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

Authors: Changjin Xu; Daxue Chen
Journal: International Journal of Differential Equations
 (2011)

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## 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.


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## Body

## 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:(1.1)ẋ1(t)=-a1(t)x1(t)+b1(t)x2(t),ẋ2(t)=a2(t)x1(t)-b2(t)x2(t)-c(t)x22(t)-β1(t)x2(t)x2(t)x3(t),ẋ3(t)=x3(t)[d(t)-e(t)x3(t)-β2(t)x2(t)],
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; ai(t),bi(t),βi(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:(1.2)ẋ1(t)=-a1(t)x1(t)+b1(t)x2(t),t≠tk,ẋ2(t)=a2(t)x2(t)-b2(t)x2(t)-c(t)x22(t)-β1(t)x2(t)x3(t),t≠tk,ẋ3(t)=x3(t)[d(t)-e(t)x3(t)-β2(t)x2(t)],t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=-γikxi(tk),i=1,2,3,k=1,2,…,q,
where Δxi(tk)=xi(tk+)-xi(tk-) are the impulses at moments tk and t1<t2<⋯ is a strictly increasing sequence such that limk→∞tk=+∞;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)
ai(t),bi(t),βi(t)(i=1,2),c(t),d(t),e(t) are all continuous ω periodic; that is, ai(t+ω)=ai(t),
bi(t+ω)=bi(t),
βi(t+ω)=βi(t)(i=1,2),
c(t+ω)=c(t),
d(t+ω)=d(t),
e(t+ω)=e(t) for any t∈R.(H2)
ai(t),bi(t),βi(t)(i=1,2),c(t),d(t),e(t) are all positive.(H3)
0<γik<1,i=1,2,3 for all k∈N, and there exists a positive integer q such that tk+q=tk+ω,γi(k+q)=γik,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 Section2, 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 system(2.1)ẋ(t)=f(t,x),t≠tk,k=1,2,…,Δx(t)∣t=tk=Ik(x(tk-)),
where x∈Rn,f:R×Rn→Rn is continuous and f(t+ω,x)=f(t,x);Ik:Rn→Rn are continuous, and there exists a positive integer q such that tk+q=tk+ω,Ik+q(x)=Ik(x) with tk∈R,tk+1>tk,limk→∞=∞,Δx(t)∣t=tk=x(tk+)-x(tk-). For tk≠0(k=1,2,…),[0,ω]∩{tk}={t1,t2,…,tq}. As we know, {tk} are called points of jump.Let us recall some definitions. For the Canchy problem,(2.2)ẋ(t)=f(t,x),t∈[0,ω],t≠tk,Δx(t)∣t=tk=Ik(x(tk-)),x(0)=x0.Definition 2.1.
A mapx:[0,ω]→Rn 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 tk∩[0,ω], and at each discontinuity point it is continuous on the left;(ii)
x(t) satisfies (2.2).Definition 2.2.
A mapx:[0,ω]→Rn 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, ifx(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 (2.3)x*(t)={x(t-jω),t∈[jω,(j+1)ω]∖{tk},x*(t)isleftcontinuousatt=tk
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:(2.4)ẋ1(t)=-a1(t)x1(t)+b1(t)x2(t),t≠tk,t∈[0,ω],k=1,2,…,q,ẋ2(t)=a2(t)x2(t)-b2(t)x2(t)-c(t)x22(t)-β1(t)x2(t)x3(t),t≠tk,t∈[0,ω],k=1,2,…,q,ẋ3(t)=x3(t)[d(t)-e(t)x3(t)-β2(t)x2(t)],t≠tk,t∈[0,ω],k=1,2,…,q,Δxi(tk)=xi(tk+)-xi(tk-)=-γikxi(tk),i=1,2,3,xi(0)=xi(ω),k=1,2,…,q.


## 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.LetX,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 KP. 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 KP(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]).
LetL be a Fredholm mapping of index zero, and let N be L-compact on Ω¯. Suppose(a)
for eachλ∈(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:(3.1)f¯=1ω∫0ωf(t)dt,fL=min⁡t∈[0,ω]f(t),fM=max⁡t∈[0,ω]f(t),|f|¯=1ω∫0ω|f(t)|dt,
where f(t) is a ω continuous periodic function. For any nonnegative integer p, let C(p)[0,ω;t1,t2,…,tq]={x:[0,ω]→Rm∣x(p)(t) exist for t≠t1,…,tq;x(p)(t+0), and let x(p)(t-0) exist at t1,t2,…,tq, and x(j)(tk)=x(j)(tk-0),k=1,…,m,j=0,1,2,…,p} with the norm ∥x∥p=max{supt∈[0,ω]∥x(j)(t)∥}j=1p, where ∥·∥ is any norm of Rm. It is easy to see that C(p)[0,ω;t1,t2,…,tq] 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:
(3.2)(H4)min⁡{P1,P2,P3}>0,(H5)a¯1ω>∑k=1qln⁡(1-γ1k),(H6)d¯ω+∑k=1qln⁡(1-γ3k)>β¯2ωeB4,β¯1β¯2≠c¯e¯,
where
(3.3)P1=|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-c¯ωeB6,P2=|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-β¯1ωeB29,P3=|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-c¯ω,
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 Section2, 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
(3.4)u1(t)=ln⁡[x1(t)],u2(t)=ln⁡[x2(t)],u3(t)=ln⁡[x3(t)],
then system (2.4) can be translated to
(3.5)u̇1(t)=-a1(t)+b1(t)e(u2(t)-u1(t)),t≠tk,t∈[0,ω],k=1,2,…,q,u̇2(t)=a2(t)-b2(t)-c(t)eu2(t)-β1(t)eu3(t),t≠tk,t∈[0,ω],k=1,2,…,q,u̇3(t)=d(t)-e(t)eu3(t)-β2(t)eu2(t),t≠tk,t∈[0,ω],k=1,2,…,q,Δui(tk)=ln⁡(1-γik),i=1,2,3,ui(0)=ui(ω).
