On the existence of positive periodic solutions for n−species Lotka-Volterra competitive systems with distributed delays and impulses

On the existence of positive periodic solutions for n−species Lotka-Volterra competitive systems with distributed delays and impulses Mimia Benhadria, Tom´as Caraballob aLAMAHIS Lab, Faculty of Sciences, Department of Mathematics, Univ Skikda, P.O. Box 26, Skikda 21000, Algeria. emails: [email protected] bDepartamento de Ecuaciones Difererenciales y An´alisis Num´erico, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla (Spain) email: [email protected] Abstract In this paper, we investigate the existence of positive periodic solutions for an n-species Lotka-Volterra system with distributed delays and impulsive effect. In the process we use integrating factors and convert the given Lotka-Volterra differential equation into an equivalent integral equation. Then we construct appropriate mappings and use Krasnoselskii’s fixed point theorem to show the existence of a positive periodic solution of this system. In particular, the results improve some previous ones in the literature. Finally, as an application, we exhibit an example to illustrate the effectiveness of our abstract results. AMS Subject Classifications: 34K20, 34K13, 92B20 Keywords: Krasnoselskii’s fixed point theorem; positive periodic solutions; Lotka-Volterra competition systems; Variable delays; impulses. 1 Introduction It is well known that the theory of impulsive differential equations has become an important area of scientific activity. Many evolution processes are characterized by the fact that at certain moments of time they experience an abrupt change of state. These short term perturbations act instantaneously, that is in the form of impulses. Equations of this kind are found in almost every domain of applied sciences, numerous examples can be found in, e.g., [2, 3, 18, 19, 27]. For example, many biological phenomena involving fields such as economics, 1 mechanics, electronics, telecommunications, medicine and biology, etc. (see [19]). Thus, impulsive differential equations appear as a natural description of observed evolution phenomena of several real world problems. However, besides impulsive effects, time delay is present in many fields in our society. In recent years, non-autonomous delay differential equations have been used in the study of population ecology and infectious diseases, population dynamics. Indeed, a famous model for population dynamics is the Lotka-Volterra competition system. Due to its theoretical and practical significance, Lotka-Volterra systems have been extensively and intensively studied for the past few years (see, [5, 6, 14, 23, 24, 25, 26, 29, 30]). On the other hand, a very basic and important qualitative problem is the study of periodic solutions of delay differential equations with or without impulsive effects which has attracted the interest of many mathematicians (we refer the reader to [1, 7, 10, 12, 15, 16, 17, 20, 21, 22, 28]). For instance, in 2006, by using the method of Krasnoselskii’s fixed point theorem, Tang and Zhou [25] investigated the existence of positive periodic solutions of the following system with deviating arguments: x0 i(t) = xi(t) ri(t)− n X j=1 aij(t)xj(t−τij (t)) , i = 1,2, ..., n. (1.1) By the same method as the one in [25], Zhang et al. investigated in [30] the existence and global attractivity of positive periodic solutions of 3-species Lotka- Volterra predator-prey systems with infinite delays as follows:                            x0 1(t) = x1(t)r1(t)−c11(t)x1(t)−c12(t)Rt −∞ K12 (s−t)x2(s)ds +c13(t)Rt −∞ K13 (s−t)x3(s)ds, x0 2(t) = x2(t)r2(t)−c21(t)Rt −∞ K21 (s−t)x1(s)ds −c22(t)x2(t) +c23(t)Rt −∞ K23 (s−t)x3(s)ds, x0 3(t) = x3(t)r3(t) + c31(t)Rt −∞ K31 (s−t)x1(s)ds +c32(t)Rt −∞ K32 (s−t)x2(s)ds −c33(t)x3(t). (1.2) Very recently, Benhadri et al. improved