# Multiple Periodic Solutions to a Kind of Lotka-Volterra Food-Chain System with Delays and Impulses on Time Scales

Authors: Kaihong Zhao; Liang Ding; Fengzao Yang
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

By using Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least2n periodic solutions for a kind of n-species Lotka-Volterra food-chain system with delays and impulses on time scales. One example is given to illustrate the effectiveness of our results.


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## Body

## 1. Introduction

The food-chain phenomenon is universal and interesting in ecosystem. It is one of important methods to investigate this phenomenon by establishing the model of differential equations. The importance of studying the dynamics of food chain was pointed by Rosenzweig. In his famous paper on the paradox of enrichment [1], he wrote “Man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” Hereafter, many scholars investigated various kinds of food-chain systems (see [2–9]).In [10], Li and Zhao have studied the existence of multiple positive periodic solutions for the following Lotka-Volterra food-chain system:
(1.1)x˙1(t)=x1(t)(a1(t)-b1(t)x1(t)-c12(t)x2(t))-h1(t),⋮x˙i(t)=xi(t)(-di(t)-bi(t)xi(t)+ci,i-1(t)xi-1(t)-ci,i+1(t)xi+1(t))-hi(t),2≤i≤n-1,⋮x˙n(t)=xn(t)(-dn(t)-bn(t)xn(t)+cn,n-1(t)xn(t))-hn(t).
As we know, in population dynamics, many evolutionary processes experience short-time rapid chance after undergoing relatively long sooth variation. Examples include stocking of species and annual immigration. Incorporating these phenomena give us impulsive differential equations. For the theory of impulsive differential equations, we refer the reader to [11, 12]. In addition, there are few results on the existence of multiple periodic solutions for the delay food-chain system with impulsive effects in literatures. This motivates us to modify system (1.1) to the form
(1.2)x˙1(t)=x1(t)(a1(t)-b1(t)x1(t-τ11(t))-c12(t)x2(t-τ12(t)))-h1(t),t≠tk,⋮x˙i(t)=xi(t)(-di(t)-bi(t)xi(t-τii(t))+ci,i-1(t)xi-1(t-τi,i-1(t))-ci,i+1(t)xi+1(t-τi,i+1(t)))-hi(t),2≤i≤n-1,t≠tk,⋮x˙n(t)=xn(t)(-dn(t)-bn(t)xn(t-τnn(t))+cn,n-1(t)xn(t-τn,n-1(t)))-hn(t),t≠tk,Δxi(tk)=xi(tk+)-xi(tk-)=gikxi(tk),1≤i≤n,
where gik∈(-1,+∞)(i=1,2,…,n;k∈ℕ={1,2,…,n}).  {tk}k∈ℕ is a strictly increasing sequence with t1>0 and limt→∞tk=∞.  xi(t)(i=1,2,…,n) is the ith species population density. a1(t)>0 is the growth rate of the only producer. bi(t)>0(i=1,2,…,n) and hi(t)>0(i=1,2,…,n) stand for the ith species intraspecific competition rate and harvesting rate, respectively. di(t)>0(i=2,3,…,n) is the ith species death rate. ci,i+1(t)>0(i=1,2,…,n-1) represents the (i+1)th species predation rate on the ith species. ci,i-1(t)>0(i=2,3,…,n) stands for the transformation rate from the (i-1)th species to the ith species. τi,i+1(t)≥0(i=1,2,…,n-1) stands for the time-lag in the process of the (i+1)th species predation rate on the ith species. τi,i-1(t)≥0(i=2,3,…,n) stands for the time-lag in the process of transformation from the (i-1)th species to the ith species. τii(t)≥0(i=1,2,…,n) represents the time-lag in the process of intraspecific competition. In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.), which leads us to assume that ai(t), bi(t), cij(t), τij(t), and hi(t)(i,j=1,2,…,n) are all continuous ω-periodic functions. For impulsive effects, we further assume that there exists a q∈ℕ such that gi(k+q)=gik(i=1,2,…,n) and tk+q=tk+ω.However, dynamics in each equally spaced time interval may vary continuously. So, it may be more realistic to assume that the population dynamics involves the hybrid discrete-continuous processes. For example, Gamarra and Solé pointed out that such hybrid processes appear in the population dynamics of certain species that feature nonoverlapping generations: the change in population from one generation to the next is discrete and so is modeled by a difference equation, while within-generation dynamics vary continuously (due to mortality rates, resource consumption, predation, interaction, etc.) and thus are described by a differential equation [13]. The theory of calculus on time scales (see [14, 15] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 [16] in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. This motivates us to consider Lotka-Volterra food-chain system with delays and impulses on time scales 𝕋 of the following form:
(1.3)u1Δ(t)=a1(t)-b1(t)eu1(t-τ11(t))-c12(t)eu2(t-τ12(t))-h1(t)e-u1(t),t∈𝕋,t≠tk,⋮uiΔ(t)=-di(t)-bi(t)eui(t-τii(t))+ci,i-1(t)eui-1(t-τi,i-1(t))-ci,i+1(t)eui+1(t-τi,i+1(t))-hi(t)e-ui(t),2≤i≤n-1,t∈𝕋,t≠tk,⋮unΔ(t)=-dn(t)-bn(t)eun(t-τnn(t))+cn,n-1(t)eun(t-τn,n-1(t))-hn(t)e-un(t),t∈𝕋,t≠tk,Δui(tk)=ui(tk+)-ui(tk-)=ln(1+gik),1≤i≤n,
where 𝕋 is a ω-periodic time scale. gik∈(-1,+∞)(i=1,2,…,n;k∈ℕ={1,2,…,n}) is a constant. {tk}k∈ℕ is a strictly increasing sequence with t1>0 and limk→∞tk=∞.  a1(t)>0, bi(t)>0, hi(t)>0, di(t)>0, ci,i+1(t)>0, ci,i-1(t)>0, τi,i+1(t), τi,i-1(t), τii(t)≥0 are rd-continuous ω-periodic functions. For impulsive effects, we further assume that there exists a integer q∈ℕ such that gi(k+q)=gik(i=1,2,…,n) and tk+q=tk+ω. We denote by κ=min{[0,+∞)∩𝕋}, Iω=[κ,κ+ω]∩𝕋. Without loss of generality, we also assume that Iω∩{tk:k∈ℕ}={t1,t2,…,tq}.Remark 1.1.
In (1.3), let xi(t)=eui(t),  i=1,2,…,n. If 𝕋=ℝ (the set of all real numbers), then (1.3) reduces to (1.2).To the best our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Our main purpose of this paper is by using Mawhin’s continuation theorem of coincidence degree theory [17], to establish the existence of at least 2n periodic solutions for system (1.3). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer to [18–25].The organization of the rest of this paper is as follows. In the Section2, we will introduce some basic notations and lemmas which are used in what follows. In Section 3, by employing the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least 2n periodic solutions of system (1.3). In the final Section, one example is given to illustrate the effectiveness of our results.


## 2. Preliminaries on Time Scales

In this section, we briefly recall some basic definitions and lemmas on time scales which are used in what follows. For more details, one can see [14–16].Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ, ρ:𝕋→𝕋 and the graininess μ:𝕋→ℝ+ are defined, respectively, by
(2.1)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},μ(t)=σ(t)-t.A pointt∈𝕋 is called left-dense if t>inf𝕋 and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<sup𝕋 and σ(t)=t, and right-scattered if σ(t)>t. If 𝕋 has a left-scattered maximum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋. If 𝕋 has a right-scattered minimum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋.Letω>0. Throughout this paper, the time scale 𝕋 is assumed to be ω-periodic, that is, t∈𝕋 implies t+ω∈𝕋. In particular, the time scale 𝕋 under consideration is unbounded above and below.Definition 2.1.
