutions

In this section, based on the Mawhin’s continuation theorem, we shall study the existence of at least one periodic solution of (1.1). To do so, we shall make some preparations.

Let 𝑋,𝑌 be normed vector spaces; 𝐿∶Dom𝐿⊂𝑋→𝑌 is a linear mapping; 𝑁∶𝑋→𝑌 is a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝑌. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 such that Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄), it follows that 𝐿∣Dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of that map by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exist isomorphisms 𝐽∶Im𝑄→Ker𝐿.

Now we introduce Mawhin’s continuation theorem [31] as follows.

Lemma 3.1 (Continuation Theorem [31]). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿-compact on Ω. Suppose(a)for each 𝜆∈(0,1), every solution 𝑥 of 𝐿𝑥=𝜆𝑁𝑥 is such that 𝑥∉𝜕Ω.(b)𝑄𝑁𝑥≠0 for each 𝑥∈Ker𝐿⋂𝜕Ω, and deg{JQN,Ω⋂𝜕Ker𝐿,0}≠0.Then the equation 𝐿𝑥=𝑁𝑥 has at least one solution lying in Dom𝐿⋂Ω.

For convenience and simplicity in the following discussion, we always use the notations below throughout the paper: 𝑓=1𝜔^𝜔0𝑓(𝑡)𝑑𝑡,𝑓^𝐿=min_{[]𝑡∈0,𝜔}𝑓(𝑡),𝑓^𝑀=max_{[]𝑡∈0,𝜔}𝑓(𝑡),||𝑓||=1𝜔^𝜔0||𝑓(𝑡)||𝑑𝑡,(3.1) where 𝑓(𝑡) is a 𝜔 continuous periodic function. For any nonnegative integer 𝑝, let 𝐶^(𝑝)[0,𝜔;𝑡₁,𝑡₂,…,𝑡_𝑞]={𝑥∶[0,𝜔]→𝑅^𝑚∣𝑥^(𝑝)(𝑡) exist for 𝑡≠𝑡₁,…,𝑡_𝑞;𝑥^(𝑝)(𝑡+0), and let 𝑥^(𝑝)(𝑡−0) exist at 𝑡₁,𝑡₂,…,𝑡_𝑞, and 𝑥^(𝑗)(𝑡_𝑘)=𝑥^(𝑗)(𝑡_𝑘−0),𝑘=1,…,𝑚,𝑗=0,1,2,…,𝑝} with the norm ‖𝑥‖_𝑝=max{sup_{𝑡∈[0,𝜔]}‖𝑥^(𝑗)(𝑡)‖}^𝑝𝑗=1, where ‖⋅‖ is any norm of 𝑅^𝑚. It is easy to see that 𝐶^(𝑝)[0,𝜔;𝑡₁,𝑡₂,…,𝑡_𝑞] is a Banach space.

Now we are now in a position to state and prove the existence of periodic solutions of (2.4).

Theorem 3.2. In addition to (H₁),(H₂),(H₃), assume further that the following hold: HH₄min𝑃₁,𝑃₂,𝑃₃>0,₅H₆𝑎₁𝜔>^𝑞_𝑘=1ln1−𝛾_1𝑘,𝑑𝜔+^𝑞_𝑘=1ln1−𝛾_3𝑘>𝛽₂𝜔𝑒^𝐵4,𝛽₁𝛽₂≠𝑐𝑒,(3.2) where 𝑃₁=||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1𝑃₂ln1−𝛾_2𝑘−𝑐𝜔𝑒^𝐵6,=||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1𝑃₃ln1−𝛾_2𝑘−𝛽₁𝜔𝑒^𝐵29,=||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘−𝑐𝜔,(3.3) and 𝐵₄,𝐵₆,𝐵₂₉ are defined by (3.27), (3.32), and (3.61), respectively. Then the system (1.2) has at least a 𝜔 periodic solution.

