# A New Existence Theory for Positive Periodic Solutions to a Class of Neutral Delay Model with Feedback Control and Impulse

Authors: Zhenguo Luo; Jianhua Huang; Binxiang Dai
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

We acquire some sufficient and realistic conditions for the existence of
positive periodic solution of a general neutral impulsiven-species competitive model with feedback
control by applying some analysis techniques and a new existence theorem, which is different from
Gaines and Mawhin's continuation theorem and abstract continuation theory for k-set contraction. As applications, we also examine some special cases, which have been studied extensively in the literature, some known results are improved and generalized.


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## Body

## 1. Introduction

In this paper, we consider the existence of the positive periodic solution of the following impulsiven-species competition system with multiple delays and feedback control:
(1)Ni′(t)=Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)(t)(t)2-∑j=1n‍bij(t)∫-∞t‍kij(t-s)Nj(s)ds(t)(t)2-∑j=1n‍cij(t)Nj(t-γij(t))(t)(t)2-∑j=1n‍dij(t)Nj′(t-τij(t))-ei(t)ui(t)(t)(t)2(t)(t)2-fi(t)ui(t-δi(t))∑j=1n‍],(t)ui(t)ui(t)ui22i=1,2,…,n,t≠tk,ui′(t)=-αi(t)ui(t)+βi(t)Ni(t)+θi(t)Ni(t-σi(t)),t≥0,ΔNi(tk)=(pik+qik)Ni(tk),i=1,2,…,n,k=1,2,…,
with the following initial conditions:
(2)Ni(ξ)=ϕi(ξ),Ni′(ξ)=ϕi′(ξ),ξ∈[-τ,0],ϕi(0)>0,ϕi∈C([-τ,0),R+)⋂‍C1([-τ,0),R+)ui(ξ)=φi(ξ),ξ∈[-τ,0],φi(0)>0,φi∈C([-τ,0),R+),i=1,2,3,…,n,
where aij, bij, cij, ei, fi, αi, βi, θi∈C(R,[0,+∞)), dij∈C1(R,[0,+∞)), γij, δi∈C1(R,R), and τij∈C2(R,R) are continuous ω-periodic functions; ri∈C(R,R) are continuous  ω-periodic functions with ∫0ω‍ri(t)dt>0. The growth functions ri are not necessarily positive; since the environment fluctuates randomly, in some conditions, ri may be negative. Consider the following: τ=maxt∈[0,ω]{γij(t),τij(t),δi(t),σi(t),1≤i,j≤n}; and ∫0∞‍kij(s)ds=1, ∫0+∞‍skij(s)ds<+∞, and i,j=1,2,…,n. And pik and qik represent the birth rate and the harvesting (or stocking) rate of Ni at time tk, respectively. When qik>0, it stands for harvesting, while qik<0 means stocking. For the ecological justification of (1) and the similar types, refer to [1–14].In 1991, in [1], Gopalsamy et al. have established the existence of a positive periodic solution for a periodic neutral delay logistic equation
(3)dNdt=r(t)N(t)[1-N(t-mT)-c(t)N′(t-mT)K(t)],
where K(t),r(t), and c(t) are positive continuous T-periodic functions with T>0 and m is a positive integer. In 1993, in [2], Kuang proposed an open problem (Open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation:
(4)dNdt=N(t)[(t)N′(t-τ(t))a(t)-β(t)N(t)-b(t)N(t-τ(t))-c(t)N′(t-τ(t))].
In [3], Li tried to give an affirmative answer to the previous open problem; however, there is a mistake in the proof of Theorem 2 in  [3]. With the aim of giving a right answer to the previous open problem,  [4–6] also have investigated the previous question. However, it is more complex to check the sufficient conditions of the system  [5, 6].  Moreover, in  [7], Li studied the existence of positive periodic solution of the neutral Lotka-Volterra equation with several delays
(5)N′(t)=N(t)[a(t)-∑i=1n‍bi(t)N(t-τi)-∑i=1n‍ci(t)N′(t-γi)],
where a(t),bi(t), and ci(t) are positive continuous T-periodic functions and τi,γi(i=1,…,n) are nonnegative constants. Recently, in [8], Lu and Ge investigated a neutral delay population model with multiple delays:
(6)dNdt=N(t)[a(t)-β(t)N(t)-∑j=1n‍bj(t)N(t-σj(t))-∑i=1m‍ci(t)N′(t-τi(t))].
They applied the theory of abstract continuous theorem of k-set contractive operator and some analysis techniques to obtain some sufficient conditions for the existence of positive periodic solutions of the model (6).It is of course very interesting to study the neutral delay population model for higher dimensional systems. In fact, in  [9], Li has studied the neutral Lotka-Volterra system with constant delays
(7)Ni′(t)=Ni(t)[ai(t)-∑j=1n‍bij(t)Nj(t-τij)-∑j=1n‍cij(t)Nj′(t-γij)],
where i=1,…,n, and ai(t), bij(t), and cij(t) are positive continuous T-periodic functions, and τij,γij are nonnegative constants. He obtained sufficient conditions that guarantee the existence of positive periodic solution of the system (7), by applying a continuation theorem based on Gaines and Mawhin's coincidence degree. Noticing that delays arise frequently in practical applications, it is difficult to measure them precisely. In population dynamics, it is clear that a constant delay is only a special case. In most situations, delays are variable, and so in  [10], Liu and Chen investigated the following general neutral Lotka-Volterra system with unbounded delays:
(8)Ni′(t)=Ni(t)[ai(t)-∑j=1n‍βij(t)Nj(t)-∑j=1n‍bij(t)Nj(t-τij(t))-∑j=1n‍cij(t)Nj′(t-γij(t))],i=1,2,…,n.
They introduced a new existence theorem to obtain a set of sufficient conditions for the existence of positive periodic solutions for the system (8), and their results improved and generalized some known results.Moreover, in some situations,  people may wish to change the position of the existing periodic solution but keep its stability. This is of significance in the control of ecology balance. One of the methods for its realization 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 schemes or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively; see [11, 15–23]. Recently, in [11], Chen considered the following neutral Lotka-Volterra competition model with feedback control of the form:
(9)yi′(t)=yi(t)[ri(t)-∑j=1n‍aij(t)yj(t)-∑j=1n‍bij(t)yj(t-τij(t))-∑j=1n‍cij(t)yj′(t-γij(t))-fi(t)ui(t)-ei(t)ui(t-σi(t))∑j=1n‍],ui′(t)=-αi(t)ui(t)+βi(t)yi(t)+γi(t)yi(t-δi(t)),i=1,2,…,n.
With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, he established easily verifiable criteria for the global existence of positive periodic solutions of the system (9), and his results extended and improved existing results.On the other hand, there are some other perturbations in the real world, such as fires and floods, that are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations  involving impulsive differential equations have attracted the interest of many researchers in the past twenty years, see [12–14, 24–30], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, and optimal control. For details, see [31–33].In  [12], Huo studied the following neutral impulsive delay Lotka-Volterra system:
(10)Ni′(t)=Ni(t)[αi(t)-∑j=1n‍βij(t)Nj(t-τij(t))-∑j=1n‍cij(t)Nj′(t-γij(t))],i=1,2,…,n,t≠tk,ΔNi(t)=Ni(t+)-Ni(tk)=bikNi(tk),i=1,2,…,n,k=1,2,….