A functionf:𝕋→ℝ is called regulated provided its right-side limits exist (finite) at all right-side points in 𝕋 and its left-side limits exist (finite) at all left-side points in 𝕋.Definition 2.2.
A functionf:𝕋→ℝ is called rd-continuous provided it is continuous at right-dense point in 𝕋 and its left-side limits exist (finite) at left-dense points in 𝕋. The set of rd-continuous functions f:𝕋→ℝ will be denoted by Crd=Crd(𝕋)=Crd(𝕋,ℝ).Definition 2.3.
Assumef:𝕋→ℝ and t∈𝕋k. Then we define fΔ(t) to be the number (if it exists) with the property that given any ε>0 there exists a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ>0) such that
(2.2)|[f(σ(t))-f(s)]-fΔ(t)[σ(t)-s]|<ε|σ(t)-s|
for all s∈U. we call fΔ(t) the delta (or Hilger) derivative of f at t. The set of functions f:𝕋→ℝ that are differentiable and whose derivative is rd-continuous is denoted by Crd1=Crd1(𝕋)=Crd1(𝕋,ℝ).Iff is continuous, then f is rd-continuous. If f is rd-continuous, the f is regulated. If f is delta differentiable at t, then f is continuous at t.Lemma 2.4.
Letf be regulated, then there exists a function F which is delta differentiable with region of differentiation D such that
(2.3)FΔ(t)=f(t)∀t∈D.Definition 2.5.
Assumef:𝕋→ℝ is a regulated function. Any function F as in Lemma 2.4 is called a Δ-antiderivative of f. We define the indefinite integral of a regulated function f by
(2.4)∫f(t)Δt=F(t)+C,
where C is an arbitrary constant and F is a Δ-antiderivative of f. We define the Cauchy integral by
(2.5)∫ab‍f(s)Δs=F(b)-F(a)∀a,b∈𝕋.
A function F:𝕋→ℝ is called an antiderivative of f:𝕋→ℝ provided
(2.6)FΔ(t)=f(t)∀t∈𝕋k.Lemma 2.6.
Ifa,b∈𝕋, α,β∈ℝ and f,g∈C(𝕋,ℝ), then (i)
∫ab‍[αf(t)+βg(t)]Δt=α∫ab‍f(t)Δt+β∫ab‍g(t)Δt;(ii)
iff(t)≥0 for all a≤t<b, then ∫ab‍f(t)Δt≥0;(iii)
if|f(t)|≤g(t) on [a,b):={t∈𝕋:a≤t<b}, then |∫ab‍f(t)Δt|≤∫ab‍g(t)Δt.For convenience, we now introduce some notation to be used throughout this paper. Let(2.7)κ=min{[0,+∞)∩𝕋},Iω=[κ,κ+ω]∩𝕋,gM=supt∈Iωg(t),gl=inft∈Iωg(t),g-=1ω∫Iω‍g(s)Δs=1ω∫κκ+ω‍g(s)Δs,
where g∈Crd(𝕋) is an ω-periodic real function, that is, g(t+ω)=g(t) for all t∈𝕋.Lemma 2.7 (see [26]).
Assume that{fn}n∈N is a function on J such that (i)
{fn}n∈N is uniformly bounded on J,(ii)
{fnΔ}n∈N is uniformly bounded on J. Then there is a subsequence of {fn}n∈N which converges uniformly on J.


## 3. Existence of at Least2n Periodic Solutions

In this section, by using Mawhin’s continuation theorem and the skills of inequalities, we will show the existence of periodic solutions of (1.3). To do so, we need to make some preparations.LetX and Z be real normed vector spaces. Let L:DomL⊂X→Z be a linear mapping and N:X×[0,1]→Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim KerL=codimImL<∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exists continuous projectors P:X→X and Q:Z→Z such that ImP=KerL and KerQ=ImL=Im(I-Q), and X=KerL⊕KerP, Z=ImL⊕ImQ. It follows that L|DomL∩KerP:(I-P)X→ImL is invertible and its inverse is denoted by KP. If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω-×[0,1], if QN(Ω-×[0,1]) is bounded and KP(I-Q)N:Ω-×[0,1]→X is compact. Because ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.The Mawhin’s continuous theorem [17, page 40] is given as follows.Lemma 3.1 (see [17]).
LetL be a Fredholm mapping of index zero and let N be L-compact on Ω-×[0,1]. Assume (a)
for eachλ∈(0,1), every solution x of Lx=λN(x,λ) is such that x∉∂Ω∩DomL;(b)
QN(x,0)x≠0 for each x∈∂Ω∩KerL;(c)
deg(JQN(x,0),Ω∩KerL,0)≠0. Then Lx=N(x,1) has at least one solution in Ω¯∩DomL.For simplicity, we need to introduce some notations as follows:(3.1)gi=12∑k=1q‍ln(1+gik)(1≤i≤n),Gi=12∑k=1q‍[ln(1+gik)+|ln(1+gik)|](1≤i≤n),l1±=(a-1+(2g1/ω))ea-1ω+G1±((a-1+(2g1/ω))ea-1ω+G1)2-4b-1h-1ea-1ω+G12b-1,A1±=(a-1+(2g1/ω)-c-12l2+)±(a-1+(2g1/ω)-c-12l2+)2-4b-1h-1ea-1ω+G12b-1,B1±=(a-1+(2g1/ω)-c-12l2+)e-(a-1ω+G1)±((a-1+(2g1/ω)-c-12l2+)e-(a-1ω+G1))2-4b-1h-1e-(a-1ω+G1)2b-1,li±=(c-i,i-1+(2gi/ω))ec-i,i-1ω+Gi±((c-i,i-1+(2gi/ω))ec-i,i-1ω+Gi)2-4b-ih-iec-i,i-1ω+Gi2b-i(2≤i≤n),Ai±=((2gi/ω)-d-i-c-i,i+1li+1+)±((2gi/ω)-d-i-c-i,i+1li+1+)2-4b-ih-iec-i,i-1ω+Gi2b-i(2≤i≤n-1),Bi±=12b-i[±((2giω-d-i-c-i,i+1li+1+)e-(c-i,-1ω+Gi))2-4b-ih-ie-(c-i,i-1ω+Gi)+(2giω-d-i-c-i,i+1li+1+)e-(c-i,i-1ω+Gi)((2giω-d-i-c-i,i+1li+1+)e-(c-i,-1ω+Gi))2-4b-ih-ie-(c-i,i-1ω+Gi)](2≤i≤n-1),Bn±=((2gn/ω)-d-n)e-(c-n,n-1ω+Gn)±(((2gn/ω)-d-n)e-(c-n,n-1ω+Gn))2-4b-nh-ne-(c-n,n-1ω+Gn)2b-n,An±=((2gn/ω)-d-n)±((2gn/ω)-d-n)2-4b-nh-nec-n,n-1ω+Gn2b-n,H1=l1-e-(a-1ω+G1),Hi=li-e-(c-i,i-1ω+Gi)(2≤i≤n).
Throughout this paper, we need the following assumptions:(H1)
(a-1+(2g1/ω)-c-12l2+)e-(1/2)(a-1ω+G1)>2b-1h-1,(H2)
((2gi/ω)-d-i-c-i,i+1li+1+)e-(1/2)(c-i,i-1ω+Gi)>2b-ih-i(2≤i≤n-1),(H3)
((2gn/ω)-d-n)e-(1/2)(c-n,n-1ω+Gn)>2b-nh-n. The following results will play an important role in the proof of our main result.Lemma 3.2.