Proof. According to the discussion above in Section 2, we need only to prove that the boundary value problem (2.4) has a solution. Since solutions of (2.4) remained positive for all 𝑡≥0, we let 𝑢₁(𝑡)=ln𝑥₁(𝑡),𝑢₂(𝑡)=ln𝑥₂(𝑡),𝑢₃(𝑡)=ln𝑥₃(𝑡),(3.4) then system (2.4) can be translated to 𝑢̇₂𝑢̇₁(𝑡)=−𝑎₁(𝑡)+𝑏₁(𝑡)𝑒^{(𝑢2(𝑡)−𝑢1(𝑡))},𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,(𝑡)𝑢̇₃=𝑎₂(𝑡)−𝑏₂(𝑡)−𝑐(𝑡)𝑒^𝑢2(𝑡)−𝛽₁(𝑡)𝑒^𝑢3(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,(𝑡)=𝑑Δ(𝑡)𝑢_𝑖−𝑒(𝑡)𝑒^𝑢3(𝑡)−𝛽₂(𝑡)𝑒^𝑢2(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,𝑡_𝑘=ln1−𝛾_𝑖𝑘,𝑖=1,2,3,𝑢_𝑖(0)=𝑢_𝑖(𝜔).(3.5) It is easy to see that if system (3.5) has one 𝜔 periodic solution (𝑢^∗1(𝑡),𝑢^∗2(𝑡),𝑢^∗3(𝑡))^𝑇, then (𝑥^∗1(𝑡),𝑥^∗2(𝑡),𝑥^∗3(𝑡))^𝑇=(𝑒^{𝑢∗1(𝑡)},𝑒^{𝑢∗2(𝑡)},𝑒^{𝑢∗3(𝑡)})^𝑇 is a positive solution of system (1.2). Therefore, to complete the proof, it suffices to show that system (3.5) has at least one 𝜔 periodic solution.
In order to use the continuation theorem of coincidence degree theory, we take𝑋=𝑢∈𝐶0,𝜔;𝑡₁,𝑡₂,…,𝑡_𝑞,𝑌=𝑋×𝑅^3×(𝑞+1).(3.6) Then 𝑋 is a Banach space with norm ‖⋅‖₀, and 𝑌 is also a Banch space with norm ‖𝑧‖=‖𝑥‖₀+‖𝑦‖,𝑥∈𝑋,𝑌∈𝑅^3𝑞.
Let the following hold:dom𝐿𝐿=∶Do𝑥m=𝐿𝑢⊂¹,𝑋𝑢₂⟶,𝑢₃𝑌^𝑇,∈𝑥𝐶⟶[0,𝜔]𝑥;𝑡_1,𝑡₂,…,𝑡_𝑞,𝑁𝑁𝑢=∶𝑋⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝⟶−𝑎₁𝑌,,Δ𝑥𝑡_𝑘^𝑞𝑘=1,𝑎(𝑡)+𝑏₁(𝑡)exp𝑢₂(𝑡)−𝑢₁(𝑡)₂(𝑡)−𝑏₂(𝑡)−𝑐(𝑡)𝑒^𝑢2(𝑡)−𝛽₁(𝑡)𝑒^𝑢3(𝑡)𝑑(𝑡)−𝑒(𝑡)𝑒^𝑢3(𝑡)−𝛽₂(𝑡)𝑒^𝑢2(𝑡)⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝llnn11−−𝛾𝛾₁₁₂₁ln1−𝛾₃₁⎛⎜⎜⎜⎜⎝ln1−𝛾₁₂⎞⎟⎟⎟⎟⎠,ln1−𝛾₂₂ln1−𝛾₃₂⎞⎟⎟⎟⎟⎠,…,⎛⎜⎜⎜⎜⎝ln1−𝛾^1𝑞ln1−𝛾_2𝑞ln1−𝛾_3𝑞⎞⎟⎟⎟⎟⎠,0⎞⎟⎟⎟⎟⎠(3.7) Obviously, KIemr𝐿𝐿==𝑢𝑧∶=𝑢(𝑡𝑓),=𝑎₁ℎ∈𝑅³,𝑡∈[0,𝜔],,𝑎₂,…,𝑎_𝑞,𝑑∈𝑌∶^𝜔0𝑓(𝑠)𝑑𝑠+^𝑞_𝑘=1=𝑋×𝑅_3×𝑞𝑎_𝑘+𝑑=0dimKer𝐿=3=codim×Im{𝐿0}.,(3.8) So, Im𝐿 is closed in 𝑌; 𝐿 is a Fredholm mapping of index zero. Define two projectors 𝑃𝑥=1𝜔^𝜔0𝑄𝑧=𝑄𝑓,𝑎₁𝑥(𝑡)𝑑𝑡,,𝑎₂,…,𝑎_𝑞,𝑑=1𝜔^𝜔0𝑓(𝑠)𝑑𝑠+^𝑞_𝑘=1𝑎_𝑘+𝑑,,0,0,…,0.(3.9) It is easy to show that 𝑃 and 𝑄 are continuous and satisfy Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄).
Further, by direct computation, we can find that the inverse 𝐾_𝑃 of 𝐿, 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 has the following form:𝐾_𝑃(𝑧)=^𝑡0𝑓(𝑠)𝑑𝑠+_{𝑡𝑘<𝑡}𝑎_𝑘−1𝜔^𝜔0^𝑡0𝑓(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1𝑎_𝑘+1𝜔^𝑞_𝑘=1𝑎_𝑘𝑡_𝑘.(3.10) Moreover, it is easy to check that 𝑄𝑁𝑢=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝1𝜔^𝑡0𝐹₁(𝑠)𝑑𝑠+1𝜔^𝑞_𝑘=11ln1−𝛾_1𝑘𝜔^𝑡0𝐹₂(𝑠)𝑑𝑠+1𝜔^𝑞_𝑘=11ln1−𝛾_2𝑘𝜔^𝑡0𝐹₃(𝑠)𝑑𝑠+1𝜔^𝑞_𝑘=1𝐾ln1−𝛾_3𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,0,0,…,0_𝑃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(𝐼−𝑄)𝑁𝑢=⎛⎜⎜⎜⎜⎜⎜⎜⎝^𝑡0_𝑡0𝐹₁(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}ln1−𝛾_1𝑘_𝑡0𝐹₂(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}ln1−𝛾_2𝑘−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝1𝐹₃(𝑠)𝑑𝑠+_{𝑡>𝑡𝑘}ln1−𝛾_3𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎠𝜔^𝜔0^𝑡0𝐹₁(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1ln1−𝛾_1𝑘+1𝜔^𝑞_𝑘=11ln1−𝛾_1𝑘𝜔^𝜔0^𝑡0𝐹₂(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1ln1−𝛾_2𝑘+1𝜔^𝑞_𝑘=11ln1−𝛾_2𝑘𝜔^𝜔0^𝑡0𝐹₃(𝑠)𝑑𝑠𝑑𝑡−^𝑞_𝑘=1ln1−𝛾_3𝑘+1𝜔^𝑞_𝑘=1−𝑡ln1−𝛾_3𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠𝜔−12⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝^𝜔0𝐹₁(𝑠)𝑑𝑠+^𝑞_𝑘=1_𝜔0ln1−𝛾_1𝑘𝐹₂(𝑠)𝑑𝑠+^𝑞_𝑘=1_𝜔0ln1−𝛾_2𝑘𝐹₃(𝑠)𝑑𝑠+^𝑞_𝑘=1ln1−𝛾_3𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.11) where 𝐹₂𝐹₁(𝑠)=−𝑎₁(𝑠)+𝑏₁(𝑠)𝑒^{(𝑢2(𝑠)−𝑢1(𝑠))},(𝑠𝐹)₂=𝑎₂(𝑠)−𝑏₂(𝑠)−𝑐(𝑠)𝑒^𝑢2(𝑠)−𝛽₁(𝑠)𝑒^𝑢3(𝑠),(𝑠)=𝑑(𝑠)−𝑒(𝑠)𝑒^𝑢3(𝑠)−𝛽₂(𝑠)𝑒^𝑢2(𝑠).(3.12) Obviously, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. Using the Ascoli-Arzela theorem, it is not difficult to show that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋. Moreover, 𝑄𝑁(Ω) is bounded. Thus, 𝑁 is 𝐿-compact on Ω with any open bounded set Ω⊂𝑋.