By using some techniques of Mawhin’s coincidence degree theory, he obtained sufficient conditions for the existence of periodic positive solutions of the system (10).In  [13], Wang and Dai investigated the following periodic neutral population model with delays and impulse:
(11)N′(t)=N(t)[a(t)-e(t)N(t)-∑j=1n‍bj(t)N(t-σj(t))-∑i=1m‍ci(t)N′(t-τi(t))⋃∑j=1n‍],t≠tk,N(t+)=(1+θk)N(tk),k=1,2,…
They obtained some sufficient conditions for the existence of positive periodic solutions of the model (11) by using the theory of abstract continuous theorem of k-set contractive operator and some analysis techniques.Recently, in  [14], Luo et al. studied the following n-species competition system with general periodic neutral delay and impulse:
(12)Ni′(t)=Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)-∑j=1n‍bij(t)×∫-∞t‍kij(t-s)Nj(s)ds-∑j=1n‍cij(t)Nj(t-τij(t))-∑j=1n‍dij(t)Nj′(t-γij(t))],i=1,2,…,n,t≠tk,ΔNi(tk)=Ni(tk+)-Ni(tk)=θikNi(tk),i=1,2,…,n,k=1,2,….
They obtained some sufficient and realistic conditions for the existence of positive periodic solutions of the system (12), by using a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for k-set contraction.However, to this day, no scholars had done works on the existence of positive periodic solution of the system (1). One could easily see that systems (3)–(12) are all special cases of the system (1). Therefore, we propose and study the system (1) in this paper.For the sake of generality and convenience, we make the following notations and assumptions: letω>0 be a constant andCω={x∣x∈C(R,R),x(t+ω)=x(t)}, with the norm defined by |x|0=maxt∈[0,ω]|x(t)|;Cω1={x∣x∈C1(R,R),x(t+ω)=x(t)}, with the norm defined by ∥x∥=maxt∈[0,ω]{|x|0,|x′|0};PC={x∣x:R→R+,lims→tx(s) = x(t),ift≠tk,limt→tk-x(t) = x(tk),limt→tk+x(t)exists,k∈Z+};PC1={x∣x:R→R+,x′∈PC};PCω={x∣x∈PC,x(t+ω)=x(t)}, with the norm defined by |x|0=maxt∈[0,ω]|x(t)|;PCω1={x∣x∈PC1,x(t+ω)=x(t)}, with the norm defined by ∥x∥=maxt∈[0,ω]{|x|0,|x′|0}.Then, the previous spaces are all Banach spaces. We also denote that
(13)g¯=1ω∫0ω‍g(t)dt,gL=mint∈[0,ω]g(t),foranyg∈PCω,Δik=1+pik+qik,i=1,2,…,n,k=1,2,…,
and make the following assumptions: (A)
aij, bij, cij, ei, fi, αi, βi, θi∈C(R,[0,+∞)), dij∈C1(R,[0,+∞)), γij,δi∈C1(R,R), and τij∈C2(R,R) are continuous ω-periodic functions, and ∫0∞‍kij(s)ds=1, ∫0+∞‍skij(s)ds<+∞,  and  i,j=1,2,…,n;(B)
[tk]k∈N satisfies 0<t1<t2<⋯<tk<⋯ and limk→∞tk=+∞;(C)
{Δik} is a real sequence such that Δik>0, ∏0<tk<t‍Δik(i=1,2,…,n,k=1,2,…) are ω-periodic functions.The organization of this paper is as follows. In the following section, we introduce some lemmas and an important existence theorem developed in [34, 35]. In the third section, we derive some sufficient conditions, which ensure the existence of positive periodic solution of system (1) by applying this theorem and some other techniques. Finally, we study some special cases of system (1), which have been studied extensively in the literature. These examples show that our sufficient conditions are new, and some known results can be improved and generalized.


## 2. Preliminaries

In this section, in order to obtain the existence of a periodic solution for system (1) and (2), we will give some concepts and results from [35], and we will state an existence theorem and some lemmas.For a fixedτ>0, let C=:C([-τ,0];Rn), if x∈C([-τ,0];Rn) for some δ>0 and η∈R, then xt∈C for t∈[η,η+δ] is defined by xt(θ)=x(t+θ) for θ∈[-τ,0]. The supremum norm in C is denoted by ∥·∥, that is, ∥ϕ∥=maxt∈[-τ,0]|ϕ(θ)| for ϕ∈C, where |·| denotes the norm in Rn and |u|=∑j=1n|ui| for u=(u1,…,un)∈Rn. Consider the following neutral functional differential equation:
(14)ddt[x(t)-b(t,xt)]=f(t,xt),
where f:R×C→Rn is completely continuous, and b:R×C→Rn is continuous. Moreover, we assume the following: (1)
there existsω>0 such that for every (t,ϕ)∈R×C, we have b(t+ω,ϕ)=b(t,ϕ) and f(t+ω,ϕ)=f(t,ϕ);(2)
there exists a constantk<1 such that |b(t,ϕ)-b(t,φ)|≤k∥ϕ-φ∥, for t∈R and ϕ,φ∈C.By using the continuation theorem for composite coincidence degree, in [34], Erbe et al. proved the following existence theorem (see also Theorem 4.7.1 in [35]).Lemma 1.
Assume that there exists a constantM>0 such that(i)
for anyλ∈(0,1) and any ω-periodic solution x of the system
(15)ddt[x(t)-λb(t,xt)]=f(t,xt).One has that|x(t)|<M for t∈R;(ii)
h(u)=:∫0ω‍f(s,u^)ds≠0 for u∈∂BM(Rn), where BM(Rn)={u∈Rn:|u|<M}, and u^ denotes the constant mapping from [-τ,0] to Rn with the value u∈Rn;(iii)
deg(h,BM(Rn))≠0. Then, there exists at least one ω-periodic solution of the system (14) that satisfies supt∈R|x(t)|<M.The following remark is introduced by Fang (see Remark 1 in [36]).Remark 2.
Lemma1 remains valid if the assumption (ii) is replaced by the following:
(ii*) there exists a constantk<1 such that |b(t,ϕ)-b(t,φ)|≤k∥ϕ-φ∥ for t∈R and ϕ,φ∈{ϕ∈C:∥ϕ∥<M} with M as given in condition (i) of Lemma 1.We will also need the following lemmas.Lemma 3 (see [8, 13]).
Suppose thatσ∈Cω1 and σ′(t)<1,t∈[0,ω]. Then, the function t-σ(t) has a unique inverse μ(t) satisfying μ∈C(R,R) with μ(a+ω)=μ(a)+ω, ∀a∈R, and if h∈PCω,σ′(t)<1, and t∈[0,ω], then h(μ(t))∈PCω.Lemma 4 (see [27]).
Suppose thatx(t) is a differently continuous ω-periodic function on R with (ω>0). Then, to any t*∈R,maxt*≤t≤t*+ω|x(t)|≤|x(t*)|+(1/2)∫0ω‍|x′(t)|dt.Lemma 5.
Consider thatR+2n={(Ni(t),ui(t)):Ni(0)>0,ui(0)>0,i=1,2,…,n} is the positive invariable region of the system (1) and (2).Proof.
