# Periodic Solutions of a Lotka-Volterra System with Delay and Diffusion

Authors: Lin Li; Mingxing Luo; Zhijie Nan; Sihong Shi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Our purpose is to prove the existence of periodic solutions for a competition Lotka-Volterra system on time scales, and one example is given to illustrate our results.


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## Body

## 1. Introduction

Denote𝕋 as an arbitrary nonempty closed subset of the real numbers ℝ. The Lotka-Volterra system is mainly devoted to the study of population dynamics in mathematics. The classical two classes of species can be modeled as
(1.1)x1′(t)=x1(t)(r11(t)+r12(t)x1(t)+r13(t)x2(t)),x2′(t)=x2(t)(r21(t)+r22(t)x1(t)+r23(t)x2(t)),
which are viewed in terms of different situations. For example, it is named with predator-prey system if r13(t)r22(t)<0, while competition system when r13(t)<0, r22(t)<0, also a reciprocal system if r13(t)>0, r22(t)>0. Moreover, in order to reflect the seasonal fluctuations, the Lotka-Volterra system with periodic coefficients is also considered in [1]. The time delay effect, density regulation, and diffusion between patches in many ecological systems have been investigated for its ecological significance in [2–4].The recently interest on ratio-dependent predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, that is, that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Ratio-dependent predator-prey models are favored by many animal ecologists recently involving a searching process.Our concern in this paper is to consider both the periodic variations of the environment and the density regulation of the predators by considering account delay effect and diffusion between patches. The environments of most natural populations undergo temporal variation, causing changes in the growth characteristics of populations. One method of incorporating temporal nonuniformity of the environments in models is to assume that the parameters are periodic with the same period of the time variable. It can be modeled with the following dynamic system:(1.2)x1δ(t)=r1(t)-f1(t)ex1(t)-g1(t)ey1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)+p1(t)(ex2(t)-x1(t)-1),x2δ(t)=r2(t)-f2(t)ex2(t)+p2(t)(ex1(t)-x2(t)-1),y1δ(t)=r3(t)-f3(t)ey1(t)-g2(t)ex1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)-h1(t)ey2(t)ey1(t)+β2(t)ey2(t),y2δ(t)=r4(t)-f4(t)ey2(t)+h2(t)ey1(t-τ)ey1(t-τ)+β2(t)ey2(t-τ),
where y1(t) and x1(t) denote the population density of species y and species x in patch 1, y2(t) and x2(t) represent the density of species y and species x in patch 2. Species x and y can be diffused between two patches and species y is confined to compete with species x. τ>0 is a delay due to gestation. pi(t)>0  (i=1,2) are rd-continuous ω-periodic functions and denote the dispersal rate of species y in the ith patch (i=1,2), respectively. ri(t)>0, fi(t)>0  (i=1,2,3,4), gj(t)>0, hj(t)>0, βj(t)>0  (j=1,2) are rd-continuous ω-periodic functions.With the transition variableXi=exi(t), Y(t)=ey1(t), and y2(t)≡0, the system (1.2) reduces to
(1.3)X1′(t)=X1(t)(r1(t)-f1(t)X1(t)-g1(t)Y(t-τ)X1(t-τ)+β(t)Y(t-τ))-p1(t)X1(t)-p1(t)X2(t),X2′(t)=X2(t)r2(t)-f2(t)X22(t)+p2(t)X1(t)-p2(t)X2(t),Y′(t)=Y(t)r3(t)-f3(t)Y2(t)-g2(t)X1(t-τ)Y(t)X1(t-τ)+β(t)Y(t-τ),
which was introduced by Hilger in [5] who firstly proposed the theory of time scales. There are many related studies of positive solutions for delayed equation [6–8], dynamic equation (1.3) on time scales [5, 9–17], and the uniform persistence, global asymptotic stability, and periodicity of system (1.3); see [18–23].Recently, various continuation theorems in coincidence degree have played an important role in study the existence of periodic solutions of the Lotka-Volterra system (see, e.g., [11, 23–28]). In this paper, by using the well-known Gains and Mawhin’s theorem, we prove the existence of periodic solutions of competition Lotka-Volterra dynamic system (1.2) with time delay and diffusion on time scales.This paper is organized as follows. In Section2, we present some basic definitions and results of topological degree theory. Section 3 is contributed to the proof of the main results while the last section goes to one example.