Let𝕋 be a ω-periodic time scale. {tk}1∞⊂𝕋 is a strictly increasing sequence with t1>0 and limk→∞tk=∞.  κ=min{[0,∞)∩𝕋}, Iω=[κ,κ+ω]∩𝕋, Iω∩{tk}k=1∞={t1,t2,…,tq}. Suppose ψ:𝕋→ℝ be a ω-periodic function which is rd-continuous for t≠tk, exists the right limit ψ(tk+) and left limit ψ(tk-)=ψ(tk) in the sense of time scales. Then
(3.2)0≤sups∈Iωψ(s)-infs∈Iωψ(s)≤12[∫κκ+ω‍|ψΔ(s)|Δs+∑k=1q‍|Δψ(tk)|].Proof.
First, taket*∈Iω such that ψ(t*)=infs∈Iωψ(s). Next, take t*∈[t*,t*+ω]∩𝕋 such that ψ(t*)=sups∈Iωψ(s). Then we have
(3.3)ψ(t*)=ψ(t*)+(ψ(t*)-ψ(t*))=ψ(t*)+(ψ(t*)-∑t*≤tk<t*‍Δψ(tk)-ψ(t*))+∑t*≤tk<t*‍Δψ(tk)=ψ(t*)+∫t*t*‍ψ(s)Δs+∑t*≤tk<t*‍Δψ(tk)≤ψ(t*)+∫t*t*‍|ψ(s)|Δs+∑t*≤tk<t*‍|Δψ(tk)|.
Noticing that ψ(t*)=ψ(t*-ω)=ψ(t*)+(ψ(t*)-ψ(t*-ω)), similar to the above, we can get
(3.4)ψ(t*)≤ψ(t*)+∫t*-ωt*‍|ψ(s)|Δs+∑t*-ω≤tk<t*‍|Δψ(tk)|.
From (3.3) and (3.4), we get
(3.5)2ψ(t*)≤2ψ(t*)+∫t*t*‍|ψ(s)|Δs+∫t*-ωt*‍|ψ(s)|Δs+∑t*≤tk<t*‍|Δψ(tk)|+∑t*-ω≤tk<t*‍|Δψ(tk)|=2ψ(t*)+∫0ω‍|ψ(s)|Δs+∑0≤tk<ω‍|Δψ(tk)|,
that is
(3.6)ψ(t*)≤ψ(t*)+12[∫κκ+ω‍|ψ(s)|Δs+∑k=1q‍|Δψ(tk)|],
which implies that (3.2) holds. This completes the proof.Lemma 3.3.
Letx>0, y>0, z>0 and x>2yz, for the functions f(x,y,z)=(x+x2-4yz)/2z and g(x,y,z)=(x-x2-4yz)/2z, the following assertions hold. (1)
f(x,y,z) and g(x,y,z) are monotonically increasing and monotonically decreasing on the variable x∈(0,∞), respectively.(2)
f(x,y,z) and g(x,y,z) are monotonically decreasing and monotonically increasing on the variable y∈(0,∞), respectively.(3)
f(x,y,z) and g(x,y,z) are monotonically decreasing and monotonically increasing on the variable z∈(0,∞), respectively.Proof.
In fact, for allx>0, y>0, z>0, we have
(3.7)∂f∂x=x+x2-4yz2zx2-4yz>0,∂g∂x=x2-4yz-x2zx2-4yz<0,∂f∂y=-1x2-4yz<0,∂g∂y=1x2-4yz>0,∂f∂z=-(x+x2-4yz)24z2x2-4yz<0,∂g∂z=(x-x2-4yz)24z2x2-4yz>0.
By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 3.3 is complete.Lemma 3.4.
Assume that(H1), (H2), and (H3) hold, then we have the following inequalities: (1)
lnli+>lnAi+>lnAi->lnli->lnHi(1≤i≤n);(2)
lnli+>lnBi+>lnBi->lnli->lnHi(1≤i≤n);(3)
min{lnAi+,lnBi+}>max{lnAi-,lnBi-}(1≤i≤n).Proof.
(1) Since(3.8)(a-1+2g1ω)ea-1ω+G1>a-1+2g1ω-c-12l2+>(a-1+2g1ω-c-12l2+)e-(1/2)(a-1ω+G1)>2b-1h-1>0,(c-i,i-1+2giω)ec-i,i-1ω+Gi>2giω-d-i-c-i,i+1li+1+>(2giω-d-i-c-i,i+1li+1+)e-(1/2)(c-i,i-1ω+Gi)>2b-ih-i>0,(c-n,n-1+2gnω)ec-n,n-1ω+Gn>2gnω-d-n>(2gnω-d-n)e-(1/2)(c-n,n-1ω+Gn)>2b-nh-n>0,0<b-i,0<h-i,i=1,2,…,n.
Applying Lemma 3.3, we have
(3.9)0<H1=l1-e-(a-1ω+G1)<l1-=g((a-1+2g1ω)ea-1ω+G1,b-1,h-1ea-1ω+G1)<g(a-1+2g1ω-c-12l2+,b-1,h-1ea1ω+G1)=A1-<A1+=f(a-1+2g1ω-c-12l2+,b-1,h-1ea-1ω+G1)<f((a-1+2g1ω)ea-1ω+G1,b-1,h-1ea-1ω+G1)=l1+,0<Hi=li-e-(c-i,i-1ω+Gi)<li-=g((c-i,i-1+2giω)ec-i,i-1ω+Gi,b-i,h-iec-i,i-1ω+Gi)<g(2giω-d-i-c-i,i+1li+1+,b-i,h-iec-i,i-1ω+Gi)=Ai-<Ai+=f(2giω-d-i-c-i,i+1li+1+,b-i,h-iec-i,i-1ω+Gi)<f((c-i,i-1+2giω)ec-i,i-1ω+Gi,b-i,h-iec-i,i-1ω+Gi)=li+,i=2,3,…,n-1,0<Hn=ln-e-(c-n,n-1ω+Gn)<ln-=g((c-n,n-1+2gnω)ec-n,n-1ω+Gn,b-n,h-nec-n,n-1ω+Gn)<g(2gnω-d-n,b-n,h-nec-n,n-1ω+Gn)=An-<An+=f(2gnω-d-n,b-n,h-nec-n,n-1ω+Gn)<f((c-n,n-1+2gnω)ec-n,n-1ω+Gn,b-n,h-nec-n,n-1ω+Gn)=ln+.
Thus, we havelnHi<lnli-<lnAi-<lnAi+<lnli+, i=1,2,…,n. The proof of (1) of Lemma 3.4 is complete. For (2) of Lemma 3.4, we similarly have
(3.10)0<H1=l1-e-(a-1ω+G1)<l1-=g((a-1+2g1ω)ea-1ω+G1,b-1ea-1ω+G1,h-1)<g(a-1+2g1ω-c-12l2+,b-1ea1ω+G1,h-1)=B1-<B1+=f(a-1+2g1ω-c-12l2+,b-1ea-1ω+G1,h-1)<f((a-1+2g1ω)ea-1ω+G1,b-1ea-1ω+G1,h-1)=l1+,0<Hi=li-e-(c-i,i-1ω+Gi)<li-=g((c-i,i-1+2giω)ec-i,i-1ω+Gi,b-iec-i,i-1ω+Gi,h-i)<g(2giω-d-i-c-i,i+1li+1+,b-iec-i,i-1ω+Gi,h-i)=Bi-<Bi+=f(2giω-d-i-c-i,i+1li+1+,b-iec-i,i-1ω+Gi,h-i)<f((c-i,i-1+2giω)ec-i,i-1ω+Gi,b-iec-i,i-1ω+Gi,h-i)=li+,i=2,3,…,n-1,0<Hn=ln-e-(c-n,n-1ω+Gn)<ln-=g((c-n,n-1+2gnω)ec-n,n-1ω+Gn,b-nec-n,n-1ω+Gn,h-n)<g(2gnω-d-n,b-nec-n,n-1ω+Gn,h-n)=Bn-<Bn+=f(2gnω-d-n,b-nec-n,n-1ω+Gn,h-n)<f((c-n,n-1+2gnω)ec-n,n-1ω+Gn,b-nec-n,n-1ω+Gn,h-n)=ln+,
which imply that lnHi<lnli-<lnBi-<lnBi+<lnli+, i=1,2,…,n; that is, the inequality (2) of Lemma 3.4 holds. Let us now to prove the inequality (3) of Lemma 3.4. In fact, since
(3.11)max{lnAi-,lnBi-}=lnAi-+lnBi-+|lnAi--lnBi-|2(1≤i≤n),min{lnAi+,lnBi+}=lnAi++lnBi+-|lnAi+-lnBi+|2(1≤i≤n),
we have
(3.12)min{lnAi+,lnBi+}-max{lnAi-,lnBi-}=12[lnAi++lnBi+-lnAi--lnBi--|lnAi--lnBi-|-|lnAi+-lnBi+|].