In view of biological population, we obtainNi(0)>0,ui(0)>0. By the system (1) and (2), we have
(16)Ni(t)=Ni(0)exp{∫0t‍[ri(ξ)-∑j=1n‍aij(ξ)Nj(ξ)Nj(ξ)Nj(ξ)Nj(ξ)2-∑j=1n‍bij(ξ)Nj(ξ)Nj(ξ)Nj(ξ)2×∫-∞t‍Kij(ξ-s)Nj(s)dsNj(ξ)Nj(ξ)Nj(ξ)2-∑j=1n‍cij(ξ)Nj(ξ-γij(ξ))Nj(ξ)Nj(ξ)Nj(ξ)2-∑j=1n‍dij(ξ)Nj′(ξ-τij(ξ))(ξ)Nj(ξ)(ξ)Nj(ξ)(ξ)2(ξ)-ei(ξ)ui(ξ)-fi(ξ)uiNj(ξ)(ξ)Nj(ξ)Nj(ξ)(ξ)2×(ξ-σi(ξ))∑j=1n‍]dξ∑j=1n‍},Nj(ξ)Nj(ξ)(ξ)Nj2(ξ)t∈[0,t1],i=1,2,…,n,Ni(t)=Ni(tk)exp{∫tkt‍[ri(ξ)-∑j=1n‍aij(ξ)Nj(ξ)NiNiNiNiNiNiNiNiNi-∑j=1n‍bij(ξ)NiNiNiNiNiNiNiNiNi×∫-∞t‍Kij(ξ-s)Nj(s)dsNiNiNiNiNiNiNiNiNi-∑j=1n‍cij(ξ)Nj(ξ-γij(ξ))NiNiNiNiNiNiNiNiNi-∑j=1n‍dij(ξ)Nj′(ξ-τij(ξ))NiNiNiNiNiNiNiNiNiNiNi-ei(ξ)ui(ξ)-fi(ξ)uiNiNiNiNiNiNiNiNiNiNiNi×(ξ-σi(ξ))∑j=1n‍]dξ},NiNiNiNiNiNit∈(tk,tk+1],i=1,2,…,n,k≥1,Ni(tk+)=e(pik+qik)Ni(tk)>0,k∈N,i=1,2,…,n,ui(t)=∫tt+ω‍Gi(t,s)[βi(s)Ni(s)+ϑi(s)NiNiNiNiNiNiNi×(s-γi(s))]ds∶=(ϕiNi)(t),
where
(17)Gi(t,s)=exp{∫ts‍αi(ξ)dξ}exp{∫ts‍αi(ξ)dξ}-1.
Then, the solution of the systems (1) and (2) is positive.Definition 6.
A functionNi:[-τ,0]→[0,+∞)(i=1,2,…,n) is said to be a positive solution of (1) and (2) on [-τ,∞], if the following conditions are satisfied: (a)
Ni(t) is absolutely continuous on each (tk,tk+1);(b)
for eachk∈Z+,Ni(tk+) and Ni(tk-) exist, and Ni(tk-)=Ni(tk);(c)
Ni(t) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0,∞]∖{tk} and satisfies Ni(tk+)=ΔikNi(tk) for t=tk,k∈Z+={1,2,…}.Consider the following nonimpulsive delay differential equation:(18)yi′(t)=yi(t)[ri(t)-∑j=1n‍Aij(t)yj(t)-∑j=1n‍Bij(t)×∫-∞t‍kij(t-s)yj(s)ds-∑j=1n‍Cij(t)yj(t-γij(t))-∑j=1n‍Dij(t)yj′(t-τij(t))-ei(t)ui(t)-fi(t)ui(t-δi(t))∑j=1n‍],dui(t)dt=-αi(t)ui(t)+βi*(t)yi(t)+θi*(t)yi(t-σi(t)),
with the following initial conditions:
(19)yi(ξ)=ϕi(ξ),yi′(ξ)=ϕi′(ξ),ξ∈[-τ,0],ϕi(0)>0,ϕi∈C([-τ,0],R+)⋂‍C1([-τ,0],R+),ui(ξ)=φi(ξ),ξ∈[-τ,0],φi(0)>0,φi∈C([-τ,0),R+),i=1,2,3,…,n,
where
(20)Aij(t)=aij(t)∏0<tk<tΔik,Bij(t)=bij(t)∏0<tk<tΔik,Cij(t)=cij(t)∏0<tk<t-γij(t)Δik,Dij(t)=dij(t)∏0<tk<t-τij(t)Δik,τ=maxt∈[0,ω]{γij(t),τij(t),δi(t),σi(t),1≤i,j≤n},βi*(t)=βi(t)∏0<tk<tΔik,θi*(t)=θi(t)∏0<tk<t-σi(t)Δik,i,j=1,2,…,n.
The following lemmas will be used in the proofs of our results. The proof of Lemma 7 is similar to that of Theorem 1 in [24].Lemma 7.
Suppose that (A)–(C) hold, then(i)
if(yi(t),ui(t))T(i=1,2,…,n) is a solution of (18) and (19) on [-τ,+∞), then (Ni(t),ui(t))T(i=1,2,…,n) is a solution of (1) and (2) on [-τ,+∞), where Ni(t)=∏0<tk<t‍Δikyi(t);(ii)
if(Ni(t),ui(t))T(i=1,2,…,n) is a solution of (1) and (2) on [-τ,+∞), then (yi(t),ui(t))T(i=1,2,…,n) is a solution of (18) and (19) on [-τ,+∞), where yi(t)=∏0<tk<t‍Δik-1Ni(t).Proof.
(i) It is easy to see thatNi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n) is absolutely continuous on every interval (tk,tk+1],t≠tk,k=1,2,…,
(21)Ni′(t)-Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)(t)(t)(t)(t)(t)2-∑j=1n‍bij(t)∫-∞t‍kij(t-s)Nj(s)ds(t)(t)(t)(t)(t)2-∑j=1n‍cij(t)Nj(t-γij(t))(t)(t)(t)(t)(t)2-∑j=1n‍dij(t)Nj′(t-τij(t))(t)(t)(t)(t)(t)(t)21-ei(t)ui(t)-fi(t)ui(t-δi(t))∑j=1n‍](t)(t)=∏0<tk<t‍Δikyi′(t)(t)(t)(t)-∏0<tk<t‍Δikyi(t)(t)(t)(t)(t)(t)(t)×[ri(t)-∑j=1n‍aij(t)∏0<tk<t‍Δikyj(t)(t)(t)(t)(t)(t)(t)(t)-∑j=1n‍bij(t)∏0<tk<t‍Δik∫-∞t‍kij(t-s)yj(s)ds(t)(t)(t)(t)(t)(t)(t)-∑j=1n‍cij(t)∏0<tk<t-γij(t)‍Δikyj(t-τij(t))(t)(t)(t)(t)(t)(t)(t)-∑j=1n‍dij(t)∏0<tk<t-γij(t)‍Δikyj′(t-τij(t))(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)1(t)-ei(t)ui(t)-fi(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)1(t)×ui(t-δi(t))∑j=1n‍](t)(t)=∏0<tk<t‍Δik{∑j=1n‍yi′(t)-yi(t)(t)(t)(t)(t)(t)(t)(t)×[ri(t)-∑j=1n‍Aij(t)yj(t)(t)(t)(t)1(t)(t)(t)(t)(t)-∑j=1n‍Bij(t)∫-∞t‍kij(t-s)yj(s)ds(t)(t)(t)1(t)(t)(t)(t)(t)-∑j=1n‍Cij(t)yj(t-γij(t))(t)(t)(t)1(t)(t)(t)(t)(t)-∑j=1n‍Dij(t)yj′(t-τij(t))(t)(t)(t)(t)(t)(t)(t)1(t)(t)(t)-ei(t)ui(t)-fi(t)ui(t)(t)(t)(t)(t)(t)(t)1(t)(t)(t)×(t-δi(t))∑j=1n‍]}=0,ui′(t)+αi(t)ui(t)-βi(t)Ni(t)-θi(t)Ni(t-σi(t))=ui′(t)+αi(t)ui(t)-βi*(t)yi(t)-θi*(t)yi(t-γi(t))=0.
