# Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

Authors: Tursuneli Niyaz; Ahmadjan Muhammadhaji
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

This paper studies a class of periodicn species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established.


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## Body

## 1. Introduction

Mathematical ecological system has become one of the most important topics in the study of modern applied mathematics. Its dynamical behavior includes persistence, permanence and extinction of species, global stability of systems, the existence of positive periodic solutions, positive almost periodic solutions, and strictly positive solutions. The existence of positive periodic solutions has already become one of the most interesting subjects for scholars. In the recent years, the application of fixed-point theorems to the existence of positive periodic solutions in mathematical ecology has been studied extensively, for example, Brouwer’s fixed point theorem [1–4], Schauder’s fixed-point theorem [5–8], Krasnoselskii’s fixed-point theorem [9–14], Horn’s fixed-point theorem [15, 16], and Mawhins continuation theorem [17–36], and so forth. In particular, Mawhins continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with a time-delayed problem, it is very convenient and the result is relatively simple [30]. Recently, a considerable number of mathematical models with delays have been proposed in the study of population dynamics. One of the most celebrated models for population is the Lotka-Volterra system. Subsequently, a lot of the literature related to the study of the existence of positive periodic solutions for various Lotka-Volterra-type population dynamical systems with delays by using the method of continuation theorem was published and extensive research results were obtained [17–21, 24–34].On the other hand, in some situations, people may wish to change the position of the existing periodic solution but to keep its stability. This is of significance in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure [21]. In fact, during the last decade, the existence of positive periodic solutions for the population dynamics with feedback control has been studied extensively [8, 14, 17–21, 24, 29]. To the best of our knowledge, studies on the existence of positive periodic solutions for cooperative systems with delays and feedback controls are fairly rare.In [21], the authors studied the following neutral Lotka-Volterra system with feedback controls:
(1)y˙i(t)=yi(t)[ri(t)-∑j=1n‍aijyj(t)yi(t)-∑j=1n‍bijyj(t-τij(t))-∑j=1n‍cijy˙j(t-γij(t))yi(t)-fi(t)ui(t)-ei(t)ui(t-σi(t)){∑j=1n‍}]u˙i(t)=-αi(t)ui(t)+βi(t)yi(t)+γi(t)yi(t)(t-δi(t)),i=1,2,…,n.
By using Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are established. In [24], the authors considered the following delay differential system with feedback control:
(2)dxdt=F(t,x(t-τ1(t)),…,x(t-τn(t)),u(t-δ(t)))dudt=-η(t)+a(t)x(t-σ(t)).
A set of natural and easily verifiable sufficient conditions of the existence of positive periodic solutions are established, by using Mawhin’s continuation theorem. In [29], the authors considered the following single-species periodic logistic systems with feedback regulation and infinite distributed delay:
(3)N˙(t)=r(t)N(t)×(1-1K(t)∫0∞‍H(s)N(t-s)ds-c(t)u(t))u˙(t)=-a(t)u(t)+b(t)∫0∞‍H(s)N2(t-s)ds.
The sufficient conditions for the existence of positive periodic solutions are established, based on Mawhin’s continuation theorem.Motivated by the above works, in this paper, we investigate the followingn species periodic Lotka-Volterra-type cooperative systems with continuous time delays and feedback controls:
(4)x˙i(t)=xi(t)[ri(t)-∑l=1m‍aiil(t)xi(t-τiil(t))xi(t)+∑j≠in‍∑l=1m‍aijl(t)xj(t-τijl(t))]-di(t)ui(t)-ei(t)ui(t-εi(t))u˙i(t)=-bi(t)ui(t)+βi(t)xi(t)+γi(t)xi(t-σi(t)),i=1,2,…,n.
By using the technique of coincidence degree developed by Gaines and Mawhin in [36], we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution.


