# Existence of Positive Periodic Solutions for Periodic Neutral Lotka-Volterra System with Distributed Delays and Impulses

Authors: Zhenguo Luo; Liping Luo
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

By using a fixed-point theorem of strict-set-contraction, we investigate the existence of positive periodic solutions for a class of the following impulsive neutral Lotka-Volterra system with distributed delays:xi′(t)=xi(t)[ri(t)-∑j=1naij(t)xj(t)-∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ-∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ],Δxi(tk)=-Iik(xi(tk)),i=1,2,…,n,k=1,2,…. Some verifiable criteria are established easily.


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## Body

## 1. Introduction

It is well known that natural environments are physically highly variable, and in response, birth rates, death rates, and other vital rates of populations vary greatly in time. Theoretical evidence suggests that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see [1–4]). Thus, the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in response to the physical environment. It is reasonable to study the models of population with periodic coefficients. In addition to the theoretical and practical significance, the Lotka-Volterra model is one of the famous models for dynamics of population; therefore it has been studied extensively [5–9]. In view of the above effects, by applying a fixed-point theorem of strict-set-contraction, Li [10] established criteria to guarantee the existence of positive periodic solutions of the following neutral Lotka-Volterra system with distributed delays:
(1)xi′(t)=xi(t)[ai(t)-∑j=1n‍bij(t)∫-Tij0‍Kij(θ)xj(t+θ)dθ-∑j=1n‍cij(t)∫-T^ij0‍K^ij(θ)xj′(t+θ)dθ],hhhhhhhhhhhhhhhhhhhhhi=1,2,…,n,
where ai,bij,cij∈C(R,R+)(i,j=1,2,…,n) are ω-periodic functions and Tij,T^ij∈(0,+∞)(i,j=1,2,…,n) and Kij,K^ij∈(R,R+) satisfying ∫-Tij0‍Kij(θ)dθ=1, ∫-T^ij0‍K^ij(θ)dθ,  i,j=1,2,…,n. On the other side, birth of many species is an annual birth pulse or harvesting. To have a more accurate description of many mathematical ecology systems, we need to consider the use of impulsive differential equations [11–13]. Some qualitative properties such as oscillation, periodicity, asymptotic behavior, and stability properties have been investigated extensively by many authors over the past few years [14–18]. However, to our knowledge, there are few published papers discussing the existence of periodic solutions for neutral Lotka-Volterra system with distributed delays and impulses. In this paper, we are concerned with the following neutral Lotka-Volterra system with distributed delays and impulses:
(2)xi′(t)=xi(t)[ri(t)-∑j=1n‍aij(t)xj(t)-∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ-∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ],i=1,2,…,n,t≠tk,Δxi(tk)=-Iik(xi(tk)),i=1,2,…,n,k=1,2,…,
where ri,aij,bij,cij∈C(R,R+)(i,j=1,2,…,n) are ω-periodic functions and τij,σij∈(0,+∞)(i,j=1,2,…,n) and fij,gij∈C(R,R+) satisfying ∫-τij0‍fij(ξ)dξ=1, ∫-σij0‍gij(ξ)dξ=1,i,j=1,2,…,n. Moreover, Δx(tk)=x(tk+)-x(tk) (here x(tk+) represents the right limit of x(t) at the point tk), Iik∈C(R+,R+); that is, x changes decreasingly suddenly at times tk;  ω>0 is a constant, R=(-∞,+∞), R+=[0,+∞). We assume that there exists an integer q>0 such that tk+q=tk+ω,Ii(k+q)=Iik, where 0<t1<t2<⋯<tq<ω.The main purpose of this paper is by using a fixed-point theorem of strict-set-contraction [19, 20] to establish new criteria to guarantee the existence of positive periodic solutions of the system (2).For convenience, we introduce the notation(3)fM=maxt∈[0,ω]{|f(t)|},fL=mint∈[0,ω]{|f(t)|},γi=limu→0sup∑t≤tk≤t+ω‍Iik(u)u,i=1,2,…,n,δi=e-∫0ω‍ri(t)dt,i=1,2,…,n,αij=∫0ω‍[δjaij(t)-bij(t)-cij(t)]dt,βij=∫0ω‍[aij(t)+bij(t)+cij(t)]dt,i,j=1,2,…,n,
where f is a continuous ω-periodic function.Throughout this paper, we assume that(A1)
δi=e-∫0ω‍ri(t)dt<1,i=1,2,…,n;(A2)
δjaij(t)-bij(t)-cij(t)≥0,i,j=1,2,…,n;(A3)
(1+riL)(δi2/(1-δi))αij≥maxt∈[0,ω]{aij(t)+bij(t)+cij(t)},i,j=1,2,…,n;(A4)
(riM-1)(1/δi(1-δi))βij≤mint∈[0,ω]{δjaij(t)-bij(t)-cij(t)},i,j=1,2,…,n;(A5)
max1≤i≤n{∑j=1ncijM}≤min1≤i≤n{(δi2/(1-δi))min1≤j≤n{αij}}.The paper is organized as follows. In Section2, we give some definitions and lemmas to prove the main results of this paper. In Section 3, by using a fixed-point theorem of strict-set-contraction, we established some criteria to guarantee the existence of at least one positive periodic solution of system (2). Finally, in Section 4, we give an example to show the validity of our result.