If Ai+≥Bi+, then Ai-≤Bi-, and
(3.13)min{lnAi+,lnBi+}-max{lnAi-,lnBi-}=lnBi+-lnBi->0;
If Ai+≤Bi+, then Ai-≥Bi-, and
(3.14)min{lnAi+,lnBi+}-max{lnAi-,lnBi-}=lnAi+-lnAi->0.
So, we drive min{lnAi+,lnBi+}>max{lnAi-,lnBi-}. From the above, we have completed the proof of Lemma 3.4.Theorem 3.5.
Assume that(H1), (H2), and (H3) hold. Then system (1.3) has at least 2n  ω-periodic solutions.Proof.
LetC[Iω;t1,…,tq]={u:Iω→ℝn is a piecewise continuous map with first-class discontinuity points in Iω∩{tk}k=1∞, and at each discontinuity point, it is continuous on the left}. Take
(3.15)X={u=(u1,u2,…,un)T∈C[Iω;t1,…,tq]:ui(t+ω)=ui(t)},Z=X×ℝn×(q+1)
and define
(3.16)∥u∥0=∑i=1n‍supt∈Iω|ui(t)|,u∈X,∥z∥1=∥u∥0+∑j=1q‍|ξj|,z=(u,ξ1,ξ2,…,ξq)∈Z.
Then both (X,∥·∥0) and (Z,∥·∥1) are Banach spaces. Let (3.17)DomL={u∈X:uΔ∈X},L:DomL→Z,N:X→Z,Lu=((u1Δu2Δ⋮unΔ),(Δu1(t1)Δu2(t1)⋮Δun(t1)),…,(Δu1(tq)Δu2(tq)⋮Δun(tq))),N(u,λ)=((F1(t,λ)⋮Fi(t,λ)⋮Fn(t,λ)),(ln(1+g11)⋮ln(1+gi1)⋮ln(1+gn1)),…,(ln(1+g1q)⋮ln(1+giq)⋮ln(1+gnq))),
where
(3.18)F1(t,λ)=a1(t)-b1(t)eu1(t-τ11(t))-λc12(t)eu2(t-τ12(t))-h1(t)e-u1(t),Fi(t,λ)=-λdi(t)-bi(t)eui(t-τii(t))+ci,i-1(t)eui-1(t-τi,i-1(t))-λci,i+1(t)eui+1(t-τi,i+1(t))-hi(t)e-ui(t),i=2,3,…,n-1,Fn(t,λ)=-λdn(t)-bn(t)eun(t-τnn(t))+cn,n-1(t)eun-1(t-τn,n-1(t))-hn(t)e-un(t).P:X→X,Q:Z→Z,P(u=(u1,u2,…,un)T)=(u-1,u-2,…,u-n)T,u∈X,Q((u1u2⋮un),{(m1km2k⋮mnk)}k=1q)=((u-1+1ω∑k=1q‍m1ku-2+1ω∑k=1q‍m2k⋮u-n+1ω∑k=1q‍mnk),{(00⋮0)}k=1q).
Obviously,
(3.19)KerL={u=(u1,u2,…,un)T∈X:(u1(t),u2(t),…,un(t))=(c1,c2,…,cn)∈ℝn,t∈𝕋},ImL={((u1u2⋮un),{(m1km2k⋮mnk)}k=1q)∈Z:ωu-1+∑k=1q‍m1k=0ωu-2+∑k=1q‍m2k=0⋮ωu-n+∑k=1q‍mnk=0}.
Since ImL is closed in Z, ImP=KerL, KerQ=ImL, and dimKerL=n=codimImL, we know that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP:ImL→KerP∩DomL is given by
(3.20)KP(z=((u1u2⋮un),{(m1km2k⋮mnk)}k=1q))=(∫κt‍u1(s)Δs+∑κ<tk<t‍m1k-1ω∑k=1q‍m1k-1ω∫κκ+ω‍∫κt‍u1(s)ΔsΔt∫κt‍u2(s)Δs+∑κ<tk<t‍m2k-1ω∑k=1q‍m2k-1ω∫κκ+ω‍∫κt‍u2(s)ΔsΔt⋮∫κt‍u1(s)Δs+∑κ<tk<t‍m1k-1ω∑k=1q‍mnk-1ω∫κκ+ω‍∫κt‍un(s)ΔsΔt).
Then
(3.21)QN(u,λ)=((1ω∫κκ+ω‍F1(s,λ)Δs+2g1ω⋮1ω∫κκ+ω‍Fn(s,λ)Δs+2gnω),{(0⋮0)}k=1q),KP(I-Q)N(u,λ)=(∫κt‍F1(s,λ)Δs+∑κ<tk<t‍ln(1+g1k)⋮∫κt‍Fn(s,λ)Δs+∑κ<tk<t‍ln(1+g1k))+(12-tω)(∫κκ+ω‍F1(s,λ)Δs+2g1⋮∫κκ+ω‍Fn(s,λ)Δs+2gn)-1ω(∫κκ+ω‍∫κt‍F1(s,λ)ΔsΔt+2g1⋮∫κκ+ω‍∫κt‍Fn(s,λ)ΔsΔt+2gn),
where
(3.22)F1(s,λ)=a1(s)-b1(s)eu1(s-τ11(s))-λc12(s)eu2(s-τ12(s))-h1(s)e-u1(s),Fi(s,λ)=-λdi(s)-bi(s)eui(s-τii(s))+ci,i-1(s)eui-1(s-τi,i-1(s))-λci,i+1(s)eui+1(s-τi,i+1(s))-hi(s)e-ui(s),i=2,3,…,n-1,Fn(s,λ)=-λdn(s)-bn(s)eun(s-τnn(s))+cn,n-1(s)eun-1(s-τn,n-1(s))-hn(s)e-un(s).
By the Lebesgue convergence theorem, QN and KP(I-Q)N are continuous. Next, we show that KP(I-Q)N maps bounded sets into relatively compact sets. Let Ω⊂X be an arbitrary bounded set in X, then there exists a number R>0 such that ∥u∥<R for any u⊂Ω. We prove that KP(I-Q)N(Ω¯)¯ is relatively compact. In fact, for any {uν=(u1ν,u2ν,…,unν)T}ν∈ℕ∈Ω and t∈Iω, we have
(3.23)∥KP(I-Q)N(uν,λ)∥=∑i=1n‍supt∈Iω|∫κt‍Fi(s,λ)Δs+∑κ<tk<t‍ln(1+gik)+(12-tω)×∫κκ+ω‍Fi(s,λ)Δs+2gi-1ω∫κκ+ω‍∫κt‍Fi(s,λ)ΔsΔt+2g1|.