## 2. Preliminaries

In system (4), we have that xi(t)(i=1,2,…n) represent the density of n cooperative species xi(i=1,2,…n) at time t, respectively; ri(t)(i=1,2,…n) represent the intrinsic growth rate of species xi(i=1,2,…n) at time t, respectively; aiil(t)(i=1,2,…,n, l=1,2,…,m) represent the intrapatch restriction density of species xi(i=1,2,…n) at time t, respectively; aijl(t)(l=1,2,…,m, i≠j, i,j=1,2,…,n) represent the cooperative coefficients between n species xi(i=1,2,…n) at time t, respectively. ui(t)(i=1,2,…n) represent the indirect feedback control variables [21] at time t, respectively. βi(t), ei(t), bi(t), di(t), and γi(t)(i=1,2,…n) represent the feedback control coefficients at time t, respectively. In this paper, we always assume that(H1)
τijl(t)(l=1,2,…,m, i,j=1,2,…,n), σi(t),εi(t),ri(t)  (i=1,2,…n) are continuous ω-periodic functions with τijl′(t)<1 and ∫0ω‍ri(t)dt>0. aijl(t)(i,j=1,2,  l=1,2,…,m), βi(t), ei(t), bi(t), di(t), and γi(t)(i=1,2,…n) are continuous, positive ω-periodic functions.From the viewpoint of mathematical biology, in this paper for system (4) we only consider the solution with the following initial conditions:
(5)xi(t)=ϕi(t),∀t∈[-σ,0],i=1,2,…n,ui(t)=ψi(t),∀t∈[-σ,0],i=1,2,…n,
where ϕi(t),ψi(t)(i=1,2,…n) are nonnegative continuous functions defined on [-σ,0] satisfying ϕi(0)>0, ψi(0)>0(i=1,2,…n) with σ=maxt∈[0,ω]{τijl(t),σi(t),εi(t)(i,j=1,2, l=1,2,…,m)}.In this paper, for anyω-periodic continuous function f(t) we denote
(6)fL=mint∈[0,ω]f(t),fM=maxt∈[0,ω]f(t),f-=1ω∫0ω‍f(t)dt.In order to obtain the existence of positiveω-periodic solutions of system (4), we will use the continuation theorem developed by Gaines and Mawhin in [36]. For the reader’s convenience, we will introduce the continuation theorem in the following.LetX and Z be two normed vector spaces. Let L:DomL⊂X→Z be a linear operator and let N:X→Z be a continuous operator. The operator L is called a Fredholm operator of index zero, if dimKerL= codimImL<∞ and ImL is a closed set in Z. If L is a Fredholm operator of index zero, then there exist continuous projectors P:X→X and Q:Z→Z such that ImP= Ker L and ImL= KerQ= Im(I-Q). It follows that L∣DomL∩KerP: DomL∩KerP→ ImL is invertible and its inverse is denoted by KP; denote by J:ImQ→KerL an isomorphism of ImQ onto KerL. Let Ω be a bounded open subset of X; we say that the operator N is L-compact on Ω¯, where Ω¯ denotes the closure of Ω in X, if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact.Lemma 1 (see [35]).
Supposeτ∈C1(R,R) with τ(t+ω)≡τ(t) and τ′(t)<1, ∀t∈[0,ω]. Then the function t-τ(t) has a unique inverse function μ(t) satisfying μ∈C(R,R), μ(u+ω)=μ(u)+ω,∀u∈R.Lemma 2 (see [36]).
LetL be a Fredholm operator of index zero and let N be L-compact on Ω¯. If(a)
for eachλ∈(0,1) and x∈∂Ω∩DomL, Lx≠λNx;(b)
for eachx∈∂Ω∩KerL, QNx≠0;(c)
deg{JQN,Ω∩KerL,0}≠0,then the operator equation Lx=Nx has at least one solution lying in DomL∩Ω¯.


## 3. Main Results

In order to obtain the existence of positive periodic solutions of system (4), firstly, we introduce the following lemma.Lemma 3.
Suppose that(x1*(t),x2*(t),…,xn*(t), u1*(t), u2*(t),  …,un*(t)) is an ω-periodic solution of (4) and (5); then (x1*(t),x2*(t),…,  xn*(t),u1*(t),u2*(t),  …,un*(t)) satisfies the system
(7)x˙i(t)=xi(t)(ri(t)-∑j=12‍(-1)i+j∑l=1m‍aijl(t)xj(t-τijl(t)))-di(t)ui(t)-ei(t)ui(t-εi(t)),ui(t)=∫tt+ω‍[βi(s)xi(s)+γi(s)xi(s-σi(s))]Gi(t,s)ds,[βi(s)xi(s)+γi(s)xi(-σi(s))]Gi(t)i=1,2,…,n,
where
(8)Gi(t,s)=exp{∫ts‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1,i=1,2,…,n.
The converse is also true.Proof.
By (4), (5), and the variation constants formula in ordinary differential equations, we have
(9)ui(t)=(∫0t‍(βi(s)xi(s)+γi(s)xi(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))×exp{-∫0t‍bi(θ)dθ},i=1,2,…,n.