## 2. Preliminaries

In order to obtain the existence of a periodic solution of system (2), we first introduce some definitions and lemmas.Definition 1 (see [13]).
A functionxi:R→(0,+∞) is said to be a positive solution of (2), if the following conditions are satisfied:(a)
xi(t) is absolutely continuous on each (tk,tk+1);(b)
for eachk∈Z+, xi(tk+) and xi(tk-) exist and xi(tk-)=xi(tk);(c)
xi(t) satisfies the first equation of (2) for almost everywhere in R and xi(tk) satisfies the second equation of (2) at impulsive point tk,k∈Z+.Definition 2 (see [21]).
LetX be a real Banach space and E a closed, nonempty subset of X. E is a cone provided(i)
αx+βy∈E for all x,y∈E and all α,β≥0,(ii)
x,-x∈E imply x=0.Definition 3 (see [21]).
LetA be a bounded subset in X. Define
(4)αX(A)=inf{⋃i‍δ>0:thereisafinitenumberofsubsetsAi⊂AsuchthatA=⋃i‍Aianddiam(Ai)≤δ},
where diam(Ai) denotes the diameter of the set Ai, obviously,  0≤αX(A)<∞. So αX(A) is called the Kuratowski measure of noncompactness of  X.Definition 4 (see [21]).
LetX,Y be two Banach spaces and D⊂X, a continuous and bounded map T:D→Y, is called k-set contractive if, for any bounded set S⊂D, we have
(5)αY(T(S))≤kαX(S);ϕis called strict-set-contractive if it is k-set-contractive for some 0≤k<1.Definition 5 (see [22]).
The setF∈PCω is said to be quasi-equicontinuous in [0,ω] if for any ϵ>0 there exists δ>0 such that if x∈F,  k∈N+,  t1,t2∈(tk-1,tk)∩[0,ω],  and |t1-t2|<δ, then |x(t1)-x(t2)|<ϵ.Lemma 6 (see [22]).
The setF⊂PCω is relatively compact if and only if(1)
F is bounded; that is, ∥x∥≤M,  for each x∈F, and some M>0;(2)
F is quasi-equicontinuous in [0,ω].Lemma 7.
x
i
(
t
) is anω-periodic solution of (2)  is equivalent to xi(t) is anω-periodic solution of the following equation:
(6)xi(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk)),
where
(7)Gi(t,s)=e-∫ts‍ri(θ)dθ1-e-∫0ω‍ri(θ)dθ,s∈[t,t+ω],i=1,2,…,n.Proof.
Assume thatxi(t)∈X,i=1,2,…,n, is a periodic solution of (2). Then, we have
(8)ddt[xi(t)e-∫0t‍ri(θ)dθ]=e-∫0t‍ri(θ)dθxi(t)×[-∑j=1n‍aij(t)xj(t)-∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ-∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ],hhhhhhit≠tk,i=1,2,…,n.