Now we are at the point to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation 𝐿𝑢=𝜆𝑁𝑢,𝜆∈(0,1), we have𝑢̇₂𝑢̇₁(𝑡)=𝜆−𝑎₁(𝑡)+𝑏₁(𝑡)𝑒^{(𝑢2(𝑡)−𝑢1(𝑡))},𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,(𝑢𝑡̇₃)=𝜆𝑎₂(𝑡)−𝑏₂(𝑡)−𝑐(𝑡)𝑒^𝑢2(𝑡)−𝛽₁(𝑡)𝑒^𝑢3(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,(𝑡)=𝜆𝑑Δ₁𝑢(𝑡_𝑖)−𝑒(𝑡)𝑒^𝑢3(𝑡)−𝛽₂(𝑡)𝑒^𝑢2(𝑡),𝑡≠𝑡_𝑘,𝑡∈[0,𝜔],𝑘=1,2,…,𝑞,𝑡_𝑘=𝜆ln1−𝛾_𝑖𝑘,𝑖=1,2,3,𝑢_𝑖(0)=𝑢_𝑖(𝜔).(3.13) Suppose that 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡))^𝑇∈𝑋 is an arbitrary solution of system (3.13) for a certain 𝜆∈(0,1), integrating both sides of (3.13) over the interval [0,𝜔] with respect to 𝑡, we obtain ^𝜔0𝑏₁(𝑡)𝑒^{(𝑢2(𝑡)−𝑢1(𝑡))}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘=^𝜔0𝑎₁(𝑡)𝑑𝑡,_𝜔0𝑐(𝑡_𝜔0)𝑒^𝑢2(𝑡)+𝛽₁(𝑡)𝑒^𝑢3(𝑡)𝑑𝑡=^𝜔0𝑎₂(𝑡)−𝑏₂(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_2𝑘,𝑒(𝑡)𝑒^𝑢3(𝑡)+𝛽₂(𝑡)𝑒^𝑢2(𝑡)𝑑𝑡=^𝜔0𝑑(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_3𝑘.(3.14) From (3.13) and (3.14), we obtain ^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡<=2^𝜔0𝑎₁(𝑡)𝑑𝑡+^𝜔0𝑏₁(𝑡)𝑒^{(𝑢2(𝑡)−𝑢1(𝑡))}𝑑𝑡𝑎₁𝜔−^𝑞_𝑘=1ln1−𝛾_1𝑘,(3.15)_𝜔0||𝑢̇₂(𝑡)||𝑑𝑡<=2^𝜔0𝑎₂(𝑡)+𝑏₂(𝑡)𝑑𝑡+^𝜔0𝑐(𝑡)𝑒^𝑢2(𝑡)+𝛽₁(𝑡)𝑒^𝑢3(𝑡))𝑑𝑡𝑎₂𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘,(3.16)_𝜔0||𝑢̇₃(𝑡)||𝑑𝑡<=2^𝜔0𝑑(𝑡)𝑑𝑡+^𝜔0𝑒(𝑡)𝑒^𝑢3(𝑡)+𝛽₂(𝑡)𝑒^𝑢2(𝑡)𝑑𝑡𝑑𝜔+^𝑞_𝑘=1ln1−𝛾_3𝑘.(3.17) Let the following hold: 𝑢_𝑖𝜉_𝑖=min_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡),𝑢_𝑖𝜂_𝑖=max_{𝑡∈[0,𝜔]}𝑢_𝑖(𝑡),𝑖=1,2,3.(3.18) From the second and the third equations of (3.14), we can obtain ||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘>^𝜔0𝑐(𝑡)𝑒^𝑢2(𝑡)𝑑𝑡≥^𝜔0𝑐(𝑡)𝑒^𝑢2(𝜉2)𝑑𝑡,𝑑𝜔+^𝑞_𝑘=1ln1−𝛾_3𝑘>^𝜔0𝑒(𝑡)𝑒^𝑢3(𝑡)𝑑𝑡≥^𝜔0𝑒(𝑡)𝑒^𝑢3(𝜉3)𝑑𝑡=𝑒𝜔𝑒^𝑢3(𝜉3),(3.19) then 𝑢₂𝜉₂<ln⎡⎢⎢⎣||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘𝑢𝑐𝜔⎤⎥⎥⎦,(3.20)₃𝜉₃<ln𝑑𝜔+∑^𝑞𝑘=1ln1−𝛾_3𝑘𝑒𝜔.(3.21) Thus 𝑢₂(𝑡)<=𝑢ln₂⎡⎢⎢⎣𝜉₂+^𝑡𝜉₂𝑢̇₂(𝑡)𝑑𝑡≤𝑢₂𝜉₂+^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘𝑐𝜔⎤⎥⎥⎦+2𝑎₂𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘=∶𝐵₁.(3.22)
In the following, we will consider four cases.
Case 1 (if 𝑢₁(𝑡)>0,𝑢₂(𝑡)>0). From the first equation of (3.14), we have𝑎₁𝜔<^𝜔0𝑏₁(𝑡)𝑒^𝑢2(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘≤𝑏₁𝜔𝑒^𝑢2(𝜂2)+^𝑞_𝑘=1ln1−𝛾_3𝑘,𝑎₁𝜔>^𝜔0𝑏₁(𝑡)𝑒^{−𝑢1(𝑡)}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘≥^𝜔0𝑏₁(𝑡)𝑒^{−𝑢1(𝜂1)}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘,(3.23) that is, 𝑢₂𝜂₂>ln𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘𝑢₁𝑏₁𝜔,𝜂₁>ln𝑏₁𝜔𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘.(3.24) Then 𝑢₂(𝑡)=≥𝑢ln₂𝜂₂−^𝜂2_𝑡𝑢̇₂(𝑡)𝑑𝑡≥𝑢₂𝜂₂−^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘𝑏₁𝜔−2𝑎₂𝜔−^𝑞_𝑘=1𝑢ln1−𝛾_2𝑘=∶𝐵₂,(3.25)₁(𝑡)=≥𝑢ln₁𝜂₁−^𝜂1_𝑡𝑢̇₁(𝑡)𝑑𝑡≥𝑢₁𝜂₁−^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡𝑏₁𝜔𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘−2𝑎₁𝜔−^𝑞_𝑘=1ln1−𝛾_1𝑘=∶𝐵₃.(3.26) Thus, from (3.22) and (3.25), we obtain ||𝑢₂(𝑡)||≤max||𝐵₁||,||𝐵₂||=∶𝐵₄.(3.27) By the first and the third equations of (3.14), we get ^𝜔0𝑏₁(𝑡)𝑒^{𝐵4−𝑢1(𝜉1)}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘>𝑎₁𝜔,_𝜔0𝑒(𝑡)𝑒^𝑢3(𝜂3)+𝛽₂(𝑡)𝑒^𝐵4𝑑𝑡>𝑑𝜔+^𝑞_𝑘=1ln1−𝛾_3𝑘,(3.28) then 𝑢₁𝜉₁<ln𝑏₁𝜔𝑒^𝐵4𝑢𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘,(3.29)₃𝜂₃>ln𝑑𝜔+∑^𝑞𝑘=1ln1−𝛾_3𝑘−𝛽₂𝜔𝑒^𝐵4𝑒𝜔.