## 2. Preliminaries

Several definitions and results will be presented in this section. For more details, refer to [9, 12].Letω>0. Throughout this paper, the time scales we considered are always assumed to be ω-periodic (i.e., t∈𝕋=ℤ or ℝ implies t±ω∈𝕋) and unbounded above and below (may be represented by ⋃k∈𝕋[2(k-1)ω,2kω]). We denote ϵ=min𝕋∩(ℝ-ℝ-), 𝕀ω=[ϵ,ϵ+ω]∩𝕋.Definition 2.1.
The forward jump operatorσ:𝕋→𝕋 and the backward jump operator ρ:𝕋→𝕋 are defined by
(2.1)σ(t)∶=inf⁡{s∈T:s≥t},ρ(t)∶=sup⁡{s∈T:s≤t},
respectively, for any t∈𝕋: If σ(t)=t, then t is called right dense (otherwise: right scattered), and if ρ(t)=t, then t is called left dense (otherwise left scattered).Definition 2.2.
Suppose thatf:𝕋→ℝ and fix t∈𝕋κ. Then f is called differential at t∈𝕋κ if there exists a constant c∈ℝ such that, for any given ϵ>0, there is an open neighborhood U of t such that
(2.2)|f(ρ(t))-f(s)-c(ρ(t)-s)|≤ϵ|ρ(t)-s|,s∈U.c is named with the delta (or Hilger) derivative of f at t∈𝕋κ and is denoted by c=fδ(t). Here, [a,b]κ=[a,b] if b is left dense and [a,b]κ=[a,b) if b is left scattered.As far as𝕋=ℤ or ℝ is considered, 𝕋κ=𝕋. We say that f is delta (Hilger) differential on 𝕋 if f(t) exists for all t∈𝕋. A function F:𝕋→ℝ is called an antiderivative of f:𝕋→ℝ provided that Fδ(t)=f(t) for all t∈𝕋. Then we define
(2.3)∫rs‍f(t)δt=F(s)-F(r),r,s∈T.Definition 2.3.
A functionf:𝕋→ℝ is called rd-continuous if it is continuous at right dense points in 𝕋 and its left-sided limits exist (finite) at left dense points in 𝕋. The set of rd-continuous functions f:𝕋→ℝ will be denoted by Crd(𝕋,ℝ).It is easy to see that every rd-continuous function has an antiderivative and every continuous function is rd-continuous.Lemma 2.4 (see [11]).
Ifs,t∈𝕋, α,β∈ℝ and f,g∈Crd(𝕋,ℝ), then  (1)
∫st‍[αf(u)+βg(u)]δu=α∫st‍f(u)δu+β∫st‍g(u)δu;(2)
iff(u)≥0 for all s≤u<t, then ∫st‍f(u)δu≥0;(3)
if|f(u)|≤g(u) on [s,t):={u∈𝕋:s≤u<t}, then |∫st‍f(u)δu|≤∫st‍g(u)δu.Next, we introduce some results related to the topology degree theories which are crucial in our arguments [20].LetX and Y be two Banach spaces. Consider an operator equation:
(2.4)Lx=λNx,λ∈(0,1),
where ℒ:Domℒ∩X→Y is a linear operator, 𝒩:X→Y is continuous, and λ is a parameter. Let P and Q be two projections P:X→X and Q:Y→Y such that ImP=kerℒ and Imℒ=kerQ=Im(I-Q). It is easy to see that ℒ|Domℒ∩kerP:(I-P)X→Imℒ is invertible, and thus we denote the inverse of this map by Φ. If Ω is a bounded open subset of X, the mapping 𝒩 is called ℒ-compact on Ω¯ if Q∘𝒩(Ω¯) is bounded and Φ∘(I-Q)∘𝒩:Ω¯→X is compact. Since ImQ is isomorphic to kerℒ, there exists an isomorphism Ψ:ImQ→kerℒ.Note that operatorℒ is called a Fredholm operator of index zero if dim(kerℒ)=codim(Imℒ)<∞ and Imℒ is closed in Y.Lemma 2.5 (Gains and Mawhin’s theorem [20]).