Integrating the previous equation with [t,t+ω], we can have
(9)xi(s)e-∫0s‍ri(θ)dθ|ttm1+nω+xi(s)e-∫0s‍ri(θ)dθ|tm1+nωtm2+nω+⋯+xi(s)e-∫0s‍ri(θ)dθ|tmq+nωt+ω=∫tt+ω‍e-∫0s‍ri(θ)dθxi(s)×[-∑j=1n‍aij(s)xj(s)-∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ-∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds,
where tmk+nω∈(t,t+ω), mk∈{1,2,…,q}, k=1,2,…,q, n∈Z+. Therefore,
(10)xi(t)e-∫0t‍ri(θ)dθ[1-e-∫tt+ω‍ri(θ)dθ]+∑t≤tk<t+ω‍Δxi(tmk)e-∫0tmk+nω‍ri(θ)dθ=∫tt+ω‍e-∫0s‍ri(θ)dθxi(s)×[∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds,
which can be transformed into
(11)xi(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk)),i=1,2,…,n.
Thus,xi is a periodic solution of (6). If xi(t)∈E,i=1,2,…,n, is a periodic solution of (6), for any t=tk,  from (6) we have
(12)x′(t)=ddt{∑j=1n‍cij(s)∫-σij0‍∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds}=[+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ)Gi(t,t+ω)xi(t+ω)×(∑j=1n‍aij(t+ω)xj(t+ω)+∑j=1n‍bij(t+ω)∫-τij0‍fij(ξ)xj(t+ω+ξ)dξ+∑j=1n‍cij(t+ω)∫-σij0‍gij(ξ)xj′(t+ω+ξ)dξ)-Gi(t,t)xi(t)×(∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)×∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ)]+ri(t)xi(t)=xi(t)[ri(t)-∑j=1n‍aij(t)xj(t)-∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t-ξ)dξ-∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ].
For any t=tj, j∈Z+, we have from (6) that
(13)xi(tj+)-xi(tj)=∫tjtj+ω‍[Gi(tj+,s)-Gi(tj,s)]xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σj0gij(ξ)xj′(s+ξ)dξ]ds+∑tj+≤tk<tj+ω‍Gi(tj+,tk)Iik(xi(tk))-∑tj≤tk<tj+ω‍Gi(tj,tk)Iik(xi(tk))=-Iik(x(tk)).
Hencexi(t) is a positive ω-periodic solution of (2). Thus we complete the proof of Lemma 7.Lemma 8 (see [20, 23]).
LetE be a cone of the real Banach space X and Er,R={x∈E:r≤∥x∥≤R} with 0<r<R. Assume that A:Er,R→E is strict-set-contractive such that one of the following two conditions is satisfied:(a)
Ax≰x,  for all  x∈E,∥x∥=r and Ax≱x,  for all  x∈E,∥x∥=R;(b)
Ax≱x,  for all  x∈E,∥x∥=r and Ax≰x,  for all  x∈E,∥x∥=R.
ThenA has at least one fixed point in Er,R.
In order to apply Lemma7 to system (1), one sets
(14)PC(R)={x=(x1,x2,…,xn)T:R⟶R∣x∈C((tk,tk+1),R),∃x(tk-)=x(tk),x(tk+),k∈Nx=(x1,x2,…,xn)T},PC1(R)={∃x′(tk-)=x′(tk),x(tk+)x=(x1,x2,⋯,xn)T:R⟶R∣x∈C1((tk,tk+1),R),∃x′(tk-)=x′(tk),x(tk+),k∈N,t∈R}.
Define
(15)X={x=(x1,x2,…,xn)T:x∈PC(R)∣x(t+ω)=x(t)}
with the norm defined by ∥x∥0=∑i=1n|xi|0, where |xi|0=supt∈[0,ω]{|xi(t)|},i=1,2,…,n, and
(16)Y={x=(x1,x2,…,xn)T:x∈PC1(R)∣x(t+ω)=x(t),t∈R(x1,x2,…,xn)T}
with the norm defined by ∥x∥1=∑i=1n|xi|1, where |xi|1=max{|xi|0,|xi′|0}. Then X and Y are both Banach spaces. Define the cone E in Y by
(17)E={x=(x1,x2,…,xn)T:x∈PC1(R)∣xi(t)=δj|xi(t)|1,t∈[0,ω],i,j=1,2…,n(x1,x2,…,xn)T}.