(3.30) From (3.15), (3.17), (3.21), and (3.30), we have 𝑢₁(𝑡)=<l𝑢n₁𝜉₁+^𝑡𝜉₁𝑢̇₁(𝑡)𝑑𝑡≤𝑢₁𝜉₁+^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡𝑏₁𝜔𝑒^𝐵4𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘+2𝑎₁𝜔+^𝑞_𝑘=1𝑢ln1−𝛾_1𝑘=∶𝐵₅,(3.31)₃(𝑡)=<l𝑢n₃𝜉₃+^𝑡𝜉₃𝑢̇₃(𝑡)𝑑𝑡≤𝑢₃𝜉₃+^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡𝑑𝜔+∑^𝑞𝑘=1ln1−𝛾_3𝑘𝑒𝜔+2𝑑𝜔+^𝑞_𝑘=1𝑢ln1−𝛾_3𝑘=∶𝐵₆,(3.32)₃(𝑡)=>𝑢ln₃𝜂₃−^𝜂3_𝑡𝑢̇₃(𝑡)𝑑𝑡≥𝑢₃𝜂₃−^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡𝑑𝜔+∑^𝑞𝑘=1ln1−𝛾_3𝑘−𝛽₂𝜔𝑒^𝐵4𝑒𝜔−2𝑑𝜔+^𝑞_𝑘=1ln1−𝛾_3𝑘=∶𝐵₇.(3.33) Thus, ||𝑢₁(𝑡)||≤𝐵₅,||𝑢₃(𝑡)||max||𝐵₆||,||𝐵₇||=∶𝐵₈.(3.34)Case 2 (if 𝑢₁(𝑡)>0,𝑢₂(𝑡)<0). By the first equation of (3.14), we have𝑎₁𝜔<^𝜔0𝑏₁(𝑡)𝑒^𝑢2(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘≤𝑏₁𝜔𝑒^𝑢2(𝜂2)+^𝑞_𝑘=1ln1−𝛾_3𝑘,(3.35) namely, 𝑢₂𝜂₂>ln𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘𝑏₁𝜔.(3.36) Then 𝑢₂(𝑡)=≥𝑢ln₂𝜂₂−^𝜂2_𝑡𝑢̇₂(𝑡)𝑑𝑡≥𝑢₂𝜂₂−^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘𝑏₁−2𝑎₂𝜔−^𝑞_𝑘=1ln1−𝛾_2𝑘=∶𝐵₂.(3.37) From (3.22) and (3.37), we obtain ||𝑢₂(𝑡)||≤max||𝐵₁||,||𝐵₂||=∶𝐵₃.(3.38) By the first equation of (3.14), we also have ^𝜔0𝑏₁(𝑡)𝑒^𝐵3𝑒^𝑢1(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘>𝑎₁𝜔,_𝜔0𝑏₁(𝑡)𝑒^−𝐵3𝑒^𝑢1(𝑡)𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘<𝑎₁𝜔,(3.39) Then 𝑏₁𝜔𝑒^𝐵3𝑒^𝑢1(𝜉1)+^𝑞_𝑘=1ln1−𝛾_1𝑘>𝑎₁𝜔,𝑏₁𝜔𝑒^−𝐵3𝑒^𝑢1(𝜂1)+^𝑞_𝑘=1ln1−𝛾_1𝑘<𝑎₁𝜔.(3.40) that is, 𝑢₁𝜉₁<ln𝑏₁𝜔𝑒^𝐵3𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘=∶𝐵⁸𝑢₁,𝜂₁<ln𝑏₁𝜔𝑒^𝐵3𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘=∶𝐵⁹.(3.41) Thus, 𝑢₁(𝑡)=<𝐵𝑢₁₈𝜉₁+^𝑡𝜉₁𝑢̇₁(𝑡)𝑑𝑡≤𝑢₁𝜉₁+^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡+2𝑎₁𝜔−^𝑞_𝑘=1𝑢₁ln1−𝛾_1𝑘=∶𝐵₁₀,(𝑡)=>𝑢𝐵₁₉𝜂₁+^𝜂1_𝑡𝑢̇₁(𝑡)𝑑𝑡≥𝑢₁𝜂₁+^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡−2𝑎₁𝜔−^𝑞_𝑘=1ln1−𝛾_1𝑘=∶𝐵₁₁.(3.42) From (3.42), we have ||𝑢₁(𝑡)≤max||𝐵₁₀||,||𝐵₁₁||=∶𝐵₁₂.(3.43) By the second equation of (3.14), we have ^𝜔0𝑐(𝑡)𝑒^𝐵3𝑑𝑡+^𝜔0𝛽₁(𝑡)𝑒^𝑢3(𝑡)𝑑𝑡>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1_𝜔0ln1−𝛾_2𝑘,𝑐(𝑡)𝑒^−𝐵3𝑑𝑡+^𝜔0𝛽₁(𝑡)𝑒^𝑢3(𝑡)𝑑𝑡<||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘.(3.44) Then 𝑐𝜔𝑒^𝐵3+𝛽₁𝜔𝑒^𝑢3(𝜂3)>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘,𝑐𝜔𝑒^−𝐵3+𝛽₁𝜔𝑒^𝑢3(𝜉3)<||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘=∶𝐵₁₃,(3.45) that is, 𝑢₃𝜂₃>ln⎡⎢⎢⎣||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘−𝑐𝜔𝑒^𝐵6𝑢₃𝛽₁𝜔⎤⎥⎥⎦,𝜉₃<ln⎡⎢⎢⎣||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘−𝑐𝜔𝑒^−𝐵3𝛽₁𝜔⎤⎥⎥⎦=∶𝐵¹⁴.(3.46) Therefore, we get 𝑢₃(𝑡)=<𝐵𝑢₃₁₄𝜉₃+^𝑡𝜉₃𝑢̇₃(𝑡)𝑑𝑡≤𝑢₃𝜉₃+^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡+2𝑑₁𝜔−^𝑞_𝑘=1𝑢₃ln1−𝛾_3𝑘=∶𝐵₁₅,(𝑡)=>𝑢𝐵₃₁₄𝜂₃−^𝜂3_𝑡𝑢̇₃(𝑡)𝑑𝑡≥𝑢₃𝜂₃−^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡−2𝑑₁𝜔−^𝑞_𝑘=1ln1−𝛾_3𝑘=∶𝐵₁₆.(3.47) Hence, we have ||𝑢₃(𝑡)||≤max||𝐵₁₅||,||𝐵₁₆||=∶𝐵₁₇.(3.48)Case 3 (if 𝑢₁(𝑡)<0,𝑢₂(𝑡)>0). By the first equation of (3.14), we have^𝜔0𝑏₁(𝑡)𝑒^{−𝑢1(𝑡)}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘<^𝜔0𝑎₁(𝑡)𝑑𝑡,_𝜔0𝑏₁(𝑡)𝑒^{(𝑢2(𝜂2)−𝑢1(𝑡))}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘>^𝜔0𝑎₁(𝑡)𝑑𝑡.(3.49) Then 𝑏₁𝜔𝑒^{−𝑢1(𝜂1)}+^𝑞_𝑘=1ln1−𝛾_1𝑘<𝑎₁𝜔,𝑏₁𝜔𝑒^−𝐵19𝑒^𝑢2(𝜂2)+^𝑞_𝑘=1ln1−𝛾_1𝑘>𝑎₁𝜔.(3.50) namely, 𝑢₁𝜂₁>ln𝑏₁𝜔𝑎₁−∑^𝑞𝑘=1ln1−𝛾_1𝑘=∶𝐵¹⁸𝑢₂,𝜂₂>ln𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘𝑏₁𝜔𝑒^−𝐵19=∶𝐵²⁰.(3.51) Therefore, 𝑢₁(𝑡)=>𝑢𝐵₁₁₈𝜂₁−^𝜂1_𝑡𝑢̇₁(𝑡)𝑑𝑡≥𝑢₁𝜂₁−^𝜔0||𝑢̇₁(𝑡)||𝑑𝑡−2𝑎₁𝜔−^𝑞_𝑘=1𝑢₂ln1−𝛾_1𝑘=∶𝐵₁₉,(𝑡)=<l𝑢n₂⎡⎢⎢⎣𝜉₂+^𝑡𝜉₂𝑢̇₂(𝑡)𝑑𝑡≤𝑢₂𝜉₂+^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘𝑐𝜔⎤⎥⎥⎦+2𝑎₂𝜔+^𝑞_𝑘=1𝑢₂ln1−𝛾_2𝑘=∶𝐵₂₁,(𝑡)=>𝑢𝐵₂₂₀𝜂₂−^𝜂2_𝑡𝑢̇₂(𝑡)𝑑𝑡≥𝑢₂𝜂₁−^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡−2𝑎₂𝜔−^𝑞_𝑘=1ln1−𝛾_2𝑘=∶𝐵₂₂.(3.52) So 𝐵₁₉<𝑢₁(𝑡)<0,||𝑢₂(𝑡)||≤max||𝐵₂₁||,||𝐵₂₂||=∶𝐵₂₃.