Since ∥uν∥<R, it follows from the periodicity that there exists a constant W1 such that ∥KP(I-Q)N(uν,λ)∥≤W1. For any {uν=(u1ν,u2ν,…,unν)T}ν∈ℕ∈Ω and t∈Iω∩(tk,tk+1], k=1,2,…,q-1, we have
(3.24)(KP(I-Q)N(uν,λ))Δ=(F1(s,λ)⋮Fn(s,λ))-1ω(∫κκ+ω‍F1(s,λ)Δs+2g1⋮∫κκ+ω‍Fn(s,λ)Δs+2gn).
Similarly, there has a constant W2 such that
(3.25)∑i=1n‍supt∈Iω∩(tk,tk+1]|(KP(I-Q)N(uν,λ))Δ|≤W2,k=1,2,…,q-1.
It follows from the definition of X that the right limit of uν(t) exists at each tk for k=1,2,…,q. ν∈ℕ. Without loss of generality, here, we can assume that t1=κ and tq=κ+ω, for if t1>κ or tq<κ+ω, we only need to add intervals [κ,t1]∩𝕋 or [tq,κ+ω]∩𝕋 into the consideration. Consider the following functions:
(3.26)(H(uν,λ))(t)={(KP(I-Q)N(uν,λ))(t),t∈Iω∩(tk,tk+1],k=1,…,q-1,(KP(I-Q)N(uν,λ))(tk+0),t=tk,k=1,…,q,(H(uν,λ))Δ(t)={(KP(I-Q)N(uν,λ))Δ(t),t∈Iω∩(tk,tk+1],k=1,…,q-1,(KP(I-Q)N(uν,λ))Δ(tk+0),t=tk,k=1,…,q,
where ν∈ℕ. Then on each interval [tk,tk+1], {(Hkuν)=(Huν)|[tk,tk+1]}ν∈ℕ, and {(Hkuν)Δ=(Huν)Δ|[tk,tk+1]}ν∈ℕ are uniformly bounded on [tk,tk+1]∩𝕋, k=1,2,…,q-1. By Lemma 2.7, there exists a subsequence {H1uν1(1)}ν1(1)∈ℕ of {H1uν}ν∈ℕ converges uniformly on [t1,t2]∩𝕋. Similarly, there is a subsequence {H2uν2(2)}ν2(2)∈ℕ of {H1uν1(1)}ν1(1)∈ℕ converges uniformly on [t2,t3]∩𝕋. Repeating such a process, we can obtain that there is a subsequence {Hq-1uνq-1(q-1)}νq-1(q-1)∈ℕ of {Hq-2uνq-2(q-2)}νq-2(q-2)∈ℕ converges uniformly on [tq-1,tq]∩𝕋. Obviously, {Hq-1uνq-1(q-1)}νq-1(q-1)∈ℕ is a subsequence of {Huν}ν∈ℕ that converges uniformly on Iω=[κ,κ+ω]∩𝕋. Therefore, {KP(I-Q)Nq-1uνq-1(q-1)}νq-1(q-1)∈ℕ is a subsequence of {KP(I-Q)Nuν}ν∈ℕ that converges uniformly on Iω, namely, KP(I-Q)N(Ω¯)¯ is relatively compact. Moreover, QN(Ω¯) is bounded. Thus, N is L-compact on Ω¯ for any open bounded set Ω⊂X.
In order to use Lemma3.1, we have to find at least 2n appropriate open bounded subsets of X. Considering the operator equation Lu=λN(u,λ), λ∈(0,1), we have
(3.27)u1Δ(t)=λ(a1(t)-b1(t)eu1(t-τ11(t))-λc12(t)eu2(t-τ12(t))-h1(t)e-u1(t)),⋮uiΔ(t)=λ(-λdi(t)-bi(t)eui(t-τii(t))+ci,i-1(t)eui-1(t-τi,i-1(t))-λci,i+1(t)eui+1(t-τi,i+1)-hi(t)e-ui(t)),2≤i≤n-1,⋮unΔ(t)=λ(-λdn(t)-bn(t)eun(t-τnn(t))+cn,n-1(t)eun-1(t-τn,-1(t))-hn(t)e-un(t)),Δuj(tk)=λln(1+gjk),1≤j≤n.
Assume that u∈X is an ω-periodic solution of system (3.27) for some λ∈(0,1). Integrating (3.27) from κ to κ+ω, we have
(3.28)-2g1=∫κκ+ω‍{a1(t)-b1(t)eu1(t-τ11(t))-λc12(t)eu2(t-τ12(t))-h1(t)e-u1(t)}Δt,⋮-2gi=∫κκ+ω‍{-λdi(t)-bi(t)eui(t-τii(t))+ci,i-1(t)eui-1(t-τi,i-1(t))-λci,i+1(t)eui+1(t-τi,i+1(t))-hi(t)e-ui(t)λci,i+1(t)eui+1(t-τi,i+1(t))}Δt,2≤i≤n-1,⋮-2gn=∫κκ+ω‍{-λdn(t)-bn(t)eun(t-τnn(t))+cn,n-1(t)eun-1(t-τn,n-1(t))-hn(t)e-un(t)}Δt.
Then
(3.29)∫κκ+ω‍{b1(t)eu1(t-τ11(t))+λc12(t)eu2(t-τ12(t))+h1(t)e-u1(t)}Δt=a-1ω+2g1⋮∫κκ+ω‍{λdi(t)+bi(t)eui(t-τii(t))+λci,i+1(t)eui+1(t-τi,i+1(t))+hi(t)e-ui(t)}Δt=∫κκ+ω‍ci,i-1(t)eui-1(t-τi,i-1(t))Δt+2gi,2≤i≤n-1,⋮∫κκ+ω‍{λdn(t)+bn(t)eun(t-τnn(t))+hn(t)e-un(t)}Δt=∫κκ+ω‍cn,n-1(t)eun-1(t-τn,n-1(t))Δt+2gn.
Furthermore, Note that u=(u1,u2,…,un)T∈X, and there exists ξi, ηi∈[κ,κ+ω]∩𝕋=Iω, i=1,2,…,n, such that ui(ξi)=supt∈Iωui(t), ui(ηi)=inft∈Iωui(t).
On the one hand, according to the first equation of (3.27) and (3.29), we have
(3.30)∫κκ+ω‍|u1Δ(t)|Δt<2a-1ω+2g1.
In the light of the first of equation of (3.29), (3.30), and Lemma 3.2, we get
(3.31)ωb-1eu1(ξ1)-(a-1ω+G1)+ωh-1e-u1(ξ1)≤ωb-1eu1(ξ1)-(1/2)[∫κκ+ω‍|u1Δ(s)|Δs+∑k=1q|Δu1(tk)|]+ωh-1e-u1(ξ1)<ωb-1eu1(η1)+ωh-1e-u1(ξ1)<∫κκ+ω‍{b1(t)eu1(t-τ11(t))+λc12(t)eu2(t-τ12(t))+h1(t)e-u1(t)}Δt=a-1ω+2g1,
namely,
(3.32)b-1e2u1(ξ1)-(a-1+2g1ω)ea-1ω+G1eu1(ξ1)+h-1ea-1ω+G1<0,
which implies that
(3.33)lnl1-<u1(ξ1)<lnl1+.
Applying Lemma 3.2, we drive
(3.34)lnH1=lnl1--(a-1ω+G1)<u1(η)≤u1(ξ1)<lnl1+.
According to (3.29) and (3.30), we get
(3.35)∫κκ+ω‍|u2Δ(t)|Δt<2c-21ω+2g2.