Let the map T be defined by
(18)(Tx)(t)=((T1x)(t),(T2x)(t),…,(Tnx)(t))T,
where x∈E, t∈R,
(19)(Tix)(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ∑j=1n‍aij(s)xj(s)]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk)),
where
(20)Gi(t,s)=e-∫ts‍ri(θ)dθ1-e-∫0ω‍ri(θ)dθ,s∈[t,t+ω],i=1,2,…,n.
It is obvious to see that Gi(t+ω,s+ω)=Gi(t,s),∂Gi(t,s)/∂t=ri(t)Gi(t,s),Gi(t,t+ω)-Gi(t,t)=-1, and
(21)δi1-δi≤Gi(t,s)≤11-δi,s∈[t,t+ω],i=1,2,…,n.In what follows, we will give some lemmas concerningE and A defined by (17) and (18), respectively.Lemma 9.
Assume that(A1)–(A3) hold.(i)
Ifmax{riM,i=1,2,…,n}≤1, then T:E→E is well defined.(ii)
If(A4) holds and max{riM,i=1,2,…,n}>1, then T:E→E is well defined.Proof.
For anyx∈E, it is clear that Tx∈PC1(R). From (18), for t∈[0,ω], we have
(22)(Tix)(t+ω)=∫t+ωt+2ω‍Gi(t+ω,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t+ω≤tk<t+2ω‍Gi(t+ω,tk)Iik(xi(tk))=∫tt+ω‍Gi(t+ω,u+ω)xi(u+ω)×[∑j=1n‍aij(u+ω)xj(u+ω)+∑j=1n‍bij(u+ω)×∫-τij0‍fij(ξ)xj(u+ω+ξ)dξ+∑j=1n‍cij(u+ω)∫-σij0‍gij(ξ)xj′(u+ω+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))=∫tt+ω‍Gi(t,u)xi(u)×[∑j=1n‍aij(u)xj(u)+∑j=1n‍bij(u)∫-τij0‍fij(ξ)xj(u+ξ)dξ+∑j=1n‍cij(u)∫-σij0‍gij(ξ)xj′(u+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))=(Tix)(t),i=1,2,…,n.
That is,(Tix)(t+ω)=(Tix)(t),t∈[0,ω]. So Tx∈Y. In view of (A2), for x∈E,t∈[0,ω], we have
(23)∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ≥∑j=1n‍aij(t)δj|xj|1-∑j=1n‍bij(t)|xj|1-∑j=1n‍cij(t)|xj|1=∑j=1n‍[aij(t)δj-bij(t)-cij(t)]|xj′|1≥0,i=1,2,…,n.
Therefore, forx∈E,t∈[0,ω], we find
(24)|Tix|0≤11-δi{∑t≤tk<t+ω‍∫0ω‍xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)×∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Iik(xi(tk))},(25)(Tix)(t)≥δi1-δi{∑t≤tk<t+ω‍∫tt+ω‍xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Iik(xi(tk))}=δi1-δi×{+∑t≤tk<t+ω‍Iik(xi(tk))∫0ω‍xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Iik(xi(tk))}≥δi|Tix|0,i=1,2,…,n.
Now, we show that (Tix)(t)≥δi|Tix|1, i=1,2,…,n, t∈[0,ω]. From (18), we obtain
(26)(Tix)′(t)=Gi(t,t+ω)xi(t+ω)×[∑j=1n‍aij(t+ω)xj(t+ω)+∑j=1n‍bij(t+ω)∫-τij0‍fij(ξ)xj(t+ω+ξ)dξ+∑j=1n‍cij(t+ω)∫-σij0‍gij(ξ)xj′(t+ω+ξ)dξ]-Gi(t,t)xi(t)×[∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)×∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]+ri(t)(Tix)(t)=ri(t)(Tix)(t)-xi(t)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ],hhhhhhhhhhhhhhhhhhhhhhi=1,2,…,n.