On the other hand, for any t=tk, k=1,2,…,
(22)Ni(tk+)=limt→tk+∏0<tj<t‍Δikyi(t)=∏0<tj≤tk‍Δikyi(tk),Ni(tk)=∏0<tj<tk‍Δikyi(tk),
thus,
(23)ΔNi(tk+)=Δikyi(tk),
which implies that (Ni(t),ui(t))T(i=1,2,…,n) is a solution of the system (1) and (2). Therefore, if (yi(t),ui(t))T(i=1,2,…,n) is a solution of the system (18) and (19) on [-τ,+∞), we can prove that (Ni(t),ui(t))T(i=1,2,…,n) are solutions of the system (1) and (2) on [-τ,+∞).
(ii) SinceNi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n) is absolutely continuous on every interval (tk,tk+1],t≠tk,k=1,2,…, and in view of (23), it follows that for any k=1,2,…,
(24)yi(tk+)=∏0<tj≤tk‍Δik-1Ni(tk+)=∏0<tj<tk‍Δik-1Ni(tk)=yi(tk),yi(tk-)=∏0<tj<tk‍Δik-1Ni(tk-)=∏0<tj≤tk-‍Δik-1Ni(tk-)=yi(tk),i=1,2,…,n,
which implies that (yi(t),ui(t))T(i=1,2,…,n) is continuous on [-τ,+∞). It is easy to prove that (yi(t),ui(t))T(i=1,2,…,n) is absolutely continuous on [-τ,+∞). Similar to the proof of (i), we can check that (yi(t),ui(t))T(i=1,2,…,n) is a solution of the system (18) and (19) on [-τ,+∞). The proof of Lemma 7 is completed.Lemma 8.
Consider that(yi(t),ui(t)) is a ω-periodic solution of (18) and (19) if and only if yi(t) is a ω-periodic solution of the following system:
(25)dyi(t)dt=yi(t)[ri(t)-∑j=1n‍Aij(t)yj(t)dyi(t)dt=yi(t2)-∑j=1n‍Bij(t)∫-∞t‍kij(t-s)yj(s)dsdyi(t)dt=yi(t2)-∑j=1n‍Cij(t)yj(t-γij(t))dyi(t)dt=yi(t2)-∑j=1n‍Dij(t)yj′(t-τij(t))dyi(t)dt=yi(t2)(t2)2-ei(t)(ψiyi)(t)-fi(t)(ψiyi)dyi(t)dt=yi(t2)(t2)2×(t-δi(t))∑j=1n‍],
where
(26)(ψiyi)(t)∶=∫tt+ω‍Gi(t,s)[βi*(s)yi(s)+θi*(s)yi×(s-σi(s))]ds,
and Gi(t,s) is defined by (17).Proof.
The proof of Lemma8 is similar to that of Lemma  2.2 in [11], and we omit the details here.From Lemmas7 and  8, if we want to discuss the existence of positive periodic solutions of systems (1) and (2), we only discuss the existence of positive periodic solutions of systems (25) and (26).


## 3. The Main Result

Sinceγij′(t)<1,τij′(t)<1,t∈[0,ω], we see that γij(t),τij(t) all have their inverse function. Throughout the following part, we set to ϑij(t),νij(t) that represent the inverse function of t-γij(t),t-τij(t), respectively. We denote that
(27)Γij(t)=Aij(t)+Bij(t)+Cij(ϑij(t))1-γij′(ϑij(t))-Dij′(νij(t))1-τij′(νij(t)),Γi1(t)=ei(t)(ψi1)(t),Γi2(t)=fi(t)(ψi1)(t-σi(t)).Remark 9.
From Lemma3, we get that ϑij(ω)=ϑij(0)+ω,νij(ω)=νij(0)+ω,i,j=1,2,…,n, then
(28)∫0ω‍Cij(ϑij(s))1-γij′(ϑij(s))ds=∫ϑij(0)ϑij(ω)‍Cij(t)(1-γij′(t))1-γij′(t)dt=∫ϑij(0)ϑij(0)+ω‍Cij(t)dt=Cij¯ω,i,j=1,2,…,n.
Similarly,(29)∫0ω‍Dij′(νij(s))1-τij′(νij(s))ds=∫νij(0)νij(ω)‍Dij′(t)(1-τij′(t))1-τij′(t)dt=∫νij(0)νij(0)+ω‍Dij′(t)dt=0,i,j=1,2,…,n.
Thus,
(30)Γij¯ω=∫0ω‍Γij(t)dt=(Aij¯+Bij¯+Cij¯)ω.Here, we have the following notations:(31)ρij=ΓijL(1-γij′)L(1-γij′)L+|Dij|0,Ri¯=1ω∫0ω‍|ri(t)|dt,L0=max{∑i=1n‍∑j=1n‍|Dij|0,∑i=1n‍∑j=1n‍|D0,ij|0},M0=max{∑i=1n‍|lnμi*|0,H,12ωΛ*+∑i=1n‍Λi},H=maxi∈[1,n]{Hi},Hi=lnri¯ρii+∑j=1n‍ri¯ρij+(Ri¯+ri¯)ω,Λ*=(∑i=1n‍|ri|0+∑i=1n‍∑j=1n‍|Aij|0eHj+∑i=1n‍∑j=1n‍|Bij|0eHj+∑i=1n‍∑j=1n‍|Cij|0eHj+(|ei|0+|fi|0)|βi*|0+|θi*|0αiLeHi∑i=1n‍)×(1-∑i=1n‍∑j=1n‍|D0,ij|0eHj)-1,Λi=max{lnri¯-∑j=1,j≠in‍(Aij¯+Bij¯+Cij¯)eHjAii¯+Bii¯+Cii¯+(ei¯++fi¯)(|βi*|0+|θi*|0)/αiL|lnriAii¯+Bii¯+Cii¯|,maxmax|lnri¯-∑j=1,j≠in(Aij¯+Bij¯+Cij¯)eHjAii¯+Bii¯+Cii¯+(ei¯+fi¯)(|βi*|0+|θi*|0/αiL)|},
where Γij(t),Γi1(t), and Γi2(t) are defined by (27), and D0,ij(t)=Dij(t)(1-γij′(t)).Theorem 10.
Suppose that the following conditions hold:(1)
the system of algebraic equations(32)f*(μ)=(ri¯-∑j=1n‍((Γi1¯+Γi2¯)(Aij¯+Bij¯+Cij¯)μj+(Γi1¯+Γi2¯)μi)∑j=1n‍)n×1=0has a unique positive solutionμ*=(μ1*,…,μn*);(2)
Aij¯+Bij¯+Cij¯>0, ri¯>∑j=1,j≠in(Aij¯+Bij¯+Cij¯)eHj, γij′(t)<1, τij′(t)<1 and Γij(t)>0;(3)
K0=:L0eM0<1.Then the system (1) and (2) has at least one positive ω-periodic solution.To prove the previous theorem, we make the change of variables(33)yi(t)=exi(t),i=1,2,…,n.
Then, the system (25) can be rewritten in the following form:
(34)xi′(t)=ri(t)-∑j=1n‍Aij(t)exj(t)-∑j=1n‍Bij(t)∫-∞t‍kij(t-s)exj(s)ds-∑j=1n‍Cij(t)exj(t-γij(t))-∑j=1n‍D0ij(t)xj′(t-τij(t))exj(t-τij(t))-ei(t)(ψiexi)(t)-fi(t)(ψiexi)(t)(t-δi(t)).