Letℒ be a Fredholm mapping of index zero, and let 𝒩 be ℒ-compact on Ω¯. Suppose that (C1)
for eachλ∈(0,1), every solution x∈∂Ω∩Domℒ of ℒx=λ𝒩(x,λ) is such that x∉∂Ω;(C2)
Q∘𝒩x≠0 for each x∈∂Ω∩kerℒ;(C3)
deg(Ψ∘Q∘𝒩,Ω∩kerℒ,0)≠0. Then equation ℒx=𝒩x has at least one solution lying in Domℒ∩Ω¯.Throughout this paper, we take the following notations for convenience, and all the other notations are defined analogously:(2.5)f¯=1ω∫Iω‍f(t)δt,fs=min⁡t∈Iωf(t),fM=max⁡t∈Iωf(t),
where f∈Crd(𝕋,ℝ) is an ω-periodic function.


## 3. Periodic Solution

The main result is stated as follows about the existence ofω-periodic solutions.Theorem 3.1.
Suppose that(1)
(r1β1-g1)s>0;(2)
r¯1β1s-g1M>0;(3)
(r3β2-g2β2-h1)s>0;(4)
r¯3β2s-g2Mβ2s-h1M>0. Then the dynamic system (1.2) has at least one ω-periodic solution.Before proving Theorem3.1, we first give some useful lemmas.Lemma 3.2.
Supposeλ∈(0,1) is a parameter, (r1β1-g1)s>0, and (r3β2-g2β2-h1)s>0. If (x1(t),x2(t),y1(t),y2(t))T is an ω-periodic solution of the system (1.2), then |x1(t)|+|x2(t)|≤2C1 and |y1(t)|+|y2(t)|≤2C2, where
(3.1)C1∶=max⁡{|(ln⁡r1f1)M|,|(ln⁡r2f2)M|,|(ln⁡r2f2)s|,|(ln⁡r1β1-g1β1f1)s|},(3.2)C2∶=max⁡{|(ln⁡r3f3)M|,|(ln⁡r4+h2f4)M|,|(ln⁡r4f4)s|,|(ln⁡r3β2-g2β2-h1β2f3)s|}.Proof.
Corresponding to the operator equation (2.4), we have
(3.3)x1δ(t)=λ[r1(t)-f1(t)ex1(t)-g1(t)ey1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)+p1(t)(ex2(t)-x1(t)-1)],x2δ(t)=λ[r2(t)-f2(t)ex2(t)+p2(t)(ex1(t)-x2(t)-1)],y1δ(t)=λ[r3(t)-f3(t)ey1(t)-g2(t)ex1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)-h1(t)ey2(t)ey1(t)+β2(t)ey2(t)],y2δ(t)=λ[r4(t)-f4(t)ey2(t)+h2(t)ey1(t-τ)ey1(t-τ)+β2(t)ey2(t-τ)].
Define
(3.4)Γ={u=(x1(t),x2(t),y1(t),y2(t))T∈C(T,R4):xi(t+ω)=xi(t),yi(t+ω)=yi(t)},
with the norm
(3.5)‖u‖=∑i=12‍max⁡t∈Iω|xi(t)|+∑i=12‍max⁡t∈Iω|yi(t)|.
Then Γ is a Banach space. Take X=Y=Γ. Assume that u=(x1(t),x2(t),y1(t),y2(t))T∈Γ is a solution of the system (3.3) for λ∈(0,1). It only needs to be proven that there exists a M1>0 such that ∥u∥<M1.