It follows from (23) and (25) that if (Tix)′(t)≥0,i=1,2,…,n, then
(27)(Tix)′(t)≤ri(t)(Tix)(t)≤riM(Tix)(t)≤(Tix)(t),hhhhhhhhhhhhhhhhhhhhhhhhihhhhi=1,2,…,n.
On the other hand, from (25) and (A3), if (Tix)′(t)<0,i=1,2,…,n, then
(28)-(Tix)′(t)=-ri(t)(Tix)(t)+xi(t)×[∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]≤|xi|1∑j=1n‍[aij(t)+bij(t)+cij(t)]|xi|1-riL(Tix)(t)≤(1+riL)δi21-δi|xi|1×∑j=1n‍{∫0ω‍[δjaij(s)-bij(s)-cij(s)]|xi|1ds}-riL(Tix)(t)=(1+riL)∫tt+ω‍δi1-δiδi|xi|1×∑j=1n‍[δjaij(s)-bij(s)-cij(s)]|xi|1ds-riL(Tix)(t)≤(1+riL){∑t≤tk<t+ω‍∫tt+ω‍∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)×∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))}-riL(Tix)(t)=(1+riL)(Tix)(t)-riL(Tix)(t)=(Tix)(t),i=1,2,…,n.
It follows from (27) and (28) that |(Tix)′|0≤|Tix|0,i=1,2,…,n. So |Tix|1=|Tix|0,i=1,2,…,n. By (25), we have (Tix)(t)≥δi|Tix|1,i=1,2,…,n. Hence, Tx∈E. This completes the proof of (i). In view of the proof of (ii), we only need to prove that (Tix)′(t)≥0,i=1,2,…,n, implies
(29)(Tix)′(t)≤(Tix)(t),i=1,2,…,n.
From (23), (26),  (A2), and (A4), we have
(30)(Tix)′(t)=ri(t)(Tix)(t)-xi(t)×[∑j=1n‍aij(t)xj(t)-∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ-∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]≤ri(t)(Tix)(t)-δi|xi|1×∑j=1n‍[δjaij(t)-bij(t)-cij(t)]|xj|1≤riM(Tix)(t)-δi|xi|1riM-1δi(1-δi)×∑j=1n‍{∫0ω‍[aij(s)+bij(s)+cij(s)]|xj|1ds}=riM(Tix)(t)-(riM-1)∫tt+ω‍11-δi|xi|1×∑j=1n‍[aij(s)+bij(s)+cij(s)]|xj|1ds≤riM(Tix)(t)-(riM-1)×{∑t≤tk<t+ω‍∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)×∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))}=riM(Tix)(t)-(riM-1)(Tix)(t)=(Tix)(t),hhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,…,n.
The proof of (ii) is complete. Thus we complete the proof of Lemma 9.Lemma 10.
Assume that(A1)–(A3) hold and Rmax1≤j≤n{∑j=1nbijM}<1.(i)
Ifmax{riM,i=1,2,…,n}≤1, then T:E⋂‍ΩR¯→E is strict-set-contractive.(ii)
If(A4) holds and min{riM,i=1,2,…,n}>1, then T:E⋂‍ΩR¯→E is strict-set-contractive, where ΩR={x∈Y:|x|1<R}.Proof.
We only need to prove (i), since the proof of (ii) is similar. It is easy to see thatT is continuous and bounded. Now we prove that a αY(T(S))≤Rmax1≤j≤n{∑j=1nbijM}αY(S) for any bounded set S∈Ω¯R. Let η=αY(S). Then, for any positive number ϵ<Rmax1≤j≤n{∑j=1nbijM}η, there is a finite family of subsets {Si} satisfying S=⋃i‍Si with diam(Si)≤η+ϵ. Therefore
(31)|x-y|1≤η+ϵ,foranyx,y∈Si.
As S and Si are precompact in X, it follows that there is a finite family of subsets {Sij} of Si such that Si=⋃j‍Sij and
(32)|x-y|0≤ϵ,foranyx,y∈Sij.