From (9), we obtain
(10)ui(t+ω)=(∫0t+ω(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0ω‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))×exp{-∫0t+ω‍bi(θ)dθ},i=1,2,…,n,ui*(t)=(∫0t‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))×exp{-∫0t‍bi(θ)dθ},i=1,2,…,n.
Considering that (x1*(t),  x2*(t),…,xn*(t), u1*(t),   u2*(t),   …,un*(t)) is an ω-periodic solution of system (4) and (5), we obtain
(11)(∫0t‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))exp{-∫0t‍bi(θ)dθ}=(∫0t+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0ω‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))×exp{-∫0t+ω‍bi(θ)dθ},i=1,2,…,n.
Then
(12)(∫0t‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))exp{∫0ω‍bi(θ)dθ}=(∫0t+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0ω‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))=∫0t‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0)+∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0ω‍×exp{∫0s‍bi(θ)dθ}ds,i=1,2,…,n,
which implies
(13)(∫0t‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫0t‍×exp{∫0s‍bi(θ)dθ}ds+ψi(0))(eb-ω-1)=∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫tω‍×exp{∫0s‍bi(θ)dθ}ds,i=1,2,…,n.
That is,
(14)∫0t‍(βi(s)xi*(s)+γi(i)xi*(s-σi(s)))×exp{∫0s‍bi(θ)dθ}ds+ψi(0)=ui*(t)exp{∫0t‍bi(θ)dθ}=1eb-ω-1∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))1eb-ω-1∫tω‍×exp{∫0s‍bi(θ)dθ}ds,i=1,2,…,n.
Hence
(15)ui*(t)=∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫tω‍×exp{∫ts‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1ds,i=1,2,…,n.
On the other hand, assume that (x1*(t),x2*(t), …,xn*(t), u1*(t),u2*(t), …,un*(t)) is an ω-periodic solution of system (7), then
(16)ui*(t)=∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))∫tω‍×exp{∫ts‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1ds,i=1,2,…,n.
By a direct calculation, we have
(17)du*(t)dt=-bi(t)∫tt+ω‍(βi(s)xi*(s)+γi(s)xi*(s-σi(s)))-bi(t)∫tω‍×exp{∫ts‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1ds+({(t+ω-σi(t+ω))}βi(t+ω)xi*(t+ω)+γi(t+ω)xi*(t+ω-σi(t+ω)))×exp{∫tt+ω‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1-(βi(t)xi*(t)+γi(t)xi*(t-σi(t)))×1exp{∫0ω‍bi(θ)dθ}-1=-bi(t)ui(t)+βi(t)xi*(t)+γi(t)xi*(t-σi(t)),=-bi(t)ui(t)+βi(t)xi*(t)+γi(t)xi*i=1,2,…,n.
This completes the proof.It is easy to see that system (7) is equivalent to the following system:
(18)x˙i(t)=xi(t)(ri(t)-∑j=12‍(-1)i+j∑l=1m‍aijl(t)xj(t-τijl(t)))-di(t)ui(t)-ei(t)ui(t-εi(t)),ui(t)=∫tt+ω‍K(xi)Gi(t,s)ds,i=1,2,…,n,
where
(19)s-σi(s)Gi(t,s)=exp{∫ts‍bi(θ)dθ}exp{∫0ω‍bi(θ)dθ}-1,K(xi)=βi(s)xi(s)+γi(s)xi(s-σi(s)),K(xi)=(s)xi(s)+γi(s)xii=1,2,…,n.
It is clear that in order to prove that systems (4) and (5) have at least one ω-periodic solution, we only need to prove that system (18) has at least one ω-periodic solution.Now, for convenience of statements we denote the functions(20)aij(t)=∑l=1m‍aijl(t),i,j=1,2,…,n.
The following theorem is about the existence of positive periodic solutions of system (4).Theorem 4.
Suppose that assumption (H1) holds and there exists a constantθi>0,  ζi>0, i=1,2,…,n, such that
(21)mint∈[0,ω]{∑l=1m‍[δiil(t)θi-∑j≠in‍δjil(t)θj]}=:ζi>0,i=1,2,…,n,
where
(22)δijl(t)=aijl(φijl(t))1-τijl′(φijl(t)),i,j=1,2,…,n,l=1,2,…,m,
and the algebraic equation
(23)r-i-H-i-a-iivi+∑j≠in‍a-ijvj=0,i=1,2,…,n,
where
(24)Hi=∫0ω‍(βi(s)+γi(s))∫0ω‍×(di(t)Gi(t,s)+ei(t)Gi(t-εi(t),s))ds,di(t)Gi(t,s)+ttt-εiei(t)Gi∫0ω‍i=1,2,…,n,
has a unique positive solution. Then system (4) has at least one positive ω-periodic solution.Proof.