It is easy to see that if system (3.5) has one ω periodic solution (u1*(t),u2*(t),u3*(t))T, then (x1*(t),x2*(t),x3*(t))T=(eu1*(t),eu2*(t),eu3*(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 take(3.6)X={u∈C[0,ω;t1,t2,…,tq]},Y=X×R3×(q+1).
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∈R3q.
Let the following hold:(3.7)dom⁡L={x=(u1,u2,u3)T∈C[0,ω];t1,t2,…,tq},L:Dom⁡L⊂X⟶Y,x⟶(x′,Δx(tk)k=1q),N:X⟶Y,Nu=((-a1(t)+b1(t)exp⁡(u2(t)-u1(t))a2(t)-b2(t)-c(t)eu2(t)-β1(t)eu3(t)d(t)-e(t)eu3(t)-β2(t)eu2(t)),(ln⁡(1-γ11)ln⁡(1-γ21)ln⁡(1-γ31)),(ln⁡(1-γ12)ln⁡(1-γ22)ln⁡(1-γ32)),…,(ln⁡(1-γ1q)ln⁡(1-γ2q)ln⁡(1-γ3q)),0(-a1(t)+b1(t)exp⁡(u2(t)-u1(t))a2(t)-b2(t)-c(t)eu2(t)-β1(t)eu3(t)d(t)-e(t)eu3(t)-β2(t)eu2(t)))
Obviously,
(3.8)Ker⁡L={u:u(t)=h∈R3,t∈[0,ω]},Im⁡L={z=(f,a1,a2,…,aq,d)∈Y:∫0ωf(s)ds+∑k=1qak+d=0}=X×R3×q×{0},dimKerL=3=codimImL.
So, ImL is closed in Y; L is a Fredholm mapping of index zero. Define two projectors
(3.9)Px=1ω∫0ωx(t)dt,Qz=Q(f,a1,a2,…,aq,d)=(1ω[∫0ωf(s)ds+∑k=1qak+d,],0,0,…,0).
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 inverseKP of L, KP:ImL→KerP∩DomL has the following form:(3.10)KP(z)=∫0tf(s)ds+∑tk<tak-1ω∫0ω∫0tf(s)dsdt-∑k=1qak+1ω∑k=1qaktk.
Moreover, it is easy to check that
(3.11)QNu=((1ω∫0tF1(s)ds+1ω∑k=1qln⁡(1-γ1k)1ω∫0tF2(s)ds+1ω∑k=1qln⁡(1-γ2k)1ω∫0tF3(s)ds+1ω∑k=1qln⁡(1-γ3k)),0,0,…,0),KP(I-Q)Nu=(∫0tF1(s)ds+∑t>tkln⁡(1-γ1k)∫0tF2(s)ds+∑t>tkln⁡(1-γ2k)∫0tF3(s)ds+∑t>tkln⁡(1-γ3k))-(1ω∫0ω∫0tF1(s)dsdt-∑k=1qln⁡(1-γ1k)+1ω∑k=1qln⁡(1-γ1k)1ω∫0ω∫0tF2(s)dsdt-∑k=1qln⁡(1-γ2k)+1ω∑k=1qln⁡(1-γ2k)1ω∫0ω∫0tF3(s)dsdt-∑k=1qln⁡(1-γ3k)+1ω∑k=1qln⁡(1-γ3k))-(tω-12)(∫0ωF1(s)ds+∑k=1qln⁡(1-γ1k)∫0ωF2(s)ds+∑k=1qln⁡(1-γ2k)∫0ωF3(s)ds+∑k=1qln⁡(1-γ3k)),
where
(3.12)F1(s)=-a1(s)+b1(s)e(u2(s)-u1(s)),F2(s)=a2(s)-b2(s)-c(s)eu2(s)-β1(s)eu3(s),F2(s)=d(s)-e(s)eu3(s)-β2(s)eu2(s).