(3.53) By the third equation of (3.14), we obtain ^𝜔0𝑒(𝑡)𝑒^𝑢3(𝜂3)𝑑𝑡+^𝜔0𝛽₂(𝑡)𝑒^𝐵23𝑑𝑡>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘,(3.54) that is, 𝑢₃𝜂₃>ln⎡⎢⎢⎣||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘−𝛽₂𝜔𝑒^𝐵23𝑒𝜔⎤⎥⎥⎦=∶𝐵²⁴.(3.55) Thus, ||𝑢₃(𝑡)||≤max||𝐵₂₃||,||𝐵₂₄||=∶𝐵₂₅.(3.56)Case 4 (if 𝑢₁(𝑡)<0,𝑢₂(𝑡)<0). By the second equation of (3.14), we have^𝜔0𝑐(𝑡)𝑑𝑡+^𝜔0𝛽₁(𝑡)𝑒^𝑢3(𝑡)𝑑𝑡>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘.(3.57) Then 𝑐𝜔+𝛽₁𝜔𝑒^𝑢3(𝜂3)>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘,(3.58) that is, 𝑢₃𝜂₃≥ln|𝑎₂−𝑏₂|𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘−𝑐𝜔𝛽₁𝜔=∶𝐵²⁶.(3.59) Therefore, 𝑢₃(𝑡)=<l𝑢n₃𝜉₃+^𝑡𝜉₃𝑢̇₃(𝑡)𝑑𝑡≤𝑢₃𝜉₃+^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡𝑑𝜔+∑^𝑞𝑘=1ln1−𝛾_3𝑘𝑒𝜔+2𝑑𝜔+^𝑞_𝑘=1𝑢₃ln1−𝛾_3𝑘=∶𝐵₂₇,(𝑡)=>𝐵𝑢₃₂₆𝜂₃−^𝜂3_𝑡𝑢̇₃(𝑡)𝑑𝑡≥𝑢₃𝜂₃−^𝜔0||𝑢̇₃(𝑡)||𝑑𝑡−2𝑑𝜔−^𝑞_𝑘=1ln1−𝛾_3𝑘=∶𝐵₂₈.(3.60) Thus, ||𝑢₃(𝑡)||≤max||𝐵₂₇||,||𝐵₂₈||=∶𝐵₂₉.(3.61) By the second equation of (3.14), we obtain ^𝜔0𝑐(𝑡)𝑒^𝑢2(𝜂2)𝑑𝑡+^𝜔0𝛽₁(𝑡)𝑒^𝐵29𝑑𝑡>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘,(3.62) that is, 𝑐𝑒^𝑢2(𝜂2)+𝛽₁𝜔𝑒^𝐵29>||𝑎₂−𝑏₂||𝜔+^𝑞_𝑘=1ln1−𝛾_2𝑘.(3.63) Thus, 𝑢₂𝜂₂>ln⎡⎢⎢⎣||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘−𝛽₁𝜔𝑒^𝐵29𝑐𝜔⎤⎥⎥⎦.(3.64) Then, from (3.16) and (3.20), we get 𝑢₂(𝑡)=<l𝑢n₂⎡⎢⎢⎣𝜉₂+^𝑡𝜉₂𝑢̇₂(𝑡)𝑑𝑡≤𝑢₂𝜉₂+^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡||𝑎₂−𝑏₂||𝜔+∑^𝑞𝑘=1ln1−𝛾_2𝑘𝑐𝜔⎤⎥⎥⎦+2𝑎₂𝜔+^𝑞_𝑘=1𝑢₂ln1−𝛾_2𝑘=∶𝐵₃₁,(𝑡)=>𝑢𝐵₂₃₀𝜂₂−^𝜂2_𝑡𝑢̇₂(𝑡)𝑑𝑡≥𝑢₂𝜂₂−^𝜔0||𝑢̇₂(𝑡)||𝑑𝑡−2𝑎₂𝜔𝜔−^𝑞_𝑘=1ln1−𝛾_2𝑘=∶𝐵₃₂.(3.65) Thus, ||𝑢₂(𝑡)||≤max||𝐵₃₁||,||𝐵₃₂||=∶𝐵₃₃.(3.66) By the first equation of (3.14), we have ^𝜔0𝑏₁(𝑡)𝑒^{𝑢2(𝑡)−𝑢1(𝜂1)}𝑑𝑡+^𝑞_𝑘=1ln1−𝛾_1𝑘<^𝜔0𝑎₁(𝑡)𝑑𝑡.(3.67) Then 𝑏₁𝜔𝑒^−𝐵33𝑒^{−𝑢1(𝜂1)}+^𝑞_𝑘=1ln1−𝛾_1𝑘<𝑎₁𝜔.(3.68) Thus, 𝑢₁𝜂₁>ln𝑏₁𝜔𝑒^−𝐵32𝑎₁𝜔−∑^𝑞𝑘=1ln1−𝛾_1𝑘=∶𝐵³⁴.(3.69) Hence, we have 𝐵₃₄<𝑢₁(𝑡)<0.(3.70) Based on the discussion above, we can easily obtain 𝑢₁𝑢(₂𝑡)≤max𝐵₅,𝐵₁₂,||𝐵₁₉||,||𝐵₃₄||,𝑢₃(𝑡)≤max𝐵₃,𝐵₄,𝐵₂₃,𝐵₃₃,(𝑡)≤max𝐵₈,𝐵₁₇,𝐵₂₅,𝐵₂₉.(3.71) Obviously, 𝐵_𝑖(𝑖=1,2,…,34) are independent of 𝜆∈(0,1). Similar to the proof of Theorem 2.1 of [17], we can easily find a sufficiently large 𝑀>0 so that we denote the set Ω=𝑢(𝑡)=𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡)^𝑇∈𝑥∶‖𝑢‖<𝑀,𝑢𝑡^+𝑘∈Ω,𝑘=1,2,…,𝑞.(3.72) It is clear that Ω satisfies the requirement (a) in Lemma 3.1.
When(𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡))^𝑇∈𝜕Ω∩Ker𝐿=𝜕Ω∩𝑅³ and 𝑢={(𝑢₁,𝑢₂,𝑢₃)^𝑇} is a constant vector in 𝑅³ with ‖𝑢‖=‖(𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡))^𝑇‖=𝑀, then we have𝑄𝑁𝑢=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎₁+𝑏₁𝑒^{𝑢2−𝑢1}+1𝜔^𝑞_𝑘=1ln1−𝛾_1𝑘𝑎₂−𝑏₂−𝑐𝑒^𝑢2𝛽₁𝑒^𝑢3+1𝜔^𝑞_𝑘=1ln1−𝛾_2𝑘𝑑−𝑒𝑒^𝑢3−𝛽₂𝑒^𝑢2+1𝜔^𝑞_𝑘=1ln1−𝛾_3𝑘⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,0,…,0⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠≠0.(3.73) Letting 𝐽∶Im𝑄→Ker𝐿,(𝑟,0,…,0,0)→𝑟 and, by direct calculation, we get deg=𝐽s𝑄ig𝑁nde𝑢t₁⎛⎜⎜⎜⎜⎝,−𝑢₂,𝑢₃^𝑇;𝜕Ωker𝐿;0𝑏₁𝑒0^{𝑢2−𝑢1}𝑏−₁𝑒^{𝑢2−𝑢1}0𝑐𝑒^𝑢2−𝛽₁𝑒^𝑢30−𝛽₂𝑒^𝑢2−𝑒𝑒^𝑢3=sign⎞⎟⎟⎟⎟⎠𝑏₁𝛽₁𝛽₂−𝑏₁𝑐𝑒𝑒^{2𝑢2−𝑢1+𝑢3}≠0.(3.74) This proves that condition (b) in Lemma 3.1 is satisfied. By now, we have proved that Ω verifies all requirements of Lemma 3.1, then it follows that 𝐿𝑢=𝑁𝑢 has at least one solution (𝑢₁(𝑡),𝑢₂(𝑡),𝑢₃(𝑡))^𝑇 in Dom𝐿∩Ω; that is, to say, (3.5) has at least one 𝜔 periodic solution in Dom𝐿∩Ω. Then we know that ((𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇=(𝑒^𝑢1(𝑡),𝑒𝑥^𝑢2(𝑡),𝑒^𝑢3(𝑡))^𝑇 is an 𝜔 periodic solution of system (2.4) with strictly positive components. This completes the proof.