By (3.29) and (3.35), we have
(3.36)ωb-2eu2(ξ2)-(c-21ω+G2)+ωh-2e-u2(ξ2)≤ωb-2eu2(ξ2)-(1/2)[∫κκ+ω‍|u2Δ(s)|Δs+∑k=1q|Δu2(tk)|]+ωh-2e-u2(ξ2)<ωb-2eu2(η2)+ωh-2e-u2(ξ2)<∫κκ+ω‍{λd2(t)+b2(t)eu2(t-τ22(t))+λc23(t)eu3(t-τ23(t))+h2(t)e-u2(t)}Δt=∫κκ+ω‍c21(t)eu1(t-τ21(t))Δt+2g1<ωc-21l1++2g2,
that is
(3.37)b-2e2u2(ξ2)-(c-21+2g2ω)ec-21ω+G2eu2(ξ2)+h-2ec-21ω+G2<0,
which implies that
(3.38)lnl2-<u2(ξ2)<lnl2+.
Applying Lemma 3.2, we drive
(3.39)lnH2=lnl2--(c-21ω+G2)<u2(η)≤u2(ξ2)<lnl2+.
By deducing for i=3,4,…,n, we obtain
(3.40)∫κκ+ω‍|uiΔ(t)|Δt<2c-i,i-1ω+2gi,ωb-ieui(ξi)-(c-i,i-1ω+Gi)+ωh-ie-ui(ξi)≤ωb-ieui(ξi)-(1/2)[∫κκ+ω‍|uiΔ(s)|Δs+∑k=1q|Δui(tk)|]+ωh-ie-ui(ξi)<ωb-ieui(ηi)+ωh-ie-ui(ξi)<∫κκ+ω‍{λdi(t)+bi(t)eui(t-τii(t))+λci,i+1(t)eui+1(t-τi,i+1(t))+hi(t)e-ui(t)}Δt=∫κκ+ω‍ci,i-1(t)eui-1(t-τi,i-1(t))Δt+2gi-1<ωc-i,i-1li-1++2gi,
namely,
(3.41)b-ie2ui(ξi)-(c-i,i-1+2giω)ec-i,i-1ω+Gieui(ξi)+h-iec-i,i-1ω+Gi<0,
which implies that
(3.42)lnli-<ui(ξi)<lnli+,i=3,4,…,n.
Applying Lemma 3.2, we drive
(3.43)lnHi=lnli--(c-i,i-1ω+Gi)<ui(η)≤ui(ξi)<lnli+,i=3,4,…,n.
On the other hand, in the light of the first equation of (3.29) and (3.30), we have
(3.44)a-1ω+2g1=∫κκ+ω‍{b1(t)eu1(t-τ11(t))+λc12(t)eu2(t-τ12(t))+h1(t)e-u1(t)}Δt<b-1ωeu1(ξ1)+c-12ωl2++h-1ωe-u1(η1)<b-1ωeu1(ξ1)+c-12ωl2++h-1ωe-u1(ξ1)+(1/2)[∫κκ+ω‍|u1Δ(s)|Δs+∑k=1q|Δu1(tk)|]<b-1ωeu1(ξ1)+c-12ωl2++h-1ωe-u1(ξ1)+a1ω+G1,
namely,
(3.45)b-1e2u1(ξ1)-(a-1+2g1ω-c-12l2+)eu1(ξ1)+h-1ea-1ω+G1>0,
which implies that
(3.46)lnA1+<u1(ξ1)oru1(ξ1)<lnA1-,a-1ω+2g1=∫κκ+ω‍{b1(t)eu1(t-τ11(t))+λc12(t)eu2(t-τ12(t))+h1(t)e-u1(t)}Δt<ω(b-1eu1(ξ1)+c-12l2++h-1e-u1(η1))<ω(b-1eu1(η1)+(1/2)[∫κκ+ω‍|u1Δ(s)|Δs+∑k=1q|Δu1(tk)|]+c-12l2++h-1e-u1(η1))<b-1ωeu1(η1)+a-1ω+G1+c-12ωl2++h-1ωe-u1(η1),
that is,
(3.47)b-1e2u1(η1)-(a-1+2g1ω-c-12l2+)e-(a-1ω+G1)eu1(η1)+h-1e-(a-1ω+G1)>0,
which implies that
(3.48)lnB1+<u1(η1)oru1(η1)<lnB1-.
By (3.29) and (3.35), we have
(3.49)2g2<∫κκ+ω‍c21(t)eu1(t-τ21(t))Δt+2g2=∫κκ+ω‍{λd2(t)+b2(t)eu2(t-τ22(t))+λc23(t)eu3(t-τ23(t))+h2(t)e-u2(t)}Δt<b-2ωeu2(ξ2)+(d-2+c-23l3+)ω+h-2ωe-u2(η2)<b-2ωeu2(ξ2)+(d-2+c-23l3+)ω+h-2ωe-u2(ξ2)+(1/2)[∫κκ+ω‍|u2Δ(s)|Δs+∑k=1q|Δu2(tk)|]<b-2ωeu2(ξ2)+(d-2+c-23l3+)ω+h-2ωe-u2(ξ2)+c-21ω+G2,
namely,
(3.50)b-2e2u2(ξ2)-(2g2ω-d-2-c-23l3+)eu2(ξ2)+h-2ec-21ω+G2>0,
which implies that
(3.51)lnA2+<u2(ξ2)oru2(ξ2)<lnA2-,2g2<∫κκ+ω‍c21(t)eu1(t-τ21(t))Δt+2g2=∫κκ+ω‍{λd2(t)+b2(t)eu2(t-τ22(t))+λc23(t)eu3(t-τ23(t))+h2(t)e-u2(t)}Δt<b-2ωeu2(ξ2)+(d-2+c-23l3+)ω+h-2ωe-u2(η2)<b-2ωeu2(η2)+(1/2)[∫κκ+ω‍|u2Δ(s)|Δs+∑k=1q|Δu2(tk)|]+(d-2+c-23l3+)ω+h-2ωe-u2(η2)<b-2ωeu2(η2)+c-21ω+G2+(d-2+c-23l3+)ω+h-2ωe-u2(η2),
that is,
(3.52)b-2e2u2(η2)-(2g2ω-d-2-c-23l3+)e-(c-21ω+G2)eu2(η2)+h-2e-(c-21ω+G2)>0,
which implies that
(3.53)lnB2+<u2(η2)oru2(η2)<lnB2-.
By deducing for i=3,4,…,n-1, we obtain
(3.54)2gi<∫κκ+ω‍ci,i-1(t)eui-1(t-τi,i-1(t))Δt+2gi=∫κκ+ω‍{λdi(t)+bi(t)eui(t-τii(t))+λci,i+1(t)eui+1(t-τi,+1(t))+hi(t)e-ui(t)}Δt<b-iωeui(ξi)+(d-i+c-i,i+1li+1+)ω+h-iωe-ui(ηi)<b-iωeui(ξi)+(d-i+c-i,i+1li+1+)ω+h-iωe-ui(ξi)+(1/2)[∫κκ+ω‍|uiΔ(s)|Δs+∑k=1q|Δui(tk)|]<b-iωeui(ξi)+(d-i+c-i,i+1li+1+)ω+h-iωe-ui(ξi)+c-i,i-1ω+Gi,
namely,
(3.55)b-ie2ui(ξi)-(2giω-d-i-c-i,i+1li+1+)eui(ξi)+h-iec-i,i-1ω+Gi>0,
which implies that
(3.56)lnAi+<ui(ξi)orui(ξi)<lnAi-,2gi<∫κκ+ω‍ci,i-1(t)eu1(t-τi,-1(t))Δt+2gi=∫κκ+ω‍{λdi(t)+bi(t)eui(t-τii(t))+λci,i+1(t)eui+1(t-τi,i+1(t))+hi(t)e-ui(t)}Δt<b-iωeui(ξi)+(d-i+c-i,i+1li+1+)ω+h-2ωe-ui(ηi)<b-iωeui(ηi)+(1/2)[∫κκ+ω‍|uiΔ(s)|Δs+∑k=1q|Δui(tk)|]+(d-i+c-i,i+1li+1+)ω+h-iωe-ui(ηi)<b-iωeui(ηi)+c-i,i-1ω+Gi+(d-i+c-i,i+1li+1+)ω+h-iωe-ui(ηi),
that is,
(3.57)b-ie2ui(ηi)-(2giω-d-i-c-i,i+1li+1+)e-(c-i,i-1ω+Gi)eui(ηi)+h-ie-(c-i,i-1ω+Gi)>0,
which implies that
(3.58)lnBi+<ui(ηi)orui(ηi)<lnBi-.