In fact, sinceu∈Γ, there exists ti∈𝕀ω  (i=1,2,3,4) such that xi(ti)=maxt∈𝕀ωxi(t) and yi(ti+2)=maxt∈𝕀ωyi(t)  (i=1,2). Thus xiδ(ti)=yiδ(ti+2)=0, for i=1,2. Consequently, it follows from the system (3.3) that(3.6a)r1(t1)-f1(t1)ex1(t1)-p1(t1)+p1(t1)ex2(t1)-x1(t1)-g1(t1)ey1(t1-τ)ex1(t1-τ)+β1(t1)ey1(t1-τ)=0,(3.6b)r2(t2)-f2(t2)ex2(t2)-p2(t2)+p2(t2)ex1(t2)-x2(t2)=0,(3.6c)r3(t3)-f3(t3)ey1(t3)-g2(t3)ex1(t3-τ)ex1(t3-τ)+β1(t3)ey1(t3-τ)-h1(t3)ey2(t3)ey1(t3)+β2(t3)ey2(t3)=0,(3.6d)r4(t4)-f4(t4)ey2(t4)+h2(t4)ey1(t4-τ)ey1(t4-τ)+β2(t4)ey2(t4-τ)=0.
Whenx1(t1)≥x2(t2), then x1(t1)≥x2(t1). From (3.6a) we get
(3.7)f1(t1)ex1(t1)=r1(t1)-p1(t1)+p1(t1)ex2(t1)-x1(t1)-g1(t1)ey1(t1-τ)ex1(t1-τ)+β1(t1)ey1(t1-τ)≤r1(t1).
It follows that
(3.8)x2(t2)≤x1(t1)≤ln⁡r1(t1)f1(t1)≤(ln⁡r1f1)M.
When x1(t1)<x2(t2), then x1(t2)<x2(t2). From (3.6b) we obtain
(3.9)f2(t2)ex2(t2)=r2(t2)-p2(t2)+p2(t2)ex1(t2)-x2(t2)≤r2(t2),
which means
(3.10)x1(t1)≤x2(t2)≤ln⁡r2(t2)f2(t2)≤(ln⁡r2f2)M.
As far as (3.6c) and (3.6d) are concerned, with analogue arguments above, we get
(3.11)y1(t3)≤r3(t3)f3(t3)≤(ln⁡r3f3)M,y2(t4)≤ln⁡r4(t4)+h2(t4)f4(t4)≤(ln⁡r4+h2f4)M.
Now chooseκi∈𝕀ω  (i=1,2) such that x1(κ1)=mint∈𝕀ωx1(t), x2(κ2)=mint∈𝕀ωx2(t), then xiδ(κi)=0. Thus we obtain that(3.12a)r1(κ1)-f1(κ1)ex1(κ1)-p1(κ1)+p1(κ1)ex2(κ1)-x1(κ1)-g1(κ1)ey1(κ1-τ)ex1(κ1-τ)+β1(κ1)ey1(κ1-τ)=0,(3.12b)r2(κ2)-f2(κ2)ex2(κ2)-p2(κ2)+p2(κ2)ex1(κ2)-x2(κ2)=0.
Whenx1(κ1)<x2(κ2), then x1(κ1)<x2(κ2)≤x2(κ1). From (3.12a), we have
(3.13)f1(κ1)ex1(κ1)=r1(κ1)-p1(κ1)+p1(κ1)ex2(κ1)-x1(κ1)-g1(κ1)ey1(κ1-τ)ex1(κ1-τ)+β1(κ1)ey1(κ1-τ)≥r1(κ1)-g1(κ1)β1(κ1),
and thus
(3.14)x2(κ2)>x1(κ1)≥ln⁡r1(κ1)β(κ1)-g1(κ1)f1(κ1)β1(κ1)≥(ln⁡r1β1-g1β1f1)s,
according to the hypothesis (r1β1-g1)s>0.
Whenx1(κ1)≥x2(κ2), then x1(κ2)≥x1(κ1)≥x2(κ2). From (3.12b), we get
(3.15)f2(κ2)ex2(κ2)=r2(κ2)-p2(κ2)+p2(κ2)ex1(κ2)-x2(κ2)≥r2(κ2),
which yields
(3.16)x1(κ1)≥x2(κ2)≥ln⁡r2(κ2)f2(κ2)≥(ln⁡r2f2)s.
Combing the inequalities (3.8) and (3.10) with (3.14) and (3.16), from (3.1) it easily gets that
(3.17)|x1(t)|+|x2(t)|≤2C1.