In addition, for any x∈S and t∈[0,ω], we have
(33)(Tix)(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))≤R21-δi∫0ω‍[∑j=1n‍aij(s)+∑j=1n‍bij(s)+∑j=1n‍cij(s)]ds+11-δi∑t≤tk<t+ω‍Iik(xi(tk)):=Bi,i=1,2,…,n,(34)|(Tix)′(t)|=|+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]ri(t)(Tix)(t)-xi(t)×[∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]|≤riMBi+R2∑j=1n‍(aijM+bijM+cijM),i=1,2,…,n.
Hence,
(35)∥(Tx)∥0≤∑i=1n‍Bi,∥(Tx)′∥0≤∑i=1n‍[BiriMBi+R2∑j=1n‍(aijM+bijM)].
Applying the Arzela-Ascoli theorem, we know that T(S) is precompact in X. Then, there is a finite family of subsets {Sijk} of Sij such that Sij=⋃k‍Sijk and
(36)|Tx-Ty|0≤ϵ,foranyx,y∈Sijk.
From (23), (26), (31)–(34), and (A2), for any x,y∈Sijk, we have
(37)|(Tix)′-(Tiy)′|0=maxt∈[0,ω]{|+∑j=1n‍cij(t)∫-σij0‍gij(ξ)yj′(t+ξ)dξ]|ri(t)(Tix)(t)-ri(t)(Tiy)(t)-xi(t)×[∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ]+yi(t)[∑j=1n‍aij(t)yj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)yj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)yj′(t+ξ)dξ]|}≤maxt∈[0,ω]{|ri(t)[(Tix)(t)-(Tiy)(t)]|}+maxt∈[0,ω]{|xi[(∑j=1n‍aij(t)xj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)xj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)xj′(t+ξ)dξ)-(∑j=1n‍aij(t)yj(t)+∑j=1n‍bij(t)×∫-τij0‍fij(ξ)yj(t+ξ)dξ+∑j=1n‍cij(t)×∫-σij0‍gij(ξ)yj′(t+ξ)dξ)|xi[(∑j=1n‍aij(t)xj(t)]|}+maxt∈[0,ω]{|+∑j=1n‍cij(t)∫-σij0‍gij(ξ)yj′(t+ξ)dξ)[xi(t)-yi(t)]×(∑j=1n‍aij(t)yj(t)+∑j=1n‍bij(t)×∫-τij0‍fij(ξ)yj(t+ξ)dξ+∑j=1n‍cij(t)∫-σij0‍gij(ξ)yj′(t+ξ)dξ)|}≤riM|(Tix)-(Tiy)|0+Rmaxt∈[0,ω]{∑j=1n‍[∑j=1n‍aij(t)|xj(t)-yj(t)|+bij(t)×∫-τij0‍fij(ξ)×|xi(t+ξ)-yi(t+ξ)|dξ+cij(t)×∫-σij0‍gij(ξ)×|xj′(t+ξ)-y′(t-ξ)|dξ∑j=1n‍[aij(t)]}+maxt∈[0,ω]{|×∫-σij0‍gij(ξ)yj′(t+ξ)dξ)[xi(t)-yi(t)]×(∑j=1n‍aij(t)yj(t)+∑j=1n‍bij(t)∫-τij0‍fij(ξ)yj(t+ξ)dξ+∑j=1n‍cij(t)×∫-σij0‍gij(ξ)yj′(t+ξ)dξ)|}≤riMϵ+Rϵ∑j=1n‍aijM+Rϵ∑j=1n‍bijM+|xi|0(η+ϵ)×∑j=1n‍cijM+Rϵ∑j=1n‍(aijM+bijM+cijM)=|xi|0∑j=1n‍cijMη+B~iϵ,
where
(38)B~i=riM+2R∑j=1n‍(aijM+bijM+cijM),i=1,2,…,n.
From (36) and (37) we obtain
(39)∥Tx-Ty∥1≤(∑i=1n‍|xi|0∑j=1n‍cijM)η+ϵ∑i=1n‍B~i≤Rmax1≤i≤n{∑j=1n‍cijM}η+ϵ∑i=1n‍B~i,foranyx,y∈Sijk.
As ϵ is arbitrarily small, it follows that
(40)αY(T(S))≤Rmax1≤j≤n{∑j=1n‍cijM}αY(S).
Therefore, T is strict-set-contractive. The proof of Lemma 10 is complete.