4. Uniqueness and Global Attractivity of Periodic Solutions

Under the hypotheses (H₁),(H₂),(H₃), we consider the following ordinary differential equation without impulsive:𝑧̇₁(𝑡)=𝑧₁(𝑡)−𝑎₁(𝑡)+𝑏₁(𝑡)∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)𝑧̇₂∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧₁(𝑡),(𝑡)𝑧̇₃=𝑧₂(𝑡)⎡⎢⎢⎣𝑎₂(𝑡)−𝑏₂(𝑡)−𝑐(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)−𝛽₁(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧₃(𝑡)⎤⎥⎥⎦,(𝑡)=𝑧₃(𝑡)⎡⎢⎢⎣𝑑(𝑡)−𝑒(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧₃(𝑡)−𝛽₂(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)⎤⎥⎥⎦,(4.1) with the initial conditions 𝑧_𝑖(0)>0,𝑖=1,2,3.

The following lemmas will be helpful in the proofs of our results. The proof of the following Lemma 4.1 is similar to that of Theorem 1 in [18], and it will be omitted.

Lemma 4.1. Assume that (H₁),(H₂),(H₃) hold, then one has the following.(i)If 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))^𝑇 is a solution of (4.1) on [0,+∞), then 𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}(1−𝛾_𝑖𝑘)𝑧_𝑖(𝑡)(i=1,2,3) is a solution of (2.4) on [0,+∞).(ii)If 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇 is a solution of (2.4) on [0,+∞), then 𝑧_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}(1−𝛾_𝑖𝑘)⁻¹𝑥_𝑖(𝑡)(𝑖=1,2,3) is a solution of (4.1) on [0,+∞).

Lemma 4.2. Let 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡))^𝑇denote any positive solution of system (4.1) with initial conditions 𝑧_𝑖(0)>0,𝑖=1,2,3. Assume that the following condition holds, H₇𝑎^𝑀2>𝑏^𝐿2,𝑑^𝑀>𝑒^𝐿.(4.2) Then there exists a 𝑇₃>0 such that 0<𝑧_𝑖(𝑡)≤𝑀_𝑖,(𝑖=1,2,3),for𝑡≥𝑇₃,(4.3) where 𝑀₁>𝑀^∗1=∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂𝑀₂𝑎^𝐿1∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘,>𝑀^∗2=𝑎^𝑀2−𝑏^𝐿2𝑀₃𝑐^𝐿∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘,>𝑀^∗3=𝑑^𝑀−𝑒^𝐿𝑒^𝐿∏_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘.(4.4)

Proof. From the second equation of (4.1), we can obtain 𝑧̇₂(𝑡)≤≤𝑧𝑧₂₂(𝑡)⎡⎢⎢⎣𝑎₂(𝑡)−𝑏₂(𝑡)−𝑐(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)⎤⎥⎥⎦(𝑡)⎡⎢⎢⎣𝑎^𝑀2−𝑏^𝐿2−𝑐^𝐿₀_{<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)⎤⎥⎥⎦.(4.5) By (4.5), we can derive the following.(A₁)If 𝑧₂(0)≤𝑀₂, then 𝑧₂(𝑡)≤𝑀₂,𝑡≥0.(A₂)If 𝑧₂(0)>𝑀₂, let −𝛼₁=𝑀₂[𝑎^𝑀2−𝑏^𝐿2−𝑐^𝐿∏_{0<𝑡𝑘<𝑡}(1−𝛾_2𝑘)𝑀₂],(𝛼₁>0). Then there exists 𝜀₁>0 such that 𝑡∈[0,𝜀₁), then 𝑧₂(𝑡)>𝑀₂, and also we have 𝑧̇₂(𝑡)<−𝛼₁<0.(4.6) From what has been discussed above, we can easily conclude that, if 𝑧₂(0)>𝑀₂, then 𝑧₂(𝑡) is strictly monotone decreasing with speed at least 𝛼₁. Therefore, there exists a 𝑇₁>0 such that 𝑡>𝑇₁, then 𝑧₂(𝑡)≤𝑀₂.
From the third equation of (4.1), we can obtain𝑧̇₃(𝑡)≤≤𝑧𝑧₃₃(𝑡)⎡⎢⎢⎣𝑑(𝑡)−𝑒(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧₃(𝑡)⎤⎥⎥⎦(𝑡)⎡⎢⎢⎣𝑑^𝑀−𝑒^𝐿₀_{<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧₃(𝑡)⎤⎥⎥⎦.(4.7) By (4.7), we can derive the following(B₁)If 𝑧₃(0)≤𝑀₃, then 𝑧₃(𝑡)≤𝑀₃,𝑡≥0.(B₂)If 𝑧₃(0)>𝑀₃, let −𝛼₂=𝑀₃[𝑑^𝑀−𝑒^𝐿∏_{0<𝑡𝑘<𝑡}(1−𝛾_3𝑘)𝑀₃],(𝛼₂>0). Then there exists 𝜀₂>0 such that 𝑡∈[0,𝜀₂), then 𝑧₃(𝑡)>𝑀₃, and also we have 𝑧̇₃(𝑡)<−𝛼₂<0.(4.8) From what has been discussed above, we can easily conclude that, if 𝑧₃(0)>𝑀₃, then 𝑧₃(𝑡) is strictly monotone decreasing with speed at least 𝛼₂. Therefore, there exists a 𝑇₂>0 such that 𝑡>𝑇₂, then 𝑧₃(𝑡)≤𝑀₃.
From the first equation of (4.1), we can obtain𝑧̇₁(𝑡)≤𝑧₁(𝑡)−𝑎₁(𝑡)+𝑏₁(𝑡)∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂=−𝑎₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧₁(𝑡)(𝑡)𝑧₁(𝑡)+∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂≤−𝑎_𝐿1∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧₁(𝑡)+∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘.(4.9) Then we have 𝑧₁(𝑡)≤𝑀₁,for𝑡≥𝑇₁.(4.10) Set 𝑇₃=max{𝑇₁,𝑇₂}, then we have 0<𝑧_𝑖(𝑡)≤𝑀_𝑖,(𝑖=1,2,3),for𝑡≥𝑇₃.(4.11) The proof is complete.

Lemma 4.3. Let (H₁),(H₂),(H₃) hold. Assume that the following condition holds. H₈𝑎^𝐿2>𝑏^𝑀2+𝛽^𝑀1_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘,𝑑^𝐿>𝑒^𝑀−𝛽^𝑀2_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘.(4.12) Then there exists positive constants 𝑇>0 and 𝑚_𝑖(𝑖=1,2,3) such that, for all 𝑡>𝑇, 𝑚_𝑖<𝑧_𝑖(𝑡),(𝑖=1,2,3),for𝑡≥𝑇,(4.13) in which 𝑚₁<𝑚^∗1=𝑏^𝐿1∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₂𝑚₂𝑎^𝑀1∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘,<𝑚^∗2=𝑎^𝐿2−𝑏^𝑀2−𝛽^𝑀1∏_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑚₃𝑐^𝑀∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘,<𝑚^∗3=𝑑^𝐿−𝑒^𝑀−𝛽^𝑀2∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂𝑒^𝑀∏_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘.(4.14)