in [1] the results of Zhang et al. [25] to the generalized nonimpulsive nonlinear Lotka-Volterra competition with deviating arguments of the form: x0 i(t) = xi(t)  ri(t)− n X j=1 aij(t)xj(t)− n X j=1 bij(t)fj(xj(t)) − n X j=1 cij(t)gj(xj(t−τij (t)))  ,(1.3) i= 1,2, ..., n. 2 The authors derived some sufficient conditions for the existence of positive periodic solutions of (1.3) . In this paper, motivated by the content in [1] and [25], we generalize system (1.1) to a model with variable and distributed delays and impulses, x0 i(t) = xi(t)  ri(t)− n X j=1 aij(t)hj(xj(t)) − n X j=1 bij(t)fj(xj(t−τij (t))) − n X j=1 cij(t)Zt −∞ Dij(t, s)gj(xj(s)ds  ,(1.4) i= 1,2, ..., n, t 6=tk, t > 0, xit+ k−xit− k=Iik (tk, xi(tk)) , t =tk, k ∈Z+, where x(t) = [x1(t), x2(t), .., xn(t)]T∈Rn.The expression ∆xi(tk) = xit+ k− xit− k=Iik (tik, xi(tk)) denotes the impulse at moment tk, and t1< t2< ..., is a strictly increasing sequence such that tkgoes to infinity, xit+ kand xit− k stand for the right-hand and the left-hand limits of xi(t) at the impulsive moment tkrespectively. Consider that Iik (·,·)∈C(R+×R+,R+), i = 1,2, ..., n, k= 1,2, ..., shows the impulsive perturbation at the moment tk. Since we are searching for the existence of periodic solutions for equation (1.4) ,it is natural to assume that ri, aij, bij, cij ∈C(R+,R+) are all ω−periodic functions (ω > 0) with respect to t, aij (t+ω) = aij (t), τij (t+ω) = τij (t), Dij (t+ω, s +ω) = Dij (t, s), bij (t+ω) = bij (t) and cij (t+ω) = cij (t) (1.5) for, i, j = 1,2, ..., n, with τij being scalar functions, continuous, and τij (t)≥ τ∗ ij >0 with bri=1 ωZω 0 ri(s)ds > 0, baij =Tj ωZω 0 aij(s)ds ≥0, b bij =Fj ωZω 0 bij(s)ds ≥0,bcij =RjEij ωZω 0 cij(s)ds ≥0, b βik =1 ωX 0≤tk<ω λik (tk)≥0, k ∈Z+,(1.6) for i, j = 1,2, ..., n, where Tj, Fjand λik, Rj, Eij are given in (H1)−(H5). We also assume that the functions Dij ∈C(R+×R+,R+) and fi, gi, hi∈ 3 C(R+,R+), i, j = 1,2, ..., n, f(x(t)) = [f1(x1(t)) , f2(x2(t)) , ..., fn(xn(t))]T∈Rn +, g(x(t)) = [g1(x1(t)) , g2(x2(t)) , ..., gn(xn(t))]T∈Rn +, h(x(t)) = [h1(x1(t)) , h2(x2(t)) , ..., hn(xn(t))]T∈Rn +, are positive and continuous in their respective arguments. Throughout this paper, we will assume the following hypotheses: (H1) There exist nonnegative constants Tj, Tj,such that for all x∈R+, Tjx≤hj(x)≤Tjx, j = 1,2, ..., n. (1.7) (H2) There exist nonnegative constants Fj, Fjsuch that for all x∈R+, Fjx≤fj(x)≤Fjx, j = 1,2, ..., n. (1.8) (H3) There exist nonnegative constants Rj, Rjsuch that for all x∈R+, Rjx≤gj(x)≤Rjx, j = 1,2, ..., n. (1.9) (H4) There exist nonnegative constants Eij, Eij such that for all t∈R+, Eij ≤Zt −∞ Dij(t, s)ds ≤Eij, i, j = 1,2, ..., n. (1.10) (H5) There exists an integer q > 0 such that tk+q=tk+ω, Ii(k+q)=Iik, k∈Z+,where [0, ω]∩ {tk, k = 1,2, ...}={t1, t2, ..., tq}.(1.11) For convenience, we introduce the notion fM= max t∈[0,ω]{|f(t)|} , δi=e−Rω 0ri(t)dt, i = 1,2..., where fis a continuous and ω−periodic function. The paper is organized as follows. In Section 2, we recall some results which are necessary for our analysis. The existence of positive periodic solutions of system (1.4) by using the Krasonoselskii fixed point theorem is proved in Section 3. Finally, in Section 4, we exhibit an example to show the validity of our result. 2 Preliminaries Throughout this paper, a vector x= (x1, x2, ..., xn)T∈Rnis said to be positive if xi≥0,1≤i≤n. 4 First, we recall the following definitions. Let I⊂Rbe an interval, and denote by PC(I, Rn) the set of operators x:I→Rnwhich are continuous for t∈I, t6=tkand have discontinuities of the first kind at the points tk∈I, (k∈Z+) but are continuous from the left at these points. The proofs of the main results in this paper are based on an application of Krasnoselskii’s fixed point theorem in cones. Firstly, we need to introduce some definitions and lemmas. Definition 2.1 (See [10, 18]) A function