Obviously, QN and KP(I-Q)N are continuous. Using the Ascoli-Arzela theorem, it is not difficult to show that KP(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 subsetΩ for the application of the continuation theorem. Corresponding to the operator equation Lu=λNu,λ∈(0,1), we have(3.13)u̇1(t)=λ[-a1(t)+b1(t)e(u2(t)-u1(t))],t≠tk,t∈[0,ω],k=1,2,…,q,u̇2(t)=λ[a2(t)-b2(t)-c(t)eu2(t)-β1(t)eu3(t)],t≠tk,t∈[0,ω],k=1,2,…,q,u̇3(t)=λ[d1(t)-e(t)eu3(t)-β2(t)eu2(t))],t≠tk,t∈[0,ω],k=1,2,…,q,Δui(tk)=λln⁡(1-γik),i=1,2,3,ui(0)=ui(ω).
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
(3.14)∫0ω[b1(t)e(u2(t)-u1(t))]dt+∑k=1qln⁡(1-γ1k)=∫0ωa1(t)dt,∫0ω[c(t)eu2(t)+β1(t)eu3(t)]dt=∫0ω(a2(t)-b2(t))dt+∑k=1qln⁡(1-γ2k),∫0ω[e(t)eu3(t)+β2(t)eu2(t)]dt=∫0ωd(t)dt+∑k=1qln⁡(1-γ3k).
From (3.13) and (3.14), we obtain
(3.15)∫0ω|u̇1(t)|dt<∫0ωa1(t)dt+∫0ω[b1(t)e(u2(t)-u1(t))]dt=2a¯1ω-∑k=1qln⁡(1-γ1k),(3.16)∫0ω|u̇2(t)|dt<∫0ω[a2(t)+b2(t)]dt+∫0ω[c(t)eu2(t)+β1(t)eu3(t))]dt=2a¯2ω+∑k=1qln⁡(1-γ2k),(3.17)∫0ω|u̇3(t)|dt<∫0ωd(t)dt+∫0ω[e(t)eu3(t)+β2(t)eu2(t))]dt=2d¯ω+∑k=1qln⁡(1-γ3k).
Let the following hold:
(3.18)ui(ξi)=min⁡t∈[0,ω]ui(t),ui(ηi)=max⁡t∈[0,ω]ui(t),i=1,2,3.
From the second and the third equations of (3.14), we can obtain
(3.19)|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)>∫0ωc(t)eu2(t)dt≥∫0ωc(t)eu2(ξ2)dt,d¯ω+∑k=1qln⁡(1-γ3k)>∫0ωe(t)eu3(t)dt≥∫0ωe(t)eu3(ξ3)dt=e¯ωeu3(ξ3),
then
(3.20)u2(ξ2)<ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)c¯ω],(3.21)u3(ξ3)<ln⁡[d¯ω+∑k=1qln⁡(1-γ3k)e¯ω].
Thus
(3.22)u2(t)=u2(ξ2)+∫ξ2tu̇2(t)dt≤u2(ξ2)+∫0ω|u̇2(t)|dt<ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)c¯ω]+2a¯2ω+∑k=1qln⁡(1-γ2k)=:B1.
In the following, we will consider four cases.
Case 1 (ifu1(t)>0,u2(t)>0).
From the first equation of (3.14), we have(3.23)a̅1ω<∫0ωb1(t)eu2(t)dt+∑k=1qln⁡(1-γ1k)≤b̅1ωeu2(η2)+∑k=1qln⁡(1-γ3k),a̅1ω>∫0ωb1(t)e-u1(t)dt+∑k=1qln⁡(1-γ1k)≥∫0ωb1(t)e-u1(η1)dt+∑k=1qln⁡(1-γ1k),
that is,
(3.24)u2(η2)>ln⁡[a¯1ω-∑k=1qln⁡(1-γ1k)b¯1ω],u1(η1)>ln⁡[b¯1ωa¯1ω-∑k=1qln⁡(1-γ1k)].
Then
(3.25)u2(t)=u2(η2)-∫tη2u̇2(t)dt≥u2(η2)-∫0ω|u̇2(t)|dt≥ln⁡[a¯1ω-∑k=1qln⁡(1-γ1k)b¯1ω]-2a¯2ω-∑k=1qln⁡(1-γ2k)=:B2,(3.26)u1(t)=u1(η1)-∫tη1u̇1(t)dt≥u1(η1)-∫0ω|u̇1(t)|dt≥ln⁡[b¯1ωa¯1ω-∑k=1qln⁡(1-γ1k)]-2a¯1ω-∑k=1qln⁡(1-γ1k)=:B3.
Thus, from (3.22) and (3.25), we obtain
(3.27)|u2(t)|≤max⁡⁡{|B1|,|B2|}=:B4.
By the first  and the third equations of (3.14), we get
(3.28)∫0ω[b1(t)eB4-u1(ξ1)]dt+∑k=1qln⁡(1-γ1k)>a¯1ω,∫0ω[e(t)eu3(η3)+β2(t)eB4]dt>d¯ω+∑k=1qln⁡(1-γ3k),
then
(3.29)u1(ξ1)<ln⁡[b¯1ωeB4a¯1ω-∑k=1qln⁡(1-γ1k)],(3.30)u3(η3)>ln⁡[d¯ω+∑k=1qln⁡(1-γ3k)-β2ωeB4e¯ω].