Let X denote the linear space of real value continuous ω-periodic functions on R. The linear space X is a Banach space with the usual norm ∥x∥0=maxt∈R|x(t)|=maxt∈R∑j=1n|xi(t)| for a given x=(x1,…,xn)∈X.We define the following maps:(35)b:R×C⟶Rn,b(t,ϕ)=(b1(t,ϕ),b2(t,ϕ),…,bn(t,ϕ)),bi(t,ϕ)=-∑j=1n‍Dij(t)eϕj(-γij(t)),f:R×C⟶Rn,f(t,ϕ)=(f1(t,ϕ),f2(t,ϕ),…,fn(t,ϕ)),fi(t,ϕ)=ri(t)-∑j=1n‍Aij(t)eϕj(0)-∑j=1n‍Bij(t)∫-∞0‍kij(t-s)eϕj(s)ds-∑j=1n‍Cij(t)eϕj(-τij(t))+∑j=1n‍Dij′(t)eϕj(-γij(t))-ei(t)(ψieϕi)(t)-fi(t)(ψieϕi)(t-δi(t)),i=1,2,…,n,ϕ=(ϕ1,ϕ2,…,ϕn)∈C,t∈R.
Clearly, b:R×C→Rn and f:R×C→Rn are complete continuation functions, and system (34) takes the form(36)ddt[x(t)-b(t,xt)]=f(t,xt).
In the proof of our main result below, we will use the following two important lemmas.Lemma 11.
If the assumptions of Theorem10 are satisfied and if Ω={ϕ∈C:∥ϕ∥<M}, where M>M0 such that k=L0eM<1, then |b(t,ϕ)-b(t,φ)|≤k∥ϕ-φ∥, for t∈R and ϕ,φ∈Ω.Proof.
Fort∈R and ϕ,φ∈Ω, we have
(37)|bi(t,ϕ)-bi(t,φ)|≤∑j=1n‍Dij(t)|eϕj(-γij(t))-eφj(-γij(t))|≤∑j=1n‍Dij(t)eσijϕj(-γij(t))+(1-σij)φj(-γij(t))×|ϕj(-γij(t))-φj(-γij(t))|,
for some σij∈(0,1). Then, we get
(38)|bi(t,ϕ)-bi(t,φ)|≤∑j=1n‍|Dij|0eM∥ϕ-φ∥.
Hence,
(39)|b(t,ϕ)-b(t,φ)|≤∑i=1n‍∑j=1n‍|Dij|0eM∥ϕ-φ∥≤L0eM∥ϕ-φ∥=k∥ϕ-φ∥.
The proof of Lemma 11 is thus completed.Lemma 12.
If the assumptions of Theorem10 are satisfied, then every solution x∈X of the system
(40)ddt[x(t)-λb(t,xt)]=f(t,xt),λ∈(0,1)
satisfies ∥x∥0≤M0.Proof.
Let(d/dt)[x(t)-λb(t,xt)]=f(t,xt), for x∈X, that is,
(41)[xi(t)+λ∑j=1n‍Dij(t)exj(t-τij(t))]′=λ[ri(t)-∑j=1n‍Aij(t)exj(t)-∑j=1n‍Bij(t)×∫-∞t‍kij(t-s)exj(s)ds-∑j=1n‍Cij(t)exj(t-γij(t))+∑j=1n‍Dij′(t)exj(t-τij(t))-ei(t)(ψiexi)(t)-fi(t)(ψiexi)×(t)(t-δi(t))∑j=1n‍],(t)(t)(t)(t)1i=1,2,…,n;λ∈(0,1),
which yields, after integrating from 0 to ω, that
(42)∫0ω‍∑j=1n‍[Aij(t)exj(t)+Bij(t)×∫-∞t‍kij(t-s)exj(s)ds+Cij(t)exj(t-γij(t))-Dij′(t)exj(t-τij(t))+ei(t)×(ψiexi)(t)+fi(t)(ψiexi)(t)(t-δi(t))]dt=∫0ω‍∑j=1n‍Γij(t)exj(t)dt+∫0ω‍[(t-δi(t))ei(t)(ψiexi)(t)+fi(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)(t)22×(ψiexi)(t)(t-δi(t))]dt=∫0ω‍ri(t)dt=ri¯ω,i=1,2,…,n,
where Γij(t) is defined by (27). From (41), we derive
(43)∫0ω|[xi(t)+λ∑j=1n‍Dij(t)exj(-τij(t))]|′‍dt=λ∫0ω|[ri(t)-∑j=1n‍Aij(t)exj(t)-∑j=1n‍Bij(t)‍×∫-∞t‍kij(t-s)exj(s)ds-∑j=1n‍Cij(t)exj(t-γij(t))+∑j=1n‍Dij′(t)exj(t-τij(t))-ei(t)(ψiexi)(t)-fi(t)(ψiexi)×(t)(t-δi(t))∑j=1n‍]|dt≤∫0ω‍|ri(t)|dt+∫0ω‍|∑j=1n‍[∫0ω‍Aij(t)exj(t)+Bij(t)≤∫0ω‍|ri(t)|dt+∫0ω‍222222222×∫-∞t‍kij(t-s)exj(s)ds≤∫0ω‍|ri(t)|dt+∫0ω‍222222222+Cij(t)exj(t-γij(t))≤∫0ω‍|ri(t)|dt+∫0ω‍222222222-Dij′(t)exj(t-τij(t))≤∫0ω‍|ri(t)|dt+∫0ω‍222222222+ei(t)(ψiexi)(t)≤∫0ω‍|ri(t)|dt+∫0ω‍222222222+fi(t)(ψiexi)(t)≤∫0ω‍|ri(t)|dt+∫0ω‍222222222×(t-δi(t))∫0ω‍]∑j=1n‍|dt.
It follows from (41)–(43) that
(44)∫0ω|[xi(t)+λ∑j=1n‍Dij(t)exj(-τij(t))]|′‍dt≤(Ri¯+ri¯)ω,≤∫0ω‍|ri(t)|dt+∫0ω‍2222222222222222i=1,2,…,n.
By amplification, it follows from (42) that
(45)ri¯ω≥∑j=1n‍∫0ω‍Γij(t)exj(t)dt=∑j=1n‍∫0ω‍[Γij(t)exj(t)-(ρijexj(t)+ρijDij(t)exj(t-τij(t)))+(ρijexj(t)+ρijDij(t)exj(t-τij(t)))]dt=∑j=1n‍∫0ω[Γij(t)exj(t)-(ρijexj(t)+ρijDij(t)exj(t-τij(t)))]dt‍+∑j=1n‍∫0ω‍[ρijexj(t)+ρijDij(t)exj(t-τij(t))]dt.
In view of Remark 9 and by a similar analysis, we obtain
(46)∑j=1n‍∫0ω‍[Γij(t)exj(t)-(ρijexj(t)+ρijDij(t)exj(t-τij(t)))]dt=∑j=1n‍∫0ω‍[Γij(s)-ρij-ρijDij(νij(s))1-τij′(νij(s))]exj(s)dt.
As ρij=ΓijL(1-τij′)L/((1-τij′)L+|Dij|0), it follows that Γij(s)-ρij-ρij(Dij(νij(s))/(1-τij′(νij(s))))≥0. So we find from (45) that
(47)ri¯ω≥∑j=1n‍∫0ω‍[ρijexj(t)+ρijDij(t)exj(t-τij(t))]dt.
That is,
(48)ri¯ω≥∫0ω‍∑j=1n‍[ρijexj(t)+ρijDij(t)exj(t-τij(t))]dt.
By the mean value theorem, we see that there exist points ζi∈[0,ω],(i=1,…,n) such that(49)ri¯≥∑j=1n‍ρijexj(ζi)+∑j=1n‍ρijDij(ζi)exj(ζi-τij(ζi)),i=1,…,n,
which implies that
(50)xi(ζi)≤lnri¯ρii,Dij(ζi)exj(ζi-τij(ζi))≤ri¯ρij,≤∫0ω‍|ri(t)|21i=1,…,n.