For system (18) we introduce new variables yi(t)(i=1,2,…,n) such that
(25)xi(t)=exp{yi(t)},i=1,2,…,n.
Then system (18) is rewritten in the following form:
(26)y˙i(t)=ri(t)-∑j=12‍(-1)i+j∑l=1m‍aijl(t)exp{yj(t-τijl(t))}y˙i(t)=-exp{-yi(t)}di(t)Ui(t),y˙i(t)=-exp{-yi(t)}ei(t)Ui(t-εi(t))Ui(t)=∫tt+ω‍K(eyi)Gi(t,s)ds,i=1,2,
where
(27)K(eyi)=βi(s)exp{yi(s)}+γi(s)exp{yi(s-σi(s))},i=1,2.
In order to apply Lemma 2 to system (26), we introduce the normed vector spaces X and Z as follows. Let C(R,Rn) denote the space of all continuous functions  y(t)=(y1(t),y2(t),…,yn(t)):R→Rn. We take
(28)X=Z={y(t)∈C(R,Rn):y(t)is  anω-periodicfunction},
with norm
(29)∥y∥=∑i=1n‍maxt∈[0,ω]|yi(t)|.
It is obvious that X and Z are the Banach spaces. We define a linear operator L: DomL⊂X→Z and a continuous operator N:X→Z as follows:
(30)Ly(t)=y˙(t),Ny(t)=(Ny1(t),Ny2(t),…,Nyn(t)),
where
(31)Nyi(t)=ri(t)-∑l=1m‍aiil(t)exp{yi(t-τiil(t))}+∑j≠in‍∑l=1m‍aijl(t)exp{yj(t-τijl(t))}-exp{-yi(t)}di(t)ui(t)-exp{-yi(t)}×ei(t)ui(t-εi(t)).
Further, we define continuous projectors P:X→X and Q:Z→Z as follows:
(32)Py(t)=1ω∫0ω‍y(t)dt,Qv(t)=1ω∫0ω‍v(t)dt.
We easily see ImL={v∈Z:∫0ω‍v(t)dt=0} and KerL=Rn. It is obvious that ImL is closed in Z and dimKerL=n. Since for any v∈Z there are unique v1∈Rn and v2∈ImL with
(33)v1=1ω∫0ω‍v(t)dt,v2(t)=v(t)-v1,
such that v(t)=v1+v2(t), we have codimImL=n. Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp: ImL→KerP∩DomL is given in the following form:
(34)Kpv(t)=∫0t‍v(s)ds-1ω∫0ω‍∫0t‍v(s)dsdt.
For convenience, we denote F(t)=(F1(t),F2(t),…,Fn(t)) as follows:
(35)Fi(t)=ri(t)-∑l=1m‍aiil(t)exp{yi(t-τiil(t))}+∑j≠in‍∑l=1m‍aijl(t)exp{yj(t-τijl(t))}-exp{-yi(t)}di(t)ui(t)-exp{-yi(t)}×ei(t)ui(t-εi(t)).
Thus, we have
(36)QNy(t)=1ω∫0ω‍F(t)dt,Kp(I-Q)Nu(t)=KpINu(t)-KpQNu(t)=∫0t‍F(s)ds-1ω∫0ω‍∫0t‍F(s)dsdt+(12-tω)∫0ω‍F(s)ds.
From formulas (36), we easily see that QN and Kp(I-Q)N are continuous operators. Furthermore, it can be verified that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X by using the Arzela-Ascoli theorem and QN(Ω¯) is bounded. Therefore, N is L-compact on Ω¯ for any open bounded subset Ω⊂X.
Now, we reach the position to search for an appropriate open bounded subsetΩ for the application of the continuation theorem (Lemma 2) to system (26).
Corresponding to the operator equationLy(t)=λNy(t) with parameter λ∈(0,1), we have
(37)y˙i(t)=λFi(t),i=1,2,…,n,
where Fi(t)(i=1,2,…,n) are given in (35).