From (3.15), (3.17), (3.21), and (3.30), we have
(3.31)u1(t)=u1(ξ1)+∫ξ1tu̇1(t)dt≤u1(ξ1)+∫0ω|u̇1(t)|dt<ln⁡[b¯1ωeB4a¯1ω-∑k=1qln⁡(1-γ1k)]+2a¯1ω+∑k=1qln⁡(1-γ1k)=:B5,(3.32)u3(t)=u3(ξ3)+∫ξ3tu̇3(t)dt≤u3(ξ3)+∫0ω|u̇3(t)|dt<ln⁡[d¯ω+∑k=1qln⁡(1-γ3k)e¯ω]+2d¯ω+∑k=1qln⁡(1-γ3k)=:B6,(3.33)u3(t)=u3(η3)-∫tη3u̇3(t)dt≥u3(η3)-∫0ω|u̇3(t)|dt>ln⁡[d¯ω+∑k=1qln⁡(1-γ3k)-β̅2ωeB4e¯ω]-2d¯ω+∑k=1qln⁡(1-γ3k)=:B7.
Thus,
(3.34)|u1(t)|≤B5,|u3(t)|max⁡⁡{|B6|,|B7|}=:B8.Case 2 (ifu1(t)>0,u2(t)<0).
By the first equation of (3.14), we have(3.35)a¯1ω<∫0ωb1(t)eu2(t)dt+∑k=1qln⁡(1-γ1k)≤b¯1ωeu2(η2)+∑k=1qln⁡(1-γ3k),
namely,
(3.36)u2(η2)>ln⁡[a¯1ω-∑k=1qln⁡(1-γ1k)b¯1ω].
Then
(3.37)u2(t)=u2(η2)-∫tη2u̇2(t)dt≥u2(η2)-∫0ω|u̇2(t)|dt≥ln⁡[a¯1ω-∑k=1qln⁡(1-γ1k)b¯1]-2a¯2ω-∑k=1qln⁡(1-γ2k)=:B2.
From (3.22) and (3.37), we obtain
(3.38)|u2(t)|≤max⁡{|B1|,|B2|}=:B3.
By the first equation of (3.14), we also have
(3.39)∫0ωb1(t)eB3eu1(t)dt+∑k=1qln⁡(1-γ1k)>a¯1ω,∫0ωb1(t)e-B3eu1(t)dt+∑k=1qln⁡(1-γ1k)<a¯1ω,
Then
(3.40)b¯1ωeB3eu1(ξ1)+∑k=1qln⁡(1-γ1k)>a¯1ω,b¯1ωe-B3eu1(η1)+∑k=1qln⁡(1-γ1k)<a¯1ω.
that is,
(3.41)u1(ξ1)<ln⁡[b¯1ωeB3a¯1ω-∑k=1qln⁡(1-γ1k)]=:B8,u1(η1)<ln⁡[b¯1ωeB3a¯1ω-∑k=1qln⁡(1-γ1k)]=:B9.
Thus,
(3.42)u1(t)=u1(ξ1)+∫ξ1tu̇1(t)dt≤u1(ξ1)+∫0ω|u̇1(t)|dt<B8+2a¯1ω-∑k=1qln⁡(1-γ1k)=:B10,u1(t)=u1(η1)+∫tη1u̇1(t)dt≥u1(η1)+∫0ω|u̇1(t)|dt>B9-2a¯1ω-∑k=1qln⁡(1-γ1k)=:B11.
From (3.42), we have
(3.43)|u1(t)≤max⁡{|B10|,|B11|}=:B12.
By the second equation of (3.14), we have
(3.44)∫0ωc(t)eB3dt+∫0ωβ1(t)eu3(t)dt>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k),∫0ωc(t)e-B3dt+∫0ωβ1(t)eu3(t)dt<|a2-b2|¯ω+∑k=1qln⁡(1-γ2k).
Then
(3.45)c¯ωeB3+β¯1ωeu3(η3)>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k),c¯ωe-B3+β¯1ωeu3(ξ3)<|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)=:B13,
that is,
(3.46)u3(η3)>ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-c¯ωeB6β¯1ω],u3(ξ3)<ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-c¯ωe-B3β¯1ω]=:B14.
Therefore, we get
(3.47)u3(t)=u3(ξ3)+∫ξ3tu̇3(t)dt≤u3(ξ3)+∫0ω|u̇3(t)|dt<B14+2d¯1ω-∑k=1qln⁡(1-γ3k)=:B15,u3(t)=u3(η3)-∫tη3u̇3(t)dt≥u3(η3)-∫0ω|u̇3(t)|dt>B14-2d¯1ω-∑k=1qln⁡(1-γ3k)=:B16.
Hence, we have
(3.48)|u3(t)|≤max⁡{|B15|,|B16|}=:B17.Case 3 (ifu1(t)<0,u2(t)>0).
By the first equation of (3.14), we have(3.49)∫0ωb1(t)e-u1(t)dt+∑k=1qln⁡(1-γ1k)<∫0ωa1(t)dt,∫0ω[b1(t)e(u2(η2)-u1(t))]dt+∑k=1qln⁡(1-γ1k)>∫0ωa1(t)dt.