By (44) and (50), we can see that
(51)xi(t)+λ∑j=1n‍Dij(t)exj(t-τij(t))≤xi(ζi)+λ∑j=1n‍Dij(ζi)exj(ζi-τij(ζi))+∫0ω‍|[xi(t)+λ∑j=1n‍Dij(t)exj(t-τij(t))]′|dt≤lnri¯ρii+∑j=1n‍ri¯ρij+(Ri¯+ri¯)ω=:Hi,i=1,2,…,n.
For λ∑j=1nDij(t)exj(t-τij(t))≥0, one can find that
(52)xi(t)≤Hi,i=1,…,n.
Besides, from (41), we have
(53)xi′(t)=λ[ri(t)-∑j=1n‍Aij(t)exj(t)-∑j=1n‍Bij(t)×∫-∞t‍kij(t-s)exj(s)ds-∑j=1n‍Cij(t)exj(t-τij(t))-∑j=1n‍D0,ij(t)xj′(t-γij(t))exj(t-γij(t))-ei(t)(ψiexi)(t)-fi(t)(ψiexi)(t)×(t-δi(t))∑j=1n‍],i=1,2,…,n.
Notice that for all t∈R, one has
(54)(ψi)(t)=∫tt+ω‍Gi(t,s)[βi*(s)+θi*(s)]ds=∫tt+ω‍Gi(t,s)αi(s)βi*(s)+θi*(s)αi(s)ds≤|βi*|0+|θi*|0αiL∫tt+ω‍Gi(t,s)αi(s)ds=|βi*|0+|θi*|0αiL,i=1,…,n.
Then, by (52) and (54), we get
(55)|xi′|0≤|ri(t)+∑j=1n‍Aij(t)exj(t)+∑j=1n‍Bij(t)×∫-∞t‍kij(t-s)exj(s)ds+∑j=1n‍Cij(t)exj(t-γij(t))+∑j=1n‍D0,ij(t)xj′(t-τij(t))exj(t-τij(t))+ei(t)×(ψiexi)(t)+fi(t)(ψiexi)(t)(t-δi(t))∑j=1n‍|≤|ri|0+∑j=1n‍|Aij|0eHj+∑j=1n‍|Bij|0eHj+∑j=1n‍|Cij|0eHj+∑j=1n‍|D0,ij|0|xj′|0eHj+(|ei|0+|fi|0)(t)(t)(t)(t)(t)(t)(t)2×|βi*|0+|θi*|0αiLeHi,(t)(t)(t)(t)(t)(t)(t)221(t)i=1,2,…,n.
Furthermore, we have
(56)∥x′∥0=∑i=1n‍|xi′|0≤∑i=1n‍|ri|0+∑i=1n‍∑j=1n‍|Aij|0eHj+∑i=1n‍∑j=1n‍|Bij|0eHj+∑i=1n‍∑j=1n‍|Cij|0eHj+(|ei|0+|fi|0)|βi*|0+|θi*|0αiLeHi+∑i=1n‍∑j=1n‍|D0,ij|0∥x′∥0eHj.
By the assumption (3) of Theorem 10, we see that
(57)∑i=1n‍∑j=1n‍|Dij|0eHj≤∑i=1n‍∑j=1n‍|D0,ij|0eH≤∑i=1n‍∑j=1n‍|D0,ij|0eM0<1.
Then, we have
(58)∥x′∥0≤((1-∑i=1n‍∑j=1n‍|D0,ij|0eHj)-1∑i=1n‍|ri|0+∑i=1n‍∑j=1n‍|Aij|0eHj+∑i=1n‍∑j=1n‍|Bij|0eHj+∑i=1n‍∑j=1n‍|Cij|0eHj+(|ei|0+|fi|0)|βi*|0+|θi*|0αiLeHi∑j=1n‍)×(1-∑i=1n‍∑j=1n‍|D0,ij|0eHj)-1=:Λ*.
Since x(t)=(x1(t),…,xn(t))∈X, there exists a ξi∈[0,ω] such that
(59)xi(ξi)=inft∈[0,ω]xi(t),i=1,…,n.
It follows from (42) that
(60)ri¯ω≥∑j=1n‍∫0ω‍Γij(t)exj(t)dt=∑j=1n‍exj(ξj)∫0ω‍Γij(t)dt,i=1,…,n.
It follows from (30) and (60) that
(61)ri¯≥∑j=1n‍exj(ξj)(Aij¯+Bij¯+Cij¯),i=1,…,n.
From (61), we obtain
(62)exi(ξi)(Aii¯+Bii¯+Cii¯)≤ri¯,i=1,…,n.
As Aii¯+Bii¯+Cii¯>0, it follows from the previous formula that
(63)xi(ξi)≤lnri¯Aii¯+Bii¯+Cii¯,i=1,…,n.
On the other hand, there also exists a ηi∈[0,ω] such that
(64)xi(ηi)=supt∈[0,ω]xi(t),i=1,…,n.
It follows from (42), (54), and (64) that
(65)ri¯≤∑j=1n‍Γij¯exj(ηi)+(ei¯+fi¯)|βi*|0+|θi*|0αiLexi(ηi)=∑j=1,j≠in‍Γij¯exj(ηi)+[Γii¯+(ei¯+fi¯)|βi*|0+|θi*|0αiL]exi(ηi),i=1,…,n.
From (41), (54), and (64), we can have
(66)[Aii¯+Bii¯+Cii¯+(ei¯+fi¯)|βi*|0+|θi*|0αiL]exi(ηi)≥ri¯-∑j=1,j≠in‍(Aij¯+Bij¯+Cij¯)exj(ηi)≥ri¯-∑j=1,j≠in‍(Aij¯+Bij¯+Cij¯)eHj,i=1,…,n.
That is,
(67)xi(ηi)≥lnri¯-∑j=1,j≠in(Aij¯+Bij¯+Cij¯)eHjAii¯+Bii¯+Cii¯+(ei¯+fi¯)(|βi*|0+|θi*|0/αiL),i=1,…,n.
Now, from (60) and (63) we know that there exist ςi∈[0,ω](i=1,…,n) such that
(68)|xi(ςi)|≤max{|lnri¯-∑j=1,j≠in(Aij¯+Bij¯+Cij¯)eHjAii¯+Bii¯+Cii¯+(ei¯++fi¯)(|βi*|0+|θi*|0/αiL)||lnriAii¯+Bii¯+Cii¯|,maxxxxx|lnri¯-∑j=1,j≠in(Aij¯+Bij¯+Cij¯)eHjAii¯+Bii¯+Cii¯+(ei¯+fi¯)(|βi*|0+|θi*|0/αiL)|}=:Λi,i=1,…,n.
From (54), (64), and Lemma 3, we have
(69)|xi|≤|xi(ςi)|+12∫0ω‍|xi′(t)|dt≤Λi+12∫0ω‍|xi′|dt,i=1,…,n.
Then,
(70)∥x∥0≤∑i=1n‍|xi|≤∑i=1n‍Λi+12∫0ω‍∥x′∥0dt<∑i=1n‍Λi+12Λ*ω≤M0.
Obviously, M0 is independent of λ; the proof of Lemma 12 is completed.Based on the previous results, we can now apply Lemma1 and Remark 2 to (34) and obtain a proof of Theorem 10.Proof.