On the other hand, chooseκi+2∈𝕀ω  (i=1,2) such that y1(κ3)=mint∈𝕀ωy1(t), y2(κ4)=mint∈𝕀ωy2(t), we have yiδ(κi+2)=0 and then(3.18a)r3(κ3)-f3(κ3)ey1(κ3)-g2(κ3)ex1(κ3-τ)ex1(κ3-τ)+β1(κ3)ey1(κ3-τ)-h1(κ3)ey2(κ3)ey1(κ3)+β2(κ3)ey2(κ3)=0,(3.18b)r4(κ4)-f4(κ4)ey2(κ4)+h2(κ4)ey1(κ4-τ)ey1(κ4-τ)+β2(κ4)ey2(κ4-τ)=0.
Wheny1(κ3)<y2(κ4), then y1(κ3)<y2(κ4)≤y2(κ3). From (3.18a), we get
(3.19)f3(κ3)ey1(κ3)=r3(κ3)-g2(κ3)ex1(κ3-τ)ex1(κ3-τ)+β1(κ3)ey1(κ3-τ)-h1(κ3)ey2(κ3)ey1(κ3)+β2(κ3)ey2(κ3)≥r3(κ3)-g2(κ3)-h1(κ3)β2(κ3),
which means
(3.20)y2(κ4)>y1(κ3)≥ln⁡r3(κ3)β2(κ3)-g2(κ3)β2(κ3)-h1(κ3)β2(κ3)f3(κ3)≥(ln⁡r3β2-g2β2-h1β2f3)s,
under the hypothesis that (r3f3-g2f3-h1)s>0.
Wheny1(κ3)≥y2(κ4), then y1(κ4)≥y1(κ3)≥y2(κ4). From (3.18b), we have
(3.21)f4(κ4)ey2(κ4)=r4(κ4)+h2(κ4)ey1(κ4-τ)ey1(κ4-τ)+β2(κ4)ey2(κ4-τ)≥r4(κ4),
which implies
(3.22)y1(κ3)≥y2(κ4)≥ln⁡r4(κ4)f4(κ4)≥(ln⁡r4f4)s.
Combing the inequalities (3.11) with (3.20) and (3.22), from (3.2) it easily follows that
(3.23)|y1(n)|+|y2(n)|≤2C2.
SetM1=2C1+2C2+1, then ∥u∥<M1. M1 is independent on λ∈(0,1).Lemma 3.3.
Supposeμ∈(0,1) is a parameter, r¯1β1s-g1M>0 and r¯3β2s-g2Mβ2s-h1M>0. Then any solution v=(v1,v2,v3,v4)T of the algebraic system
(3.24)0=r¯1-f¯1ev1+μ(-p¯1+p¯1ev2-v1)-μω∫Iω‍g1(t)ev3ev1+β1(t)ev3δt,0=r¯2-f¯2ev2+μ(-p¯2+p¯2ev1-v2),0=r¯3-f¯3ev3-μω∫Iω‍g2(t)ev1ev1+β1(t)ev3δt-μω∫Iω‍h1(t)ev4ev3+β2(t)ev4δt,0=r¯4-f¯4ev4+μω∫Iω‍h2(t)ev3ev3+β2(t)ev4δt
satisfies ∥v∥≤2C3+2C4, where
(3.25)C3∶=max⁡{|ln⁡r¯1β1s-g1Mf¯1β1s|,|ln⁡r¯1f¯1|,|ln⁡r¯2f¯2|},(3.26)C4∶=max⁡{|ln⁡r¯3β2s-g2Mβ2s-h1Mf¯3β2s|,|ln⁡r¯3f¯3|,|ln⁡r¯4f¯4|,|ln⁡r¯4+h2Mf¯4|}.Proof.
Whenv2≤v1, from the first two equations of (3.24) and Lemma 2.4, we obtain that
(3.27)f¯1ev1=r¯1+μ(-p¯1+p¯1ev2-v1)-μev3ω∫Iω‍g1(t)ev1+β1(t)ev3δt≤r¯1,f¯2ev2=r¯2+μ(-p¯2+p¯2ev1-v2)≥r¯2,
which implies that
(3.28)ln⁡r¯2f¯2≤v2≤v1≤ln⁡r¯1f¯1.