Then
(3.50)b¯1ωe-u1(η1)+∑k=1qln⁡(1-γ1k)<a¯1ω,b¯1ωe-B19eu2(η2)+∑k=1qln⁡(1-γ1k)>a¯1ω.
namely,
(3.51)u1(η1)>ln⁡[b¯1ωa¯1-∑k=1qln⁡(1-γ1k)]=:B18,u2(η2)>ln⁡[a¯1ω-∑k=1qln⁡(1-γ1k)b¯1ωe-B19]=:B20.
Therefore,
(3.52)u1(t)=u1(η1)-∫tη1u̇1(t)dt≥u1(η1)-∫0ω|u̇1(t)|dt>B18-2a¯1ω-∑k=1qln⁡(1-γ1k)=:B19,u2(t)=u2(ξ2)+∫ξ2tu̇2(t)dt≤u2(ξ2)+∫0ω|u̇2(t)|dt<ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)c¯ω]+2a¯2ω+∑k=1qln⁡(1-γ2k)=:B21,u2(t)=u2(η2)-∫tη2u̇2(t)dt≥u2(η1)-∫0ω|u̇2(t)|dt>B20-2a¯2ω-∑k=1qln⁡(1-γ2k)=:B22.
So
(3.53)B19<u1(t)<0,|u2(t)|≤max⁡{|B21|,|B22|}=:B23.
By the third equation of (3.14), we obtain
(3.54)∫0ωe(t)eu3(η3)dt+∫0ωβ2(t)eB23dt>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k),
that is,
(3.55)u3(η3)>ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-β¯2ωeB23e¯ω]=:B24.
Thus,
(3.56)|u3(t)|≤max⁡{|B23|,|B24|}=:B25.Case 4 (ifu1(t)<0,u2(t)<0).
By the second equation of (3.14), we have(3.57)∫0ωc(t)dt+∫0ωβ1(t)eu3(t)dt>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k).
Then
(3.58)c¯ω+β¯1ωeu3(η3)>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k),
that is,
(3.59)u3(η3)≥ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-c¯ωβ¯1ω]=:B26.
Therefore,
(3.60)u3(t)=u3(ξ3)+∫ξ3tu̇3(t)dt≤u3(ξ3)+∫0ω|u̇3(t)|dt<ln⁡[d¯ω+∑k=1qln⁡(1-γ3k)e¯ω]+2d¯ω+∑k=1qln⁡(1-γ3k)=:B27,u3(t)=u3(η3)-∫tη3u̇3(t)dt≥u3(η3)-∫0ω|u̇3(t)|dt>B26-2d¯ω-∑k=1qln⁡(1-γ3k)=:B28.
Thus,
(3.61)|u3(t)|≤max⁡{|B27|,|B28|}=:B29.
By the second equation of (3.14), we obtain
(3.62)∫0ωc(t)eu2(η2)dt+∫0ωβ1(t)eB29dt>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k),
that is,
(3.63)c¯eu2(η2)+β¯1ωeB29>|a2-b2|¯ω+∑k=1qln⁡(1-γ2k).
Thus,
(3.64)u2(η2)>ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)-β¯1ωeB29c¯ω].
Then, from (3.16) and (3.20), we get
(3.65)u2(t)=u2(ξ2)+∫ξ2tu̇2(t)dt≤u2(ξ2)+∫0ω|u̇2(t)|dt<ln⁡[|a2-b2|¯ω+∑k=1qln⁡(1-γ2k)c¯ω]+2a¯2ω+∑k=1qln⁡(1-γ2k)=:B31,u2(t)=u2(η2)-∫tη2u̇2(t)dt≥u2(η2)-∫0ω|u̇2(t)|dt>B30-2a¯2ωω-∑k=1qln⁡(1-γ2k)=:B32.
Thus,
(3.66)|u2(t)|≤max⁡{|B31|,|B32|}=:B33.
By the first equation of (3.14), we have
(3.67)∫0ωb1(t)eu2(t)-u1(η1)dt+∑k=1qln⁡(1-γ1k)<∫0ωa1(t)dt.
Then
(3.68)b¯1ωe-B33e-u1(η1)+∑k=1qln⁡(1-γ1k)<a¯1ω.
Thus,
(3.69)u1(η1)>ln⁡[b¯1ωe-B32a¯1ω-∑k=1qln⁡(1-γ1k)]=:B34.
Hence, we have
(3.70)B34<u1(t)<0.
Based on the discussion above, we can easily obtain
(3.71)u1(t)≤max⁡{B5,B12,|B19|,|B34|},u2(t)≤max⁡{B3,B4,B23,B33},u3(t)≤max⁡{B8,B17,B25,B29}.
Obviously, Bi(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
(3.72)Ω={u(t)=(u1(t),u2(t),u3(t))T∈x:‖u‖<M,u(tk+)∈Ω,k=1,2,…,q}.
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 have(3.73)QNu=((a¯1+b¯1eu2-u1+1ω∑k=1qln⁡(1-γ1k)a̅2-b2-c¯eu2β¯1eu3+1ω∑k=1qln⁡(1-γ2k)d¯-e¯eu3-β¯2eu2+1ω∑k=1qln⁡(1-γ3k)),0,…,0)≠0.