Proof. By the second equation of (4.1), It is easy to obtain that, for 𝑡≥𝑇₃, 𝑧̇₂(𝑡)≥𝑧₂(𝑡)⎡⎢⎢⎣𝑎^𝐿2−𝑏^𝑀2−𝑐^𝑀₀_{<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₂(𝑡)−𝛽^𝑀1₀_{<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑀₃⎤⎥⎥⎦,(4.15) where 𝑇₃ is defined in Lemma 4.1. (C₁)If 𝑧₂(𝑇₃)≥𝑚₂, then 𝑧₂(𝑡)≥𝑚₂,𝑡≥𝑇₃. (C₂)If 𝑧₂(𝑇₃)<𝑚₂ and let 𝜇₁=𝑧₂𝑇₃⎡⎢⎢⎣𝑎^𝐿2−𝑏^𝑀2−𝑐^𝑀₀_{<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₂⎤⎥⎥⎦,(4.16) then there exists 𝜀₃>0 such that 𝑡∈[𝑇₃,𝑇₃+𝜀₃), then 𝑧₂(𝑡)>𝑚₂, and also we have 𝑧̇₂(𝑡)>𝜇₁>0.(4.17) Then we know that if 𝑧₂(𝑇₃)<𝑚₂, 𝑧₂(𝑡) will strictly monotonically increase with speed 𝜇₂. Thus, there exists 𝑇₄>𝑇₃ such that if 𝑡≥𝑇₄, then 𝑧₂(𝑡)≥𝑚₂.
By the third equation of (4.1), It is easy to obtain that for 𝑡≥𝑇₃,𝑧̇₃(𝑡)≥𝑧₃(𝑡)⎡⎢⎢⎣𝑑^𝐿−𝑒^𝑀₀_{<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧₃(𝑡)−𝛽^𝑀2₀_{<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂⎤⎥⎥⎦,(4.18) where 𝑇₃ is defined in Lemma 4.2. (D₁)If 𝑧₂(𝑇₃)≥𝑚₃, then 𝑧₃(𝑡)≥𝑚₃,𝑡≥𝑇₃.(D₂)If 𝑧₂(𝑇₃)<𝑚₃, and let 𝜇₂=𝑧₃𝑇₃⎡⎢⎢⎣𝑑^𝐿−𝑒^𝑀₀_{<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑚₃−𝛽^𝑀2₀_{<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑀₂⎤⎥⎥⎦,(4.19) then there exists 𝜀₄>0 such that 𝑡∈[𝑇₃,𝑇₃+𝜀₄), then 𝑧₃(𝑡)>𝑚₃, and also we have 𝑧̇₃(𝑡)>𝜇₂>0.(4.20) Then we know that if 𝑧₃(𝑇₃)<𝑚₃, 𝑧₃(𝑡) will strictly monotonically increase with speed 𝜇₂. Thus, there exists 𝑇₅>𝑇₃ such that, if 𝑡≥𝑇₅, then 𝑧₃(𝑡)≥𝑚₃.
Finally, by the third equation of (4.1), we obtain𝑧̇₁(𝑡)≥𝑧₁(𝑡)−𝑎^𝑀1+𝑏^𝐿1∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₂=−𝑎_𝑀1∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧₁(𝑡)𝑧₁(𝑡)+𝑏^𝐿1∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₂∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧₁(𝑡).(4.21) Thus, we have 𝑧₁(𝑡)≥𝑚₁,(4.22) for 𝑡≥𝑇₄. Set 𝑇=max{𝑇₄,𝑇₅}, then we have 𝑧_𝑖(𝑡)>𝑚_𝑖,(𝑖=1,2,3),for𝑡≥𝑇.(4.23) In the sequel, we formulate the uniqueness and global attractivity of the 𝜔 periodic solution 𝑥^∗(𝑡) in Theorem 4.4. It is immediate that if 𝑥^∗(𝑡) is global attractivity, then 𝑥^∗(𝑡) is in fact unique.

Theorem 4.4. In addition to (H₁)–(H₈), assume further (H₉)lim_𝑡→∞inf𝐵_𝑖(𝑡)>0, where 𝐵₁(𝑡)=𝑐(𝑡)−𝛽₂(𝑡)−𝑏^𝑀𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘_{0<𝑡𝑘<𝑡}𝐵₂1−𝛾_2𝑘,(𝑡)=𝑒(𝑡)−𝛽₁(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘.(4.24) Then system (2.4) has a unique positive 𝜔 periodic solution 𝑥^∗(𝑡)=(𝑥^∗1(𝑡),𝑥^∗2(𝑡),𝑥^∗3(𝑡))^𝑇 which is global attractivity.

Proof. According to the conclusion of Theorem 3.2, we only need to show that the positive periodic solution of (2.4) is global asymptotical stable. Let 𝑥^∗(𝑡)=(𝑥^∗1(𝑡),𝑥^∗2(𝑡),𝑥^∗3(𝑡))^𝑇 be a positive 𝜔 periodic solution of system (2.4) let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡))^𝑇 be any positive solution of system (2.4). Then 𝛾𝑧^∗(𝑡)=(𝑧^∗1(𝑡),𝑧^∗2(𝑡),𝑧^∗3(𝑡))^𝑇,(𝑧^∗1(𝑡)=∏_{0<𝑡𝑘<𝑡}(1−𝛾_1𝑘)𝑥^∗1(𝑡),𝑧^∗2(𝑡)=∏_{0<𝑡𝑘<𝑡}(1−𝛾_2𝑘)𝑥^∗2(𝑡),𝑧^∗3(𝑡)=∏_{0<𝑡𝑘<𝑡}(1−_3𝑘)𝑥^∗3(𝑡)) is the positive 𝜔 periodic solution of (4.1), and 𝑧(𝑡) is the positive solution of (4.1). It follows from Lemma 4.2 and 4.3 that there exists positive constants 𝑇>0,𝑀_𝑖 and 𝑚_𝑖 (defined by Lemmas 4.2 and 4.3, resp.) such that, for all 𝑡>𝑇, 𝑚_𝑖<𝑧^∗𝑖(𝑡)≤𝑀_𝑖,𝑚_𝑖<𝑧_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,3.(4.25) Define 𝑉(𝑡)=||ln𝑧^∗1(𝑡)−ln𝑧₁(𝑡)||+||ln𝑧^∗2(𝑡)−ln𝑧₂(𝑡)||+||ln𝑧^∗3(𝑡)−ln𝑧₃(𝑡)||.(4.26) Calculating the upper-right derivative of 𝑉(𝑡) along the solution of (4.1), it follows for 𝑡≥𝑇that 𝐷⁺𝑉(𝑡)=³_𝑖=1𝑧^∗𝑖(𝑡)𝑧^∗𝑖(𝑡)−𝑧^𝑖(𝑡)𝑧_𝑖(𝑡)sgn𝑧^∗𝑖=sgn𝑧_∗1(𝑡)−𝑧_𝑖(𝑡)(𝑡)−𝑧₁(𝑡)𝑏₁(𝑡)∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧^∗2(𝑡)𝑧^∗1(𝑡)−𝑧²(𝑡)+sgn𝑧_∗2𝑧₁(𝑡)(𝑡)−𝑧₂(𝑡)⎡⎢⎢⎣−−𝑐𝛽(₁𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧^∗2(𝑡)−𝑧₂(𝑡)+sgn𝑧_∗3(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧^∗3(𝑡)−𝑧₃(𝑡)⎤⎥⎥⎦(𝑡)−𝑧₃(𝑡)⎡⎢⎢⎣−−𝑒𝛽(₂𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘𝑧^∗3(𝑡)−𝑧₃(𝑡)≤−𝑐(𝑡)_{0<𝑡𝑘<𝑡}(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧^∗2(𝑡)−𝑧₂(𝑡)⎤⎥⎥⎦−𝑒(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||+𝛽₁(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘||𝑧^∗3(𝑡)−𝑧₃(𝑡)||1−𝛾_3𝑘||𝑧^∗3(𝑡)−𝑧₃(𝑡)||+𝛽₂(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||+𝐷₁(𝑡),(4.27) where 𝐷₁(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑏¹(𝑡)∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧^∗2(𝑡)𝑧^∗1(𝑡)−𝑧²(𝑡)𝑧₁(𝑡),𝑧^∗1𝑏(𝑡)>𝑧₁(𝑡),₁(𝑡)∏^{0<𝑡𝑘<𝑡}1−𝛾_2𝑘∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧^∗2(𝑡)𝑧^∗1(𝑡)−𝑧²(𝑡)𝑧₁(𝑡),𝑧^∗1(𝑡)<𝑧₁(𝑡).(4.28) In the sequel, we will estimate 𝐷₁(𝑡) under the following two cases.(i)If 𝑧^∗1(𝑡)≥𝑧₁(𝑡), then𝐷₁(𝑡)≤𝑏¹(𝑡)∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧^∗1(𝑡)∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧^∗2≤𝑏(𝑡)−𝑧₂(𝑡)_𝑀∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||.(4.29)(ii)If 𝑧^∗1(𝑡)<𝑧₁(𝑡), then𝐷₁(𝑡)≤𝑏¹(𝑡)∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑧₁(𝑡)∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘𝑧²≤𝑏(𝑡)−𝑧^∗2(𝑡)_𝑀∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||.(4.30) Combining the conclusions of (4.29) and (4.30), we obtain 𝐷₁(𝑡)≤𝑏^𝑀∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||.(4.31) It follows from (4.31) that 𝐷⁺𝑉(𝑡)≤−−𝑒𝑐(𝑡()𝑡)_{0<𝑡𝑘<𝑡0<𝑡𝑘<𝑡}1−𝛾_2𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)||+𝛽₁(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘||𝑧^∗3(𝑡)−𝑧₃(𝑡)||+𝑏1−𝛾_3𝑘||𝑧^∗3(𝑡)−𝑧₃(𝑡)||+𝛽₂(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)_𝑀||∏_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘||𝑧^∗2=𝑏(𝑡)−𝑧₂(𝑡)_𝑀||𝑚₁∏_{0<𝑡𝑘<𝑡}1−𝛾_1𝑘−𝑐(𝑡)+𝛽²+𝛽(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_2𝑘||𝑧^∗2(𝑡)−𝑧₂(𝑡)₁||≤−𝐵(₁𝑡)−𝑒(𝑡)_{0<𝑡𝑘<𝑡}1−𝛾_3𝑘||𝑧^∗3(𝑡)−𝑧₃(𝑡)||(𝑡)||𝑧^∗2(𝑡)−𝑧₂(𝑡)||+𝐵₂(𝑡)||𝑧^∗3(𝑡)−𝑧₃(𝑡)||,(4.32) where 𝐵₁(𝑡) and 𝐵₂(𝑡) are defined in Theorem 4.4. By hypothesis (H₈), there exist constants 𝛼_𝑖,(𝑖=2,3) and 𝑇^∗>𝑇 such that 𝐵_𝑖(𝑡)≥𝛼_𝑖>0,(𝑖=2,3),for𝑡≥𝑇^∗.(4.33) Integrating both sides of (4.32) on interval [𝑇^∗,𝑡] yields 𝑉(𝑡)+³_𝑖=2^𝑡𝑇_∗𝐵_𝑖(𝑡)||𝑧^∗𝑖(𝑡)−𝑧_𝑖(𝑡)||𝑑𝑠≤𝑉𝑇^∗.(4.34) It follows from (4.33) and (4.34) that ³_𝑖=2^𝑡𝑇_∗𝐵_𝑖(𝑡)||𝑧^∗𝑖(𝑡)−𝑧_𝑖(𝑡)||𝑑𝑠≤𝑉𝑇^∗<∞,for𝑡≥𝑇^∗.(4.35) Since 𝑧^∗𝑖(𝑡) and 𝑧_𝑖(𝑡) (𝑖=2,3) are bounded for 𝑡≥𝑇^∗, so |𝑧^∗𝑖(𝑡)−𝑧_𝑖(𝑡)|(𝑖=2,3) are uniformly continuous on [𝑇^∗,∞). By Barbalat’s Lemma [32], we have lim_𝑡→∞||𝑧^∗𝑖(𝑡)−𝑧_𝑖(𝑡)||=lim_𝑡→∞⎡⎢⎢⎣_{0<𝑡𝑘<𝑡}1−𝛾_𝑖𝑘⁻¹||𝑥^∗𝑖(𝑡)−𝑥_𝑖(𝑡)||⎤⎥⎥⎦=0,(𝑖=2,3).(4.36) Thus, lim_𝑡→∞||𝑥^∗𝑖(𝑡)−𝑥_𝑖(𝑡)||=0,(𝑖=2,3).(4.37) By (4.37) and the first equation of (2.4), one can easily obtain that lim_𝑡→∞||𝑥^∗1(𝑡)−𝑥₁(𝑡)||=0.(4.38) By Theorems 7.4 and 8.2 in [33], we know that the positive periodic solution 𝑥^∗(𝑡)=(𝑥^∗1(𝑡),𝑥^∗2(𝑡),𝑥^∗3(𝑡))^𝑇 of (2.4) is uniformly asymptotically stable. The proof of Theorem 4.4 is complete.