Assume thaty(t)=(y1(t),y2(t),…,yn(t))∈X is a solution of system (37) for some parameter λ∈(0,1). By integrating system (37) over the interval [0,ω], we obtain
(38)∫0ω[ri(t)-∑l=1m‍aiil(t)exp{yi(t-τiil(t))}∫0ω‍+∑j≠in‍∑l=1m‍aijl(t)exp{yj(t-τijl(t))}-ei(t)exp{-yi(t)}∫t-εi(t)t-εi(t)+ω‍K(eyi)Gi(t-εi(t),s)ds-di(t)exp{-yi(t)}∫tt+ω‍K(eyi)Gi(t,s)ds{∑l=1m‍}]dt=0,-di(t)exp{-yi(t)}∫tt+ω‍K(eyi)Gi(t,s)dsi=1,2,…,n.
Consequently,
(39)r-iω+∫0ω‍[∑j≠in‍∑l=1m‍aijl(t)exp{yj(t-τijl(t))}]dt=∫0ω‍[∑l=1m‍aiil(t)exp{yi(t-τiil(t))}]dt+∫0ω‍ei(t)exp{-yi(t)}×∫t-εi(t)t-εi(t)+ω‍K(eyi)Gi(t-εi(t),s)dsdt+∫0ω‍di(t)exp{-yi(t)}∫tt+ω‍K(eyi)Gi(t,s)dsdt,+∫0ω‍di(t)exp{-yi(t)}∫tt+ω‍K(eyi)i=1,2,…,n.
From the continuity of y(t)=(y1(t),y2(t),…,yn(t)), there exist constants ξi,ηi∈[0,ω](i=1,2,…,n) such that
(40)yi(ξi)=maxt∈[0,ω]yi(t),yi(ηi)=mint∈[0,ω]yi(t)i=1,2,…,n.
By (39) and (40) we obtain
(41)r-iω≤∫0ω‍aii(t)exp{yi(ξi)}dt+∫0ω‍Ai(t)exp{yi(ξi)}dt,i=1,2,…,n,
where
(42)Ai(t)=ei(t)∫t-εi(t)t-εi(t)+ω‍Gi(t-εi(t),s)(βi(s)+γi(s))ds+di(t)∫tt+ω‍Gi(t,s)(βi(s)+γi(s))ds,+di(t)βi(s)∫tt+ω‍Gi(t,s)i=1,2,…,n.
Therefore, we further have
(43)yi(ξi)≥ln(r-ia-ii+A-i),i=1,2,…,n.
Let sijl(t)=t-τijl(t)(i,j=1,2,…,n,  l=1,2,…,m); then from Lemma 1 and (H1) we get that function sijl(t) has a unique ω periodic inverse function φijl(t); then, for every i,j=1,2,…,n,l=1,2,…,m, we have
(44)∫0ω‍aijl(t)exp{yi(t-τijl(t))}dt=∫-τijl(0)ω-τijl(ω)‍aijl(φijl(t))1-τijl′(φijl(t))exp{yi(t)}dt.
One can see that
(45)aijl(φijl(t))1-τijl′(φijl(t))=:δijl(t),i,j=1,2,…,n,l=1,2,…,m,
are ω periodic functions. Then, for every i,j=1,2,…,n,l=1,2,…,m, we have
(46)∫0ω‍aijl(t)exp{yi(t-τijl(t))}dt=∫0ω‍δijl(t)exp{yi(t)}dt.
From (39) and (46) we further obtain
(47)∑i=1n‍(∫0ω‍(∑l=1m‍δiil(t)exp{yi(t)}∑i=1n‍∫0ω‍+di(t)exp{-yi(t)}∫tt+ω‍K(eyi)Gi(t,s)ds∑i=1n‍∫0ω‍+ei(t)exp{-yi(t)}∑i=1n‍∫0ω‍×∫t-εi(t)t-εi(t)+ω‍K(eyi)Gi(t-εi(t),s)ds)dt)=∑i=1n‍(r-iω+∫0ω‍[∑j≠in‍∑l=1m‍δijl(t)exp{yj(t)}]dt).
From the above equality we have
(48)∑i=1n‍(∫0ω‍[∑l=1m‍δiil(t)-∑j≠in‍∑l=1m‍δjil(t)]exp{yi(t)}dt)≤∑i=1n‍r-iω,∫0ω‍[∑l=1m‍δiil(t)-∑j≠in‍∑l=1m‍δjil(t)]exp{yi(t)}dt≤∑i=1n‍r-iω,i=1,2,…,n.