Obviously, forM as given in Lemma 11, condition (i) in Lemma 1 is satisfied. Let h(μ)=(h1(μ),…,hn(μ)). Since
(71)hi(μ)=∫0ω‍fi(s,μ^)ds=∫0ω‍ri(t)dt-∑i=1n‍∫0ω‍Aij(t)dteμj-∑i=1n‍∫0ω‍Bij(t)dteμj-∑i=1n‍∫0ω‍Cij(t)dteμj-∫0ω‍[ei(t)+fi(t)]dteμi={ri¯-∑j=1n‍[Aij¯+Bij¯+Cij¯eμj+(ei¯+fi¯)eμiCij)¯eμj]∑j=1n‍}ω
and M>∑i=1n|lnμi*|, we have h(μ)≠0 for any μ∈∂BM(Rn). That is, condition (ii) in Lemma 1 holds. At last, we verify that condition (iii) of Lemma 1 also holds. By assumption (1) of Theorem 10 and the formula for the Brouwer degree (see Theorem 2.2.3 in [35, 36]), a straightforward calculation shows that
(72)deg(h,BM(Rn))=∑μ∈h-1(0)⋂‍BM(Rn)‍signdetDh(μ)=sign{+(ei¯+fi¯)eμi*(Aij¯+Bij¯+Cij¯)e∑j=1nμj*(-1)ndet×[(Aij¯+Bij¯+Cij¯)e∑j=1nμj*+(ei¯+fi¯)eμi*(Aij¯+Bij¯+Cij¯)e∑j=1nμj*]}≠0.
By now, all the assumptions required in Lemma 1 hold. It follows from Lemma 1 and Remark 2 that system (34) has an ω-periodic solution. Returning to yi(t)=exi(t), we infer that systems (18) and (19) have at least one positive ω-periodic solution. By Lemmas 7 and 8, (N*(t),u*(t))T=(N1*(t),…,Nn*(t),u1*(t),…,un*(t))T is the unique positive periodic solution of the system (1) and (2), where Ni*(t)=∏0<tk<t‍Δikyi*(t)(i=1,2,…,n). The proof of Theorem 10 is complete.Consider the following:(73)Ni′(t)=Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)-∑j=1n‍bij(t)∫-∞t‍kij(t-s)Nj(s)ds-∑j=1n‍cij(t)Nj(t-γij(t))-∑j=1n‍dij(t)Nj′(t-τij(t))-ei(t)ui(t)-fi(t)ui(t-δi(t))∑j=1n‍],ui′(t)=-αi(t)ui(t)+βi(t)Ni(t)+θi(t)Ni(t-σi(t)),i=1,2,…,n,
which is a special case of system (1) without impulse. We get easily the following result. Here, we have the following notations:
(74)ρij*=Γij*L(1-γij′)L(1-γij′)L+|dij|0,Ri¯=1ω∫0ω‍|ri(t)|dt,L*=max{∑i=1n‍∑j=1n‍|dij|0,∑i=1n‍∑j=1n‍|d0,ij|0},M*=max{∑i=1n‍|lnμi*|0,H*,12ωΘ*+∑i=1n‍Δi},H*=maxi∈[1,n]{Hi*},Hi*=lnri¯ρii*+∑j=1n‍ri¯ρij*+(Ri¯+ri¯)ω,Θ*=((1-∑i=1n‍∑j=1n‍|d0,ij|0eHj*)-1∑i=1n‍|ri|0+∑i=1n‍∑j=1n‍|aij|0eHj*+∑i=1n‍∑j=1n‍|bij|0eHj*+∑i=1n‍∑j=1n‍|cij|0eHj*+(|ei|0+|fi|0)|βi|0+|θi|0αiLeHi*∑j=1n‍)×(1-∑i=1n‍∑j=1n‍|d0,ij|0eHj*)-1,Θi=max{ri¯-∑j=1,j≠in(aij¯+bij¯+cij¯)eHj*aii¯+bii¯+cii¯+(ei¯++fi¯)(|βi|0+|θi|0/αiL)|lnriaii¯+bii¯+cii¯|,|lnri¯-∑j=1,j≠in(aij¯+bij¯+cij¯)eHj*aii¯+bii¯+cii¯+(ei¯+fi¯)(|βi|0+|θi|0/αiL)|},Γij*(t)=aij(t)+bij(t)+cij(ϑij(t))1-γij′(ϑij(t))-dij′(νij(t))1-τij′(νij(t)),Γi1(t)=ei(t)(ψi1)(t),Γi2(t)=fi(t)(ψi1)(t-σi(t)),d0,ij(t)=dij(t)(1-τij′(t)),
and ϑij(t),νij(t) represent the inverse function of t-γij(t),t-τij(t)(i,j=1,2,…,n), respectively.Corollary 13.
Suppose that the following conditions hold;(1)
the system of algebraic equations(75)f*(μ)=(ri¯-∑j=1n‍((Γi1¯+Γi2¯)(aij¯+bij¯+cij¯)μj+(Γi1¯+Γi2¯)μi)∑j=1n‍)n×1=0has a unique positive solutionμ*=(μ1*,…,μn*);(2)
aij¯+bij¯+cij¯>0, ri¯>∑j=1,j≠in(aij¯+bij¯+cij¯)eHj*, τij′(t)<1, γij′(t)<1 and Γij*(t)>0;(3)
K*=:L*eM*<1. Then, (73) has at least one positive ω-periodic solution.Proof.
Its proof is similar to the proof of Theorem10. Here, we omit it.Similarly, we can get the following results.Theorem 14.
Assume that conditions of Theorem10 hold, and then, the conclusion of Theorem 10 holds for the following system:
(76)Ni′(t)=-Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)-∑j=1n‍bij(t)∫-∞t‍kij(t-s)Nj(s)ds-∑j=1n‍cij(t)Nj(t-γij(t))-∑j=1n‍dij(t)Nj′(t-τij(t))-ei(t)ui(t)-fi(t)ui(t-δi(t))∑j=1n‍],i=1,2,…,n,t≠tk,ui′(t)=-αi(t)ui(t)+βi(t)Ni(t)+θi(t)Ni(t-σi(t)),t≥0,ΔNi(tk)=(pik+qik)Ni(tk),i=1,2,…,n,k=1,2,….Proof.
Its proof is similar to the proof of Theorem10. Here, we omit it.Corollary 15.
Assume that conditions of Corollary13 hold, and then, the conclusion of Corollary 13 holds for the following system:
(77)Ni′(t)=-Ni(t)[ri(t)-∑j=1n‍aij(t)Nj(t)-∑j=1n‍bij(t)∫-∞t‍kij(t-s)Nj(s)ds-∑j=1n‍cij(t)Nj(t-τij(t))-∑j=1n‍dij(t)Nj′(t-γij(t))],ui′(t)=-αi(t)ui(t)+βi(t)Ni(t)+θi(t)Ni(t-σi(t)),i=1,2,…,n.Proof.
Its proof is similar to the proof of Theorem10. Here, we omit it.Remark 16.
Whenαi(t)=βi(t)=θi(t)=0, we can derive some immediate corollaries from Theorems 10 and 14; thus, our results generalize the corresponding results in [14].


## 4. Applications

In order to illustrate some features of our main result, in the following, we will apply Theorem10 to some special cases, which have been studied extensively in the literature.Application 17.