By the final equation of (3.29), we get
(3.59)2gn<∫κκ+ω‍cn,n-1(t)eun-1(t-τn,n-1(t))Δt+2gn=∫κκ+ω‍{λdn(t)+bn(t)eun(t-τnn(t))+hn(t)e-un(t)}Δt<d-nω+b-nωeun(ξn)+h-nωe-un(ηn)<d-nω+b-nωeun(ξn)+h-nωe-un(ξn)+(1/2)[∫κκ+ω‍|unΔ(s)|Δs+∑k=1q|Δun(tk)|]<d-nω+b-nωeun(ξn)+h-nωe-un(ξn)+c-n,n-1ω+Gn,
namely,
(3.60)b-ne2un(ξn)-(2gnω-d-n)eun(ξn)+h-nec-n,n-1ω+Gn>0,
which implies that
(3.61)lnAn+<un(ξn)orun(ξn)<lnAn-,2gn<∫κκ+ω‍cn,n-1(t)eun-1(t-τn,n-1(t))Δt+2gn=∫κκ+ω‍{λdn(t)+bn(t)eun(t-τnn(t))+hn(t)e-un(t)}Δt<d-nω+b-nωeun(ξn)+h-nωe-un(ηn)<d-nω+b-nωeun(ηn)+(1/2)[∫κκ+ω‍|unΔ(s)|Δs+∑k=1q|Δun(tk)|]+h-nωe-un(ηn)<d-nω+b-nωeun(ηn)+c-n,n-1ω+Gn+h-nωe-un(ηn),
namely,
(3.62)b-ne2un(ηn)-(2gnω-d-n)e-(c-n,n-1ω+Gn)eun(ηn)+h-ne-(c-n,n-1ω+Gn)>0,
which implies that
(3.63)lnBn+<un(ηn)orun(ηn)<lnBn-,
In view of (3.34)–(3.63) and Lemma 3.3, we have
(3.64)min{lnAi+,lnBi+}<ui(ηi)<ui(ξi)<lnli+(1≤i≤n)
or
(3.65)lnHi<ui(ηi)<ui(ξi)<max{lnAi-,lnBi-}(1≤i≤n),
which imply that, for all t∈𝕋,
(3.66)min{lnAi+,lnBi+}<ui(t)<lnli+orlnHi-<ui(t)<max{lnAi-,lnBi-}.
For convenience, we denote
(3.67)Gi=(lnHi-,max{lnAi-,lnBi-}),Ei=(min{lnAi+,lnBi+},lnli+)(1≤i≤n).
Clearly,li±  (i=1,2,…,n), Ai±(i=1,2,…,n), and Bi±  (i=1,2,…,n) are independent of λ. For each i=1,2,…,n, we choose an interval between two intervals Gi and Ei and denote it as Δi, then define the set
(3.68){u=(u1,u2,…,un)T∈X:ui(t)∈Δi,t∈𝕋,i=1,2,…,n}.
Obviously, the number of the above sets is 2n. We denote these sets as Ωk, k=1,2,…,2n.  Ωk, k=1,2,…,2n are bounded open subsets of X, Ωi∩Ωj=ϕ, i≠j. Thus Ωk  (k=1,2,…,2n) satisfies the requirement (a) in Lemma 3.1.
Now we show that (b) of Lemma3.1 holds; that is, we prove when u∈∂Ωk∩KerL=∂Ωk∩ℝn, QN(u,0)≠(0,0,…,0)T, k=1,2,…,2n. If it is not true, then when u∈∂Ωk∩KerL=∂Ωk∩ℝn, i=1,2,…,2n, constant vector u=(u1,u2,…,un)T with u∈∂Ωk, k=1,2,…,2n, satisfies
(3.69)-2g1=∫κκ+ω‍{a1(t)-b1(t)eu1-h1(t)e-u1}Δt,⋮-2gi=∫κκ+ω‍{-bi(t)eui+ci,i-1(t)eui-1-hi(t)e-ui}Δt,2≤i≤n-1,⋮-2gn=∫κκ+ω‍{-bn(t)eun+cn,n-1(t)eun-1-hn(t)e-un}Δt,
that is,
(3.70)-2g1ω=a-1-b-1eu1-h-1e-u1,⋮-2giω=-b-ieui+c-i,i-1eui-1-h-ie-ui,2≤i≤n-1,⋮-2gnω=-b-neun+c-n,n-1eun-1-h-ne-un.
Similar to the process of (3.29)–(3.66), we obtain
(3.71)lnHi<ui-<max{lnAi-,lnBi-}<min{lnAi+,lnBi+}<ui+<lnli+(1≤i≤n).
Then u belongs to one of Ωk∩ℝn, k=1,2,…,2n. This contradicts the fact that u∈∂Ωk∩ℝn, k=1,2,…,2n. This proves that (b) in Lemma 3.1 holds.
Finally, in order to show that (c) in Lemma3.1 holds, we only prove that for u∈∂Ωk∩KerL=∂Ωk∩ℝn, k=1,2,…,2n, then it holds that deg{JQN(u,0),Ωk∩KerL,(0,0,…,0)T}≠0. To this end, we define the mapping ϕ:DomL×[0,1]→X by
(3.72)ϕ(u,μ)=μQN(u,0)+(1-μ)G(u),
here μ∈[0,1] is a parameter and G(u) is defined by (3.73)G(u)=((1ω∫κκ+ω‍(a1(s)-b1(s)eu1(s)-h1(s)e-u1(s))Δs+2g1ω⋮1ω∫κκ+ω‍(ci,i-1(s)li-1+-bi(s)eui(s)-hi(s)e-ui(s))Δs+2giω⋮1ω∫κκ+ω‍(cn,n-1(s)ln-1+-bn(s)eun(s)-hn(s)e-un(s))Δs+2gnω),{(0⋮0⋮0)}k=1q),
where i=2,3,…,n-1. We show that for u∈∂Ωk∩KerL=∂Ωk∩ℝn, k=1,2,…,2n, μ∈[0,1], then it holds that ϕ(u,μ)≠(0,0,…,0)T. Otherwise, parameter μ and constant vector u=(u1,u2,…,un)T∈ℝn satisfy ϕ(u,μ)=(0,0…,0)T, that is,
(3.74)0=μ∫κκ+ω‍[a1(s)-b1(s)eu1-h1(s)e-u1+2g1ω]Δs+(1-μ)∫κκ+ω‍[a1(s)-b1(s)eu1-h1(s)e-u1+2g1ω]Δs,⋮0=μ∫κκ+ω‍[ci,i-1(s)eui-1-bi(s)eui-hi(s)e-ui+2giω]Δs+(1-μ)∫κκ+ω‍[ci,i-1(s)li-1+-bi(s)eui-hi(s)e-ui+2giω]Δs,⋮0=μ∫κκ+ω‍[cn,n-1(s)eun-1-bn(s)eun-hn(s)e-un+2gnω]Δs+(1-μ)∫κκ+ω‍[cn,n-1(s)ln-1+-bn(s)eun-hn(s)e-un+2gnω]Δs,
where i=2,3,…,n-1, that is,
(3.75)a-1-b-1eu1-h-1e-u1=-2g1ω,⋮c-i,i-1li-1+-b-ieui-h-ie-ui=μ(c-i,i-1li-1+-c-i,i-1eui-1)-2giω,⋮c-n,n-1ln-1+-b-neun-h-ne-un=μ(c-n,n-1ln-1+-c-i,n-1eun-1)-2gnω,
where i=2,3,…,n-1. Following the argument of (3.29)–(3.66), we obtain
(3.76)lnHi<ui-<max{lnAi-,lnBi-}<min{lnAi+,lnBi+}<ui+<lnli+(1≤i≤n).