Letting J:ImQ→KerL,(r,0,…,0,0)→r and, by direct calculation, we get
(3.74)deg⁡{JQN(u1,u2,u3)T;∂Ω⋂ker⁡L;0}=signdet(-b¯1eu2-u1b¯1eu2-u100-c¯eu2-β¯1eu30-β¯2eu2-e¯eu3)=sign⁡{(b¯1β¯1β¯2-b¯1c¯e¯)e2u2-u1+u3}≠0.
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=(eu1(t),exu2(t),eu3(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:(4.1)ż1(t)=z1(t)[-a1(t)+b1(t)∏0<tk<t(1-γ2k)z2(t)∏0<tk<t(1-γ1k)z1(t)],ż2(t)=z2(t)[a2(t)-b2(t)-c(t)∏0<tk<t(1-γ2k)z2(t)-β1(t)∏0<tk<t(1-γ3k)z3(t)],ż3(t)=z3(t)[d(t)-e(t)∏0<tk<t(1-γ3k)z3(t)-β2(t)∏0<tk<t(1-γ2k)z2(t)],
with the initial conditions zi(0)>0,i=1,2,3.The following lemmas will be helpful in the proofs of our results. The proof of the following Lemma4.1 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)
Ifz(t)=(z1(t),z2(t),z3(t))T is a solution of (4.1) on [0,+∞), then xi(t)=∏0<tk<t(1-γik)zi(t)(i=1,2,3) is a solution of (2.4) on [0,+∞).(ii)
Ifx(t)=(x1(t),x2(t),x3(t))T is a solution of (2.4) on [0,+∞), then zi(t)=∏0<tk<t(1-γik)-1xi(t)(i=1,2,3) is a solution of (4.1) on [0,+∞).Lemma 4.2.
Letz(t)=(z1(t),z2(t),z3(t))Tdenote any positive solution of system (4.1) with initial conditions zi(0)>0,i=1,2,3.  Assume that the following condition holds,
(4.2)(H7)a2M>b2L,dM>eL.
Then there exists a T3>0 such that
(4.3)0<zi(t)≤Mi,(i=1,2,3),fort≥T3,
where
(4.4)M1>M1*=∏0<tk<t(1-γ2k)M2a1L∏0<tk<t(1-γ1k),M2>M2*=a2M-b2LcL∏0<tk<t(1-γ2k),M3>M3*=dM-eLeL∏0<tk<t(1-γ3k).Proof.
From the second equation of (4.1), we can obtain
(4.5)ż2(t)≤z2(t)[a2(t)-b2(t)-c(t)∏0<tk<t(1-γ2k)z2(t)]≤z2(t)[a2M-b2L-cL∏0<tk<t(1-γ2k)z2(t)].
By (4.5), we can derive the following.(A1)
Ifz2(0)≤M2, then z2(t)≤M2,t≥0.(A2)
Ifz2(0)>M2, let -α1=M2[a2M-b2L-cL∏0<tk<t(1-γ2k)M2],(α1>0). Then there exists ɛ1>0 such that t∈[0,ɛ1), then z2(t)>M2, and also we have(4.6)ż2(t)<-α1<0.
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 obtain(4.7)ż3(t)≤z3(t)[d(t)-e(t)∏0<tk<t(1-γ3k)z3(t)]≤z3(t)[dM-eL∏0<tk<t(1-γ3k)z3(t)].
By (4.7), we can derive the following(B1)
Ifz3(0)≤M3, then z3(t)≤M3,t≥0.(B2)
Ifz3(0)>M3, let -α2=M3[dM-eL∏0<tk<t(1-γ3k)M3],(α2>0). Then there exists ɛ2>0 such that t∈[0,ɛ2), then z3(t)>M3, and also we have(4.8)ż3(t)<-α2<0.
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 obtain(4.9)ż1(t)≤z1(t)[-a1(t)+b1(t)∏0<tk<t(1-γ2k)M2∏0<tk<t(1-γ1k)z1(t)]=-a1(t)z1(t)+∏0<tk<t(1-γ2k)M2∏0<tk<t(1-γ1k)≤-a1Lz1(t)+∏0<tk<t(1-γ2k)M2∏0<tk<t(1-γ1k).
Then we have
(4.10)z1(t)≤M1,fort≥T1.
Set T3=max{T1,T2}, then we have
(4.11)0<zi(t)≤Mi,(i=1,2,3),fort≥T3.
The proof is complete.Lemma 4.3.
Let(H1),(H2),(H3) hold. Assume that the following condition holds.
(4.12)(H8)a2L>b2M+β1M∏0<tk<t(1-γ3k),dL>eM-β2M∏0<tk<t(1-γ2k).
Then there exists positive constants T>0 and mi(i=1,2,3) such that, for all t>T,
(4.13)mi<zi(t),(i=1,2,3),fort≥T,
in which
(4.14)m1<m1*=b1L∏0<tk<t(1-γ2k)m2a1M∏0<tk<t(1-γ1k),m2<m2*=a2L-b2M-β1M∏0<tk<t(1-γ3k)cM∏0<tk<t(1-γ2k),m3<m3*=dL-eM-β2M∏0<tk<t(1-γ2k)M2eM∏0<tk<t(1-γ3k).Proof.