## 3. Main Results

Our main result of this paper is as follows.Theorem 11.
Assume that(A1)–(A3), (A5) hold.(i)
Ifmax{riM,i=1,2,…,n}≤1, then system (2) has at least one positive ω-periodic solution.(ii)
If(A4) holds and min{riM,i=1,2,…,n}>1, then system (2) has at least one positive ω-periodic solution.Proof.
We only need to prove (i), since the proof of (ii) is similar. Let(41)R=(mini∈[1,n]{δi21-δiminj∈[1,n]{αij}})-1,0<r<mini∈[1,n]{δi(1-δi)-γimaxj∈[1,n]{βij}}.
Then it is easy to see that 0<r<R. From Lemmas 9 and 10, we know that T is strict-set-contractive on Er,R. In view of (26), we see that if there exists x*∈E such that Tx*=x*, then x* is one positive ω-periodic solution of system (2). Now, we will prove that condition (ii) of Lemma 7 holds. First, we prove that Tx≱x, for all x∈E,∥x∥1<r. Otherwise, there exists x∈E,∥x∥1<r, such that Tx≥x. So ∥x∥0>0 and Tx-x≥0, which implies that
(42)Tix(t)-xi(t)≥δi|Tix-xi|1≥0,foranyt∈[0,ω],i=1,2,…,n.
Moreover, for t∈[0,ω], we have
(43)(Tix)(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))≤11-δi|xi|0{∑j=1n‍∫0ω‍[aij(s)+bij(s)+cij(s)]|xj|1ds+γi}≤βijr+γi1-δi|xi|0≤δi|xi|0,i=1,2,…,n.
In view of (42) and (43), we obtain
(44)∥x∥0≤∥Tx∥0≤max1≤i≤n{δi}∥x∥0<∥x∥0,
which is a contradiction. Finally, we prove that Tx≰x,  for all x∈E,  and∥x∥1=R also hold. For this case, suppose, for the sake of contradiction, that there exist x∈Eand∥x∥1=R such that Tx≤x. Furthermore, for any t∈[0,ω], we have
(45)xi(t)-Tix(t)≥δi|xi-Tix|1≥0,foranyt∈[0,ω],i=1,2,…,n.
In addition, for any t∈[0,ω], we find
(46)(Tix)(t)=∫tt+ω‍Gi(t,s)xi(s)×[∑j=1n‍aij(s)xj(s)+∑j=1n‍bij(s)∫-τij0‍fij(ξ)xj(s+ξ)dξ+∑j=1n‍cij(s)∫-σij0‍gij(ξ)xj′(s+ξ)dξ]ds+∑t≤tk<t+ω‍Gi(t,tk)Iik(xi(tk))>δi21-δi|xi|1∑j=1n‍∫0ω‍[δjaij(s)-bij(s)-cij(s)]ds|xj|1=δi21-δi|xi|1∑j=1n‍(αij|xj|1),i=1,2,…,n.
Thus, we have
(47)∥Tx∥0=∑i=1n‍|Tix|0>∑i=1n‍δi21-δi|xi|1∑j=1n‍(αij|xj|1)≥min1≤i≤n{δi2min1≤j≤n{αij}1-δi}∑i=1n‍|xi|1∑j=1n‍|xj|1≥min1≤i≤n{δi2min1≤j≤n{αij}1-δi}R2=R.
From (45) and (47), we obtain
(48)∥x∥1≥∥Tx∥1≥∥Tx∥0>R,
which is a contradiction. Therefore, condition (ii) of Lemma 7 holds. By Lemma 7, we see that T has at least one nonzero fixed point in E. Thus, the system (6) has at least one positive ω-periodic solution. Therefore, it follows from Lemma 6 that system (2) has a positive ω-periodic solution. The proof of Theorem 11 is complete.Remark 12.
IfIik(x(tk))=0, we can easily derive the corresponding results in [10]. So we extend the corresponding results in [10].


## 4. Example

In this section, we give an example to show the effectiveness of our result.Example 1.