Equation (3.76) gives that u belongs to one of Ωk∩ℝn, k=1,2,…,2n. This contradicts the fact that u∈∂Ωk∩ℝn, k=1,2,…,2n. This proves ϕ(u,μ)≠(0,0,…,0)T holds. Note that the system of algebraic equations
(3.77)a-1-b-1ex1-h-1e-x1=-2g1ω,⋮c-i,i-1li-1+-b-iexi-h-ie-xi=-2giω,⋮c-n,n-1ln-1+-b-nexn-h-ne-xn=-2gnω,
has 2n distinct solutions since (H1), (H2), and (H3) hold, (x1*,x2*,…,xn*)=(lnx^1,lnx^2,…,lnx^n), where x1±=((a-1+(2g1/ω))±(a-1+(2g1/ω))2-4b-1h-1)/2b-1, xk±=((c-k,k-1lk-1++(2gk/ω))±(c-k,k-1lk-1++(2gk/ω))2-4b-kh-k)/2b-k(k=2,3,…,n),  x^i=xi- or x^i=xi+, i=1,2,…,n. Similar to the proof of Lemma 3.3, it is easy to verify that
(3.78)lnHi-<lnxi-<max{lnBi-,lnAi-}<min{lnAi+,lnBi+}<lnxi+<lnli+,i=1,2,…,n.
Therefore,(x1*,x2*,…,xn*) uniquely belongs to the corresponding Ωk. Since KerL=ImQ, we can take J=I. A direct computation gives, for k=1,2,…,2n,
(3.79)deg{JQN(u,0),Ωk∩KerL,(0,0,…,0)T}=deg{ϕ(u,1),Ωk∩KerL,(0,0,…,0)T}=deg{ϕ(u,0),Ωk∩KerL,(0,0,…,0)T}=sign[∏i=1n‍(-b-ixi*+h-ixi*)].
Since (a-1+(2g1/ω))-b-1x1*-(h-1/x1*)=0,  (c-i,i-1li-1++(2gi/ω))-b-ixi*-(h-i/xi*)=0(i=2,3,…,n), then
(3.80)deg{JQN(u,0),Ωk∩KerL,(0,0,…,0)T}=sign[∏i=2n‍(a-1+2g1ω-2b-1x1*)(c-i,i-1li-1++2giω-2b-ixi*)]=±1,k=1,2,…,2n.
So far, we have proved thatΩk  (k=1,2,…,2n) satisfies all the assumptions in Lemma 3.1. Hence, system (1.3) has at least 2n different ω-periodic solutions. This completes the proof of Theorem 3.5.Ifn=2, then (1.3) is changed into the predator-prey system (3.81). The following Corollary 3.6 gives the existence of multiple periodic solutions for system (3.81).Corollary 3.6.
For the following predator-prey system onω-periodic time scales 𝕋,
(3.81)u1Δ(t)=a1(t)-b1(t)eu1(t-τ11(t))-c12(t)eu2(t-τ12(t))-h1(t)e-u1t,t≠tk,u2Δ(t)=-d2(t)-b2(t)eu2(t-τ22(t))+c21(t)eu1(t-τ21(t))-h2(t)e-u2(t),t≠tk,Δui(tk)=ui(tk+)-ui(tk-)=ln(1+gik),i=1,2,t=tk
if the inequalities (a-1+(2g1/ω)-c-12l2+)e-(1/2)(a-1ω+G1)>2b-1h-1 and ((2g2/ω)-d-2)e-(1/2)(c-21ω+G2)>2b-2h-2 hold, then system (3.81) at least exist four ω-periodic solutions.


## 4. Illustrative Examples

Consider the following time-delay predator-prey system with impulses on time scales:(4.1)u1Δ(t)=a1(t)-b1(t)eu1(t-τ11(t))-c12(t)eu2(t-τ12(t))-h1(t)e-u1(t),u2Δ(t)=-d2(t)-b2(t)eu2(t-τ22(t))+c21(t)eu1(t-τ21(t))-h2(t)e-u2(t),Δx1(1)=-2,Δx1(3)=2,Δx2(1)=-1,Δx2(3)=3,
where a1(t)=(3+sin(0.5πt))/2, b1(t)=(2+sin(0.5πt))/10e4, h1(t)=(4+cos(0.5πt))/20e4, τ11(t)=|sin(0.5πt)|, τ12(t)=|cos(0.5πt)|, d2(t)=(2+cos(0.5πt))/25, b2(t)=(5+cos(0.5πt))/10, h2(t)=(2+cos(0.5πt))/25e2, c21(t)=(2+sin(πt))/8, τ22(t)=|cos(πt)|, τ21(t)=|sin(πt)|.If𝕋=ℝ, then (4.1) reduces to the following system:
(4.2)u˙1(t)=a1(t)-b1(t)eu1(t-τ11(t))-c12(t)eu2(t-τ12(t))-h1(t)e-u1(t),u˙2(t)=-d2(t)-b2(t)eu2(t-τ22(t))+c21(t)eu1(t-τ21(t))-h2(t)e-u2(t),Δx1(1)=-2,Δx1(3)=2,Δx2(1)=-1,Δx2(3)=3.
Let xi(t)=eui(t)  (i=1,2), then (4.2) can be changed into the following system:
(4.3)x˙1(t)=x1(t)(a1(t)-b1(t)x1(t-τ11(t))-c12(t)x2(t-τ12(t)))-h1(t),x˙2(t)=x2(t)(-d2(t)-b2(t)x2(t-τ22(t))+c21(t)x1(t-τ21(t)))-h2(t),Δx1(1)=e-2-1,Δx1(3)=e2-1,Δx2(1)=e-1-1,Δx2(3)=e3-1.
Since
(4.4)l1±=(a-1+(2g1/ω))ea-1ω+G1±((a-1+(2g1/ω))ea-1ω+G1)2-4b-1h-1ea-1ω+G12b-1=(15e8±3225e16-16)4e-4,l2±=(c-21+(2g2/ω))ec-21ω+G2±((c-21+(2g2/ω))ec-21ω+G2)2-4b-2h-2ec-21ω+G22b-2=15e4±225e8-6420e-2.
Take c12(t)=c12(t+2π)>0, such that c-12=1/l2+, then
(4.5)(a-1+2g1ω-c-12l2+)e-(1/2)(a-1ω+G1)=12e-4>25e-4=2b-1h-1,(2g2ω-d-2)e-(1/2)(c-21ω+G2)=2125e-2>25e-2=2b-2h-2.
Therefore, all conditions of Corollary 3.6 are satisfied. By Corollary 3.6, system (4.1) has at least four 4-periodic solutions.



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*Source: 769267-2012-12-30.xml*