By the second equation of (4.1), It is easy to obtain that, for t≥T3,
(4.15)ż2(t)≥z2(t)[a2L-b2M-cM∏0<tk<t(1-γ2k)z2(t)-β1M∏0<tk<t(1-γ3k)M3],
where T3 is defined in Lemma 4.1. (C1)
Ifz2(T3)≥m2, then z2(t)≥m2,t≥T3.(C2)
Ifz2(T3)<m2 and let(4.16)μ1=z2(T3)[a2L-b2M-cM∏0<tk<t(1-γ2k)m2],
then there exists ɛ3>0 such that t∈[T3,T3+ɛ3), then z2(t)>m2, and also we have
(4.17)ż2(t)>μ1>0.
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,(4.18)ż3(t)≥z3(t)[dL-eM∏0<tk<t(1-γ3k)z3(t)-β2M∏0<tk<t(1-γ2k)M2],
where T3 is defined in Lemma 4.2. (D1)
Ifz2(T3)≥m3, then z3(t)≥m3,t≥T3.(D2)
Ifz2(T3)<m3, and let(4.19)μ2=z3(T3)[dL-eM∏0<tk<t(1-γ3k)m3-β2M∏0<tk<t(1-γ2k)M2],
then there exists ɛ4>0 such that t∈[T3,T3+ɛ4), then z3(t)>m3, and also we have
(4.20)ż3(t)>μ2>0.
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 obtain(4.21)ż1(t)≥z1(t)[-a1M+b1L∏0<tk<t(1-γ2k)m2∏0<tk<t(1-γ1k)z1(t)]=-a1Mz1(t)+b1L∏0<tk<t(1-γ2k)m2∏0<tk<t(1-γ1k)z1(t).
Thus, we have
(4.22)z1(t)≥m1,
for t≥T4. Set T=max{T4,T5}, then we have
(4.23)zi(t)>mi,(i=1,2,3),fort≥T.
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)limt→∞infBi(t)>0, where
(4.24)B1(t)=[c(t)-β2(t)-bMm1∏0<tk<t(1-γ1k)]∏0<tk<t(1-γ2k),B2(t)=[e(t)-β1(t)]∏0<tk<t(1-γ3k).
Then system (2.4) has a unique positive ω periodic solution x*(t)=(x1*(t),x2*(t),x3*(t))T which is global attractivity.Proof.
According to the conclusion of Theorem3.2, we only need to show that the positive periodic solution of (2.4) is global asymptotical stable. Let x*(t)=(x1*(t),x2*(t),x3*(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 z*(t)=(z1*(t),z2*(t),z3*(t))T,(z1*(t)=∏0<tk<t(1-γ1k)x1*(t),z2*(t)=∏0<tk<t(1-γ2k)x2*(t),z3*(t)=∏0<tk<t(1-γ3k)x3*(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,Mi and mi (defined by Lemmas 4.2 and 4.3, resp.) such that, for all t>T,
(4.25)mi<zi*(t)≤Mi,mi<zi(t)≤Mi,i=1,2,3.
Define
(4.26)V(t)=|ln⁡z1*(t)-ln⁡z1(t)|+|ln⁡z2*(t)-ln⁡z2(t)|+|ln⁡z3*(t)-ln⁡z3(t)|.
Calculating the upper-right derivative of V(t) along the solution of (4.1), it follows for t≥Tthat
(4.27)D+V(t)=∑i=13(zi*′(t)zi*(t)-zi′(t)zi(t))sgn⁡(zi*(t)-zi(t))=sgn⁡(z1*(t)-z1(t))[b1(t)∏0<tk<t(1-γ2k)∏0<tk<t(1-γ1k)(z2*(t)z1*(t)-z2(t)z1(t))]+sgn⁡(z2*(t)-z2(t))[-c(t)∏0<tk<t(1-γ2k)(z2*(t)-z2(t))-β1(t)∏0<tk<t(1-γ3k)(z3*(t)-z3(t))]+sgn⁡(z3*(t)-z3(t))[-e(t)∏0<tk<t(1-γ3k)(z3*(t)-z3(t))-β2(t)∏0<tk<t(1-γ2k)(z2*(t)-z2(t))]≤-c(t)∏0<tk<t(1-γ2k)|z2*(t)-z2(t)|+β1(t)∏0<tk<t(1-γ3k)|z3*(t)-z3(t)|-e(t)∏0<tk<t(1-γ3k)|z3*(t)-z3(t)|+β2(t)∏0<tk<t(1-γ2k)|z2*(t)-z2(t)|+D1(t),
where
(4.28)D1(t)={b1(t)∏0<tk<t(1-γ2k)∏0<tk<t(1-γ1k)(z2*(t)z1*(t)-z2(t)z1(t)),z1*(t)>z1(t),b1(t)∏0<tk<t(1-γ2k)∏0<tk<t(1-γ1k)(z2*(t)z1*(t)-z2(t)z1(t)),z1*(t)<z1(t).