We consider ann-species neutral delay competition system in a periodic environment with impulse:
(78)Ni′(t)=±Ni(t)[ai(t)-∑j=1n‍bij(t)Nj(t)(t)(t)(t)(t)-∑j=1n‍cij(t)Nj(t-τij(t))(t)(t)(t)(t)-∑j=1n‍dij(t)Nj′(t-γij(t))(t)(t)(t)(t)(t)(t)2-ei(t)ui(t)-fi(t)ui(t)(t)(t)(t)(t)(t)2×(t-δi(t))∑j=1n‍],(t)(t)(t)(t)2i=1,2,…,n,t≠tk,ui′(t)=-αi(t)ui(t)+βi(t)Ni(t)+θi(t)Ni(t-σi(t)),t≥0,ΔNi(tk)=(pik+qik)Ni(tk),i=1,2,…,n,k=1,2,…,
where bij, cij, ei, fi, αi, βi, θi∈C(R,[0,+∞)), dij∈C1(R,[0,+∞)), γij∈C1(R,R), and τij∈C2(R,R) are continuous ω-periodic functions. And ai∈C(R,R) are continuous ω-periodic functions with ∫0ω‍ai(t)dt>0. Here, we give the following notations:
(79)Ai¯=1ω∫0ω‍|ai(t)|dt,Bij(t)=bij(t)∏0<tk<t‍(1+pik+qik),Cij(t)=cij(t)∏0<tk<t-τij(t)‍(1+pik+qik),Dij(t)=dij(t)∏0<tk<t-γij(t)‍(1+pik+qik),D0,ij(t)=Dij(t)(1-γij′(t)),βi*(t)=βi(t)∏0<tk<t‍(1+pik+qik),θi*(t)=θi(t)∏0<tk<t‍(1+pik+qik),Γij(t)=Aij(t)+Bij(t)+Cij(ϑij(t))1-γij′(ϑij(t))-Dij′(νij(t))1-τij′(νij(t)),Γi1(t)=ei(t)(ψi1)(t),Γi2(t)=fi(t)(ψi1)(t-σi(t)),L1=max{∑i=1n‍∑j=1n‍|Dij|0,∑i=1n‍∑j=1n‍|D0,ij|0},M1=max{∑i=1n‍|lnμi*|0,H*,12ωΛ*+∑i=1n‍Λi},H*=maxi∈[1,n]{Hi*},Hi*=lnai¯ρii*+∑j=1n‍ai¯ρij*+(Ai¯+ai¯)ω,ρij*=Γij*L(1-γij′)L(1-γij′)L+|Dij|0,Λ*=(∑i=1n‍|ai|0+∑i=1n‍∑j=1n‍|Bij|0eHj*max+∑i=1n‍∑j=1n‍|Cij|0eHj*maxmaxxxxx+(|ei|0+|fi|0)|βi*|0+|θi*|0αiLeHi*∑j=1n‍)max×(1-∑i=1n‍∑j=1n‍|D0,ij|0eHj*)-1,Λi=max{|lnai¯-∑j=1,j≠in(Bij¯+Cij¯)eHj*Bii¯+Cii¯+((|βi*|0+|θi*|0)/αiL)||lnaiBii¯+Cii¯|,maxxxxx|lnai¯-∑j=1,j≠in(Bij¯+Cij¯)eHj*Bii¯+Cii¯+(ei¯+fi¯)((|βi*|0+|θi*|0)/αiL)|},maxmaxmaxmaxmaxmaxmaxmaxmaxmi,j=1,2,…,n,
where ϑij(t),νij(t) represent the inverse function of t-γij(t),t-τij(t), respectively. Applying Theorem 10 to (78), we can obtain the following theorem.Theorem 18.
Assume that the following conditions are satisfied:(1)
the system of algebraic equations(80)f*(μ)=(ai¯-∑j=1n‍((Bij¯+Cij¯)μj+(Γi1¯+Γi2¯)μi))n×1=0has a unique positive solutionμ*=(μ1*,…,μn*);(2)
Bij¯+Cij¯>0, ai¯>∑j=1,j≠in(Bij¯+Cij¯)eHj, γij′(t)<1, τij′(t)<1, and Γij(t)>0;(3)
K1=:L1eM1<1.
Then, (78) has at least one positive ω-periodic solution.Remark 19.
Whenαi(t)=βi(t)=θi(t)=ui(t)=0, pik+qik=0, we can derive some immediate corollaries of Theorem 18, then, Theorem 18 generalizes the corresponding results in [9, 10]. On the other hand, when pik+qik=0, we can derive an immediate corollary of Theorem 18, then, Theorem 18 generalizes the corresponding results in  [11].Application 20.
We consider the single specie neutral delay logistic equation with impulse:(81)N′(t)=±N(t)[a(t)-b(t)N(t)-∑i=1n‍ci(t)∫-∞t‍ki(t-s)N(s)ds-∑j=1m‍dj(t)N(t-γj(t))-∑l=1p‍el(t)N′(t-τl(t))-f(t)u(t)-g(t)u(t-δ(t))],t≠tk,u′(t)=-α(t)u(t)+β(t)N(t)+θ(t)N(t-σ(t)),t≥0,N(tk+)=(pk+qk)N(tk),k=1,2,….,
where b, ci, dj, f, g, α, β, θ∈C(R,[0,+∞)), el∈C1(R,[0,+∞)), γj∈C1(R,R), and τl∈C2(R,R) are continuous ω-periodic functions. And a∈C(R,R) are continuous ω-periodic functions with ∫0ω‍a(t)dt>0. Here, we have the following notations:
(82)A¯=1ω∫0ω‍|a(t)|dt,B(t)=b(t)∏0<tk<t‍(1+pk+qk),Ci(t)=ci(t)∏0<tk<t‍(1+pk+qk),Dj(t)=dj(t)∏0<tk<t-σj(t)‍(1+pk+qk),El(t)=el(t)∏0<tk<t-τl(t)‍(1+pk+qk),E0,l(t)=El(t)(1-τl′(t)),β*(t)=β(t)∏0<tk<t‍(1+pik+qik),θ*(t)=θ(t)∏0<tk<t‍(1+pik+qik),ρ=Γ1L(1-τl′)L(1-τl′)L+|El|0,L2=max{∑l=1p‍|El|0,∑l=1p‍|E0,l|0},M2=max{H*,Δ*},H*=lna¯ρ+a¯ρ+(A¯+|a¯)ω,Δ*=ω2(⋃|a|0+[|β*|0+|θ*|0αL|B|0+∑i=1n|Ci|0+∑j=1m|Dj|0+(|f|0+|g|0)×|β*|0+|θ*|0αL]eH*⋃)×(1-|E0,l|0eH*)-1+|lna¯B¯+∑i=1nCi¯+∑j=1mDj¯+Γ2¯+Γ3¯|,Γ1(t)=B(t)+∑i=1n‍Ci(t)+∑j=1m‍Dj(μj(t))1-σ′(μj(t))-∑l=1p‍El′(νl(t))1-τl′(νl(t)),Γ2(t)=f(t)(ψ1)(t),Γ3(t)=g(t)(ψ1)(t-σ(t)),andμj(t),νl(t) represent the inverse function of t-γj(t),t-τl(t), respectively. Applying Theorem 10 to (81), we can obtain the following theorem.Theorem 21.
Assume that the following conditions are satisfied:(1)
B¯+∑i=1nCi¯+∑j=1mDj¯+Γ2¯+Γ3¯>0, γj′(t)<1, τl′(t)<1 and Γ1(t)>0;(2)
K2=:L2eM2<1.
Then, system (81) has at least one positive ω-periodic solution.Remark 22.
Wheni=0, j=l=1, α(t)=β(t)=θ(t)=u(0)=0, σj(t)=τl(t), and pk+qk=0, we can derive an immediate corollary of Theorem 21, which is also an answer to the open problem 9.2 due to Kuang  [2]. On the other hand, when i=0, α(t)=β(t)=θ(t)=u(t)=0, and pk+qk=0, we can derive some immediate corollaries of Theorem 21 (that is the corresponding results in  [7, 8]), therefore, our result improves and generalizes the corresponding result in  [7, 8]. Moreover, when l=0, α(t)=β(t)=θ(t)=u(t)=0, we can see that our Theorem 21 can hold without the assumption a¯>0. When a¯<0, Wang's main result (see Theorem 3.1 in  [13]) cannot be applied. Therefore, in comparison with [13], our result improves and generalizes the result in  [13].



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