5. An Example

As an application of our main results, we consider the following system:𝑥̇₂𝑥̇₁(𝑡)=−2𝑥₁(𝑡)+𝑥₂(𝑡),𝑡≠𝑡_𝑘,(𝑡)=−(4+c1os𝑡)𝑥₂(𝑡)−(2+cos𝑡)𝑥₂(𝑡)−(1−sin𝑡)𝑥²²(𝑡)2𝑒^200𝜋+1𝑥̇₃+sin𝑡𝑥₂(𝑡)𝑥₃(𝑡),𝑡≠𝑡_𝑘,(𝑡)=𝑥₃(𝑡)50+sin𝑡−50𝑒^200𝜋+1+sin𝑡𝑥₃(𝑡)−49𝑒⁵⁰3𝜋−cos𝑡𝑥²Δ𝑥₁(𝑡),𝑡≠𝑡_𝑘,Δ𝑥₂𝑡_𝑘=𝑥₁𝑡^+𝑘−𝑥₁𝑡^−𝑘=−21𝑥¹𝑡_𝑘,𝑘=1,2,…,Δ𝑥₃𝑡_𝑘=𝑥₂𝑡^+𝑘−𝑥₂𝑡^−𝑘=−31𝑥²𝑡_𝑘,𝑘=1,2,…,𝑡_𝑘=𝑥₃𝑡^+𝑘−𝑥₃𝑡^−𝑘=−41𝑥³𝑡_𝑘,𝑘=1,2,…,(5.1) in which 𝑡_𝑘+2=𝑡_𝑘+2𝜋, [0,2𝜋]∩{𝑡_𝑘}={𝑡₁,𝑡₂}, 𝑎₁(𝑡)=2, 𝑏₁(𝑡)=1, 𝑎₂(𝑡)=4+cos𝑡, 𝑏₂(𝑡)=2+cos𝑡, 𝛽₁(𝑡)=(1/2𝑒^200𝜋+1)+sin𝑡, 𝛽₂(𝑡)=(49/𝑒⁵⁰3𝜋)−cos𝑡, 𝑐(𝑡)=1−sin𝑡, 𝑑(𝑡)=50+sin𝑡, and 𝑒(𝑡)=50𝑒^200𝜋+1+sin𝑡. By direct computation, we can obtain 𝑎₁=2,𝑎₂=4,𝑐=1,|𝑎₂−𝑏₂|=2,𝑑=50,𝛽₁=12𝑒^200𝜋+1,𝛽₂=49𝑒⁵⁰3𝜋,𝐵𝑒=50𝑒^200𝜋+1,𝑏₁=1,₁=ln4𝜋+2ln(2/3)𝐵2𝜋+16𝜋+2ln23≐50.07,₂=ln4𝜋−2ln(1/2)𝐵2𝜋−16𝜋−2ln23≐−47.07,₆=𝐵₂₇=ln100𝜋−2ln(3/4)𝐵2𝜋50𝑒^200𝜋+200𝜋+2ln34≐−0.5754,₂₆=ln2𝜋+2ln(2/3)𝜋/𝑒^200𝜋+1𝐵≐629.55,₂₈=𝐵₂₆−200𝜋−2ln34≐2.9362.(5.2) Then 𝐵₄≐50.07,𝐵₂₉≐2.9362. It is easy to check that (5.1) satisfies all the conditions of Theorems 3.2 and 4.4; hence, (5.1) has a positive 2𝜋 periodic solution which is global attractivity.

Acknowledgments

The authors would like to thank the referees for their helpful comments and valuable suggestions regarding this paper. This work is supported by the National Natural Science Foundation of China (no. 10771215), the Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560), the Doctoral Foundation of Guizhou College of Finance and Economics (2010) and the Science and technology Program of Hunan Province (no. 2010FJ6021).