In the sequel, we will estimate D1(t) under the following two cases.(i)
Ifz1*(t)≥z1(t), then(4.29)D1(t)≤b1(t)∏0<tk<t(1-γ2k)z1*(t)∏0<tk<t(1-γ1k)(z2*(t)-z2(t))≤bM∏0<tk<t(1-γ2k)m1∏0<tk<t(1-γ1k)|z2*(t)-z2(t)|.(ii)
Ifz1*(t)<z1(t), then(4.30)D1(t)≤b1(t)∏0<tk<t(1-γ2k)z1(t)∏0<tk<t(1-γ1k)(z2(t)-z2*(t))≤bM∏0<tk<t(1-γ2k)m1∏0<tk<t(1-γ1k)|z2*(t)-z2(t)|.
Combining the conclusions of (4.29) and (4.30), we obtain
(4.31)D1(t)≤bM∏0<tk<t(1-γ2k)m1∏0<tk<t(1-γ1k)|z2*(t)-z2(t)|.
It follows from (4.31) that
(4.32)D+V(t)≤-c(t)∏0<tk<t(1-γ2k)|z2*(t)-z2(t)|+β1(t)∏0<tk<t(1-γ3k)|z3*(t)-z3(t)|-e(t)∏0<tk<t(1-γ3k)|z3*(t)-z3(t)|+β2(t)∏0<tk<t(1-γ2k)|z2*(t)-z2(t)|+bM∏0<tk<t(1-γ2k)m1∏0<tk<t(1-γ1k)|z2*(t)-z2(t)|=[bMm1∏0<tk<t(1-γ1k)-c(t)+β2(t)]∏0<tk<t(1-γ2k)|z2*(t)-z2(t)|+[β1(t)-e(t)]∏0<tk<t(1-γ3k)|z3*(t)-z3(t)|≤-(B1(t)|z2*(t)-z2(t)|+B2(t)|z3*(t)-z3(t)|),
where B1(t) and B2(t) are defined in Theorem 4.4. By hypothesis (H8), there exist constants αi,(i=2,3) and T*>T such that
(4.33)Bi(t)≥αi>0,(i=2,3),fort≥T*.
Integrating both sides of (4.32) on interval [T*,t] yields
(4.34)V(t)+∑i=23∫T*tBi(t)|zi*(t)-zi(t)|ds≤V(T*).
It follows from (4.33) and (4.34) that
(4.35)∑i=23∫T*tBi(t)|zi*(t)-zi(t)|ds≤V(T*)<∞,fort≥T*.
Since zi*(t) and zi(t) (i=2,3) are bounded for t≥T*, so |zi*(t)-zi(t)|(i=2,3) are uniformly continuous on [T*,∞). By Barbalat’s Lemma [32], we have
(4.36)lim⁡t→∞|zi*(t)-zi(t)|=lim⁡t→∞[∏0<tk<t(1-γik)-1|xi*(t)-xi(t)|]=0,(i=2,3).
Thus,
(4.37)lim⁡t→∞|xi*(t)-xi(t)|=0,(i=2,3).
By (4.37) and the first equation of (2.4), one can easily obtain that
(4.38)lim⁡t→∞|x1*(t)-x1(t)|=0.
By Theorems 7.4 and 8.2 in [33], we know that the positive periodic solution x*(t)=(x1*(t),x2*(t),x3*(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:(5.1)ẋ1(t)=-2x1(t)+x2(t),t≠tk,ẋ2(t)=(4+cos⁡t)x2(t)-(2+cos⁡t)x2(t)-(1-sin⁡t)x22(t)-12e200π+1+sin⁡tx2(t)x3(t),t≠tk,ẋ3(t)=x3(t)[50+sin⁡t-(50e200π+1+sin⁡t)x3(t)-(49e503π-cos⁡t)x2(t)],t≠tk,Δx1(tk)=x1(tk+)-x1(tk-)=-12x1(tk),k=1,2,…,Δx2(tk)=x2(tk+)-x2(tk-)=-13x2(tk),k=1,2,…,Δx3(tk)=x3(tk+)-x3(tk-)=-14x3(tk),k=1,2,…,
in which tk+2=tk+2π,
[0,2π]∩{tk}={t1,t2},
a1(t)=2,
b1(t)=1,
a2(t)=4+cost,
b2(t)=2+cost,
β1(t)=(1/2e200π+1)+sint,
β2(t)=(49/e503π)-cost,
c(t)=1-sint,
d(t)=50+sint, and e(t)=50e200π+1+sint. By direct computation, we can obtain (5.2)a¯1=2,a¯2=4,c¯=1,|a2-b2|¯=2,d¯=50,β¯1=12e200π+1,β¯2=49e503π,e¯=50e200π+1,b¯1=1,B1=ln⁡4π+2ln⁡(2/3)2π+16π+2ln⁡23≐50.07,B2=ln⁡4π-2ln⁡(1/2)2π-16π-2ln⁡23≐-47.07,B6=B27=ln⁡100π-2ln⁡(3/4)2π(50e200π)+200π+2ln⁡34≐-0.5754,B26=ln⁡2π+2ln⁡(2/3)π/e200π+1≐629.55,B28=B26-200π-2ln⁡34≐2.9362.
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.



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*Source: 259805-2011-11-13.xml*