Consider the following nonimpulsive system:(49)x1′(t)=x1(t)[×∫-σ120‍g12(ξ)x2′(t+ξ)dξ1-sint24-(10+sint)x1(t)-(9-cost)x2(t)-(3-2cost)×∫-τ110‍f11(ξ)x1(t+ξ)dξ-5+sint15×∫-τ120‍f12(ξ)x2(t+ξ)dξ-(1+sint)×∫-σ110‍g11(ξ)x1′(t+ξ)dξ-4-3cost6×∫-σ120‍g12(ξ)x2′(t+ξ)dξ],x2′(t)=x2(t)[×∫-σ220‍g22(ξ)x2′(t+ξ)dξ1+cost8π-(12+3cost)x1(t)-(7-sint)x2(t)-(2+3sint)×∫-τ210‍f21(ξ)x2(t+ξ)dξ-(1-cost)×∫-τ220‍f22(ξ)x2(t+ξ)dξ-4+sint12×∫-σ210‍g21(ξ)x2′(t+ξ)dξ-5-cost10×∫-σ220‍g22(ξ)x2′(t+ξ)dξ],
where τij,σij∈(0,+∞)(i,j=1,2), and fij,gij∈C(R,R+) satisfying ∫-τij0‍fij(ξ)dξ=1, ∫-σij0‍gij(ξ)dξ,i,j=1,2. Obviously
(50)r1(t)=1-sint24,r2=1+cost8π,a11(t)=10+sint,a12(t)=9-cost,a21(t)=12+3cost,a22(t)=7-sint,b11(t)=3-2cost,b12(t)=5+sint15,b21(t)=2+3sint,b22(t)=1-cost,c11(t)=1+sint,c12(t)=4-3cost6,c21(t)=4+sint12,c22(t)=5-cost10,r1L=r2L=0.
Furthermore, we obtain
(51)δ1=e-(π/12)<1,δ2=e-(1/4)<1,α11=20πδ1-8π≈23.8007,α12=18πδ2-2π≈37.2423,α21=24πδ1-14π3≈44.0594,α22=14πδ2-3π≈24.4284,min1≤j≤2{α1j}=α11=20πδ1-8π≈23.8007,min1≤j≤2{α2j}=α22=14πδ2-3π≈24.4284,δ1a11(t)-b11(t)-c11(t)>0.009>0,δ2a12(t)-b12(t)-c12(t)>1>0,δ1a21(t)-b21(t)-c21(t)>4>0,δ2a22(t)-b22(t)-c22(t)>2>0,(1+r1L)δ12α111-δ1=e-(π/6)(20πδ1-8π)1-e-(π/12)≈65.2614>18≥max0≤t≤2π{a11(t)+b11(t)+c11(t)},(1+r1L)δ12α121-δ1=e-(π/6)(18πδ2-2π)1-e-(π/12)≈102.1184>34730≥max0≤t≤2π{a12(t)+b12(t)+c12(t)},(1+r2L)δ22α211-δ2=e-(1/2)(24πδ1-(14π/3))1-e-(1/4)≈113.3411>24512>max0≤t≤2π{a21(t)+b21(t)+c21(t)},(1+r2L)δ22α221-δ2=e-(1/2)(14πδ2-3π)1-e-(1/4)≈62.8411>535>max0≤t≤2π{a22(t)+b22(t)+c22(t)},δ12min1≤j≤2{α1j}1-δ1=e-(π/6)(20πδ1-6π)1-e-(π/12)≈65.2614,δ22min1≤j≤2{α2j}1-δ2=e-(1/2)(14πδ2-3π)1-e-(1/4)≈62.8411,min1≤i≤2{δi2min1≤j≤2{αij}1-δi}=δ22min1≤j≤2{α2j}1-δ1≈62.8411,c11M+c12M=196,c21M+c22M=6160,max1≤i≤2{∑j=12‍dijM}=196.
Therefore,
(52)196=max1≤i≤2{∑i=12‍dijM}<min1≤i≤2{δi2min1≤j≤2{αij}1-δi}≈62.8411.
Hence, (A1)–(A3), (A5) hold and aiM≤1,i=1,2. According to Theorem 11, system (49) has at least one positive 2π-periodic solution.



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*Source: 890281-2013-07-15.xml*