Analogously, when v1<v2, we have
(3.29)f¯1ev1=r¯1+μ(-p¯1+p¯1ev2-v1)-μev3ω∫Iω‍g1(t)ev1+β1(t)ev3δt≥r¯1-g1Mβ1s,f¯2ev2=r¯2+μ(-p¯2+p¯2ev1-v2)≤r¯2;
it follows that
(3.30)ln⁡r¯1β1s-g1Mf¯1β1s≤v1<v2≤ln⁡r¯2f¯2,
by the assumption that r¯1β1s-g1M>0.
Hence, from (3.25) we have |v1|+|v2|≤2M3.
On the other hand, with similar discussion above, from the last two of (3.24), we obtain
(3.31)ln⁡r¯3β2s-g2Mβ2s-h1Mf¯3β2s≤v3≤ln⁡r¯3f¯3,ln⁡r¯4f¯4≤v4≤ln⁡r¯4+h2Mf¯4,
which imply that |v3|+|v4|≤2C4 from (3.26). So, ∥v∥≤2C3+2C4.With the preparations above, we can complete the proof of Theorem3.1 as follows.Proof of Theorem3.1.
SetM2=2C3+2C4+1. By Lemma 3.3, we know that any solution v of the system (3.24) satisfies ∥u∥<M2. Take C=M1+M2, and define Ω={u∈Γ:∥u∥<C}. Due to Lemmas 3.2 and 3.3, condition (C1) in Lemma 2.5 is satisfied.
Let(3.32)L:Dom⁡L∩Γ⟶Γ,Lu(t)=(x1δ(t)x2δ(t)y1δ(t)y2δ(t)),
where Domℒ={(x1(t),x2(t),y1(t),y2(t))T∈C(𝕋,ℝ4)} and
(3.33)N:Γ⟶Γ,Nu(t)=(N1(t)N2(t)N3(t)N4(t)),
where
(3.34)N1(t)=r1(t)-f1(t)ex1(t)-g1(t)ey1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)-p1(t)+p1(t)ex2(t)-x1(t),N2(t)=r2(t)-f2(t)ex2(t)-p2(t)+p2(t)ex1(t)-x2(t),N3(t)=r3(t)-f3(t)ey1(t)-g2(t)ex1(t-τ)ex1(t-τ)+β1(t)ey1(t-τ)-h1(t)ey2(t)ey1(t)+β2(t)ey2(t),N4(t)=r4(t)-f4(t)ey2(t)+h2(t)ey1(t-τ)ey1(t-τ)+β2(t)ey2(t-τ).
With the definitions above, we obtain thatℒu=𝒩u for u∈Domℒ∩Γ with Imℒ={u∈Γ:∫𝕀ω‍xi(t)δt=0,∫𝕀ω‍yi(t)δt=0,t∈𝕋,i=1,2} and kerℒ=ℝ4 which is closed in Γ, and dim(kerℒ)=codim(Imℒ)=4. Therefore, ℒ is a Fredholm mapping of index zero. Moreover, define two projections P,Q such that ImP=kerℒ and Imℒ=kerQ=Im(I-Q), where
(3.35)P=Q:Γ⟶Γ,Pu=Qu=(1ω∫Iωx1(t)δt‍1ω∫Iωx2(t)δt‍1ω∫Iωy1(s)δs‍1ω∫Iωy2(s)δs‍).
ThenΓ=kerℒ⊕kerP=kerℒ⊕kerQ, and choose Ψ as the identity isomorphism of ImQ to kerP. Furthermore, the generalized inverse (to ℒ) exists and is given by
(3.36)Φ:Im⁡L⟶Dom⁡L∩ker⁡P,Φu=(∫ηsx1(t)δt‍-1ω∫Iω‍∫ηtx1(s)δsδt‍∫ηsx2(t)δt‍-1ω∫Iω‍∫ηtx2(s)δsδt‍∫ηty1(s)δs‍-1ω∫Iω‍∑ηty1(s)δs‍∫ηty2(s)δs‍-1ω∫Iω‍∑ηty2(s)δs‍).