From the assumptions of Theorem 4 and (48) we can obtain
(49)ζi∫0ω‍exp{yi(t)}dt≤∑i=1n‍r-iω,i=1,2,…,n.
Consequently,
(50)∫0ω‍exp{yi(t)}dt≤∑i=1nr-iωζi,i=1,2,…,n.
From (40) and (50), we further obtain
(51)yi(ηi)≤ln(∑i=1nr-iζi),i=1,2,…,n.
On the other hand, directly from system (26) we have
(52)∫0ω‍|y˙i(t)|dt≤∫0ω|ri(t)|dt+∫0ω‍(∑l=1m‍aiil(t)exp{yi(t-τiil(t))}+∑j≠in‍∑l=1m‍aijl(t)exp{yj(t-τijl(t))})dt≤∫0ω‍|ri(t)|dt+∫0ω‍∑l=1m‍(δiil(t)+∑j≠in‍δjil(t))exp{yi(t)}dt≤∫0ω‍|ri(t)|dt+DiM∫0ω‍exp{yi(t)}dt≤|r-i|ω+DiM∑i=1nr-iωζi,i=1,2,…,n,
where
(53)Di(t)=∑l=1m‍(δiil(t)+∑j≠in‍δjil(t)),i=1,2,…,n.
From (43), (51), and (52), we have for any t∈[0,ω](54)yi(t)≤yi(ηi)+∫0ω‍|y˙i(t)|dt≤ln(∑i=1nr-iζi)+|r-i|ω+DiM∑i=1nr-iωζi=:Mi,ln(∑i=1n‍r-iζi)+|r-i|+|r-i|ω+i=1,2,…,n,(55)yi(t)≥yi(ξi)-∫0ω‍|y˙i(t)|dt≥ln(r-ia-ii+A-i)-|r-i|ω-DiM∑i=1nr-iωζi=:Ni,(r-ia-ii+A-i)(r-iA-i)(r-ia-ii+A-i)i=1,2,…,n.
Therefore, from (54) and (55), we have
(56)maxt∈[0,ω]|yi(t)|≤max{|Mi|,|Ni|}=:Bi,i=1,2,…,n.
It can be seen that the constants Bi(i=1,2,…,n) are independent of parameter λ∈(0,1). For any y=(y1,y2,…,yn)∈Rn, from (31) we can obtain
(57)QNy=(QNy1,QNy2,…,QNyn)QNyi=r-i-H-i-a-iiexp{yi}+∑j≠in‍a-ijexp{yj},r-i-H-iH-i-a-iiexp{yi}+i=1,2,…,n,Hi(t)=∫0ω‍(βi(s)+γi(s))(di(t)Gi(t,s))+ei(t)Gi(t-εi(t),s)ds,i=1,2,…,n.
We consider the following algebraic equation:
(58)r-i-H-i-a-iivi+∑j≠in‍a-ijvj=0,i=1,2,…,n.
From the assumption of Theorem4, the equation has a unique positive solution v*=(v1*,v2*,…,vn*). Hence, the equation QNy=0 has a unique solution y*=(y1*,y2*,…,yn*)=(lnv1*,lnv2*,…,lnvn*)∈Rn.
Choosing constantB>0 large enough such that |y1*|+|y2*|+⋯+|yn*|<B and B>B1+B2+⋯+Bn, we define a bounded open set Ω⊂X as follows:
(59)Ω={y∈X:∥y∥<B}.
It is clear that Ω satisfies conditions (a) and (b) of Lemma 2. On the other hand, by directly calculating we can obtain
(60)deg{JQN,Ω∩KerL,(0,0,…,0)}=sgn|fy11fy21⋯fyn1fy12fy22⋯fyn2⋯⋯⋯⋯fy1nfy2n⋯fynn|,
where
(61)fyji=-a-ijexp{yj*},i=jfyji=a-ijexp{yj*},i≠ji,j=1,2,…,n.
From the assumption of Theorem 4, we have
(62)|fy11fy21⋯fyn1fy12fy22⋯fyn2⋯⋯⋯⋯fy1nfy2n⋯fynn|≠0.
From this, we finally have
(63)deg{JQN,Ω∩KerL,(0,0,…,0)}≠0.
This shows that Ω satisfies condition (c) of Lemma 2. Therefore, system (26) has an ω-periodic solution y*(t)=(y1*(t),y2*(t),…,yn*(t))∈Ω-. Finally, we have system that (4) has a positive ω-periodic solution. This completes the proof.



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*Source: 502963-2013-05-30.xml*