Thus
(3.37)Q∘Nu=(1ω∫Iω‍N1(t)δt,1ω∫Iω‍N2(t)δt,1ω∫Iω‍N3(t)δt,1ω∫Iω‍N4(t)δt)T.
Clearly, Q∘𝒩 and Φ∘(I-Q) are well defined. By the Lebesgue convergence theorem and the Arzela-Ascoli theorem, Φ∘(I-Q)(Ω¯) is relatively compact for any open-bounded set Ω⊂Γ. Moreover, Q∘𝒩(Ω¯) is bounded. Therefore, 𝒩 is ℒ-compact on Ω¯ for any open-bounded set Ω⊂Γ. When u∈∂Ω∩ℝ4 is a constant vector in ℝ4, then Q∘𝒩u≠0 since Q∘𝒩u=0 is the system (3.24) with ϵ=1. Condition (C2) in Lemma 2.5 is also satisfied.
Finally, we claim thatdeg(Ψ∘Q∘𝒩,Ω,O)≠0, where O∶=(0,0,0,0)T. In fact, consider the homotopy
(3.38)Hμv=μQ∘Nv+(1-μ)Gv,μ∈[0,1],
where Gv=(r¯1-f¯1ev1,r¯2-f¯2ev2,r¯3-f¯3ev3,r¯4-f¯4ev4)T.
Whenv∈Ω∩kerℒ=Ω∩ℝ4 is a constant vector with ∥v∥=C, from Lemma 2.5, we get that Hμv≠O on ∂Ω∩kerℒ. Since ImQ=kerℒ and (v1*,v2*,v3*,v4*)T∈Ω∩kerℒ is the unique solution of the algebraic equations Gv=(0,0,0,0)T, by the homotopy invariance of Brouwer degree, we obtain
(3.39)deg⁡(Ψ∘Q∘N,∂Ω∩ker⁡L,O)=sign⁡(-f¯1f¯2f¯3f¯4ev1*+v2*+v3*+v4*)≠0.
Therefore, all the conditions in Lemma 2.5 are fulfilled and the dynamic system (1.2) has at least one ω-periodic solution lying in Domℒ∩Ω¯.


## 4. Example

Consider the following system with20π-periodic time scale:
(4.1)x1δ(t)=4-2cos⁡t10-(5-sin⁡t10)ex1(t)-(5-cos⁡(t/10)+2sin⁡(t/10))ey1(t-1/3)ex1(t-1/3)+(5-cos⁡(t/10))ey1(t-1/3)+(53-sin⁡t10)(ex2(t)-x1(t)-1),x2δ(t)=5-2sin⁡t10-(4+cos⁡t10)ex2(t)+(54-cos⁡(t10))(ex1(t)-x2(t)-1),y1δ(t)=5+2sin⁡t10+cos⁡t10-(2-cos⁡(t/10))ey2(t)ey1(t)+(4-sin⁡(t/10))ey2(t)-(73-cos⁡t10)ey1(t)-(2+sin⁡(t/10))ex1(t-1/3)ex1(t-1/3)+(5-cos⁡(t/10))ey1(t-1/3),y2δ(t)=5+2sin⁡t10-(5+cos⁡t10)ey2(t)+(3+sin⁡(t/10))ey1(t-1/3)ey1(t-1/3)+(4-sin⁡t/10)ey2(t-1/3).
From the definition of 𝕀ω, we obtain that 𝕀ω=[0,20π]. It is straight to check that (r1β1-g1)s=mint{r1(t)β1(t)-g1(t)}=15-13cos(t/10)+2sin(t/10)+2cos2(t/10)>0, and other inequalities r¯1β1s-g1M>0, (r3β2-g2β2-h1)s>0 and r¯3β2s-g2Mβ2s-h1M>0. Hence, from Theorem 3.1, the dynamic system (4.1) has at least one 20-periodic solution on the time scale 𝕋.



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*Source: 762768-2012-07-05.xml*