xi:R→(0,+∞)is said to be a positive solution of (1.4), if the following conditions are satisfied 1) xi(t)is absolutely continuous on each (tk, tk+1) ; 2) for each k∈Z+, xit+ kand xit− kexist and xit− k=xi(tk) ; 3) xi(t)satisfies the first equation of (1.4) for almost everywhere in Rand xi(tk)satisfies the second equation of (1.4) at impulsive point tk, k ∈Z+. Definition 2.2 Let Xbe a Banach space and let Kbe a closed, nonempty subset of X.Kis a cone if i) αx +βy ∈Kfor all x, y ∈Kand all α, β ≥0; ii) x, −x∈Kimply x= 0. Theorem 2.1. (Krasnoselskii, [13]). Let Xbe a Banach space, and let K⊂X be a cone in X. Assume that Ω1and Ω2are open subsets of Xwith 0∈Ω1, Ω1⊂Ω2and let Φ : K∩Ω2\Ω1→K, be a completely continuous operator such that either a) kΦxk≤kxkfor x∈K∩∂Ω1and kΦxk≥kxkfor x∈K∩∂Ω2;or b) kΦxk ≥ kxkfor x∈K∩∂Ω1and kΦxk≤kxkfor x∈K∩∂Ω2. Then Φhas a fixed point in K∩Ω2\Ω1. 3 Existence of positive periodic solutions As we mentioned previously, one of our main objectives in this paper is to improve the work carried out in [25], and to extend it to investigate a wider class of differential equations with impulsive effects presented in (1.4). In particular, by using Kranoselskii’s fixed point theorem on cones, we will establish a sufficient condition ensuring the existence of positive ω−periodic solutions of equation (1.4). This section will be splitted into two parts: in the first one, we will focuse on the existence of positive periodic solutions when we use subquadratic impulse functions, while in the second part, we will consider the case in which the impulse effects are sublinear (most frequently used in the published literature). Let us start by obtaining an equivalent formulation for our problem (1.4). Lemma 3.1. The function x(·)is an ω−periodic solution of equation (1.4) if 5 and only if x(·)is an ω−periodic solution of the following equation: xi(t) = Zt+ω t Gi(t, s)xi(s)×  n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s))) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz  ds (3.1) +X t≤tk<t+ω Gi(t, tk)Iik (tk, xi(tk)) , where Gi(t, s) = e−Rs tri(ξ)dξ 1−e−Rω 0ri(ξ)dξ , s ∈[t, t +ω], i = 1,2, ..., n, (3.2) and we assume e−Rω 0ri(ξ)dξ 6= 1. Proof. Unlike the procedure carried out in [25], where the authors used the variation of constants formula to rewrite the original equation as an integral equation, we have to proceed in a very different way which is motivated and justified by the appearance of the impulsive terms in our problem. Assume that x= (x1, x2, ..., xn)T∈X, is a periodic solution of equation (1.4) .Then, we have d dt xi(t) exp −Zt 0 ri(s)ds = exp −Zt 0 ri(s)dsxi(t) ×  − n X j=1 aij(t)hj(xj(t)) − n X j=1 bij(t)fj(xj(t−τij (t))) − n X j=1 cij(t)Zt −∞ Dij(t, s)gj(xj(s)) ds  ,(3.3) t6=tk, i = 1,2, ..., n. 6 Integrating the above equation over [t, t +ω], we have xi(s)e−Rs 0ri(ξ)dξ tm1+nω t +xi(s)e−Rs 0ri(ξ)dξ tm2+nω tm1+nω +... +xi(s)e−Rs 0ri(ξ)dξ t+ω tmq+nω =Zt+ω t xi(s) exp −Zs 0 ri(ξ)dξ  − n X j=1 aij(s)hj(xj(s)) − n X j=1 bij(s)fj(xj(s−τij (s)) − n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)) dz  ds, where tmk+nω ∈(t, t +ω), mk∈ {1,2, ..., q}, k = 1,2, ..., q, n ∈Z+.Therefore, xi(t)e−Rt 0ri(ξ)dξ h1−e−Rt+ω tri(ξ)dξi +X t≤tk<t+ω ∆xi(tmk)e−Rtmk+nω 0ri(ξ)dξ =Zt+ω t xi(s) exp −Zs 0 ri(ξ)dξ ×   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s)) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)) dz  ds, which can be transformed into xi(t) = Zt+ω t Gi(t, s)xi(s) ×   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s)) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)) dz  ds +X t≤tk<t+ω Gi(t, tk)Iik (tk, xi(tk)) , i = 1,2, , ..., n. (3.4) Thus, xiis a periodic solution of (3.1). If x= (x1(t), x2(t), ..., xn(t))T∈K, is 7 a periodic solution of (3.1), for any t=tk, from (3.1) we obtain x0 i(t) = Gi(t, t +ω)xi(t+ω) ×  n X j=1 aij(t+ω)hj(xj(t+ω)) + n X j=1 bij(t+ω)fj(xj(t+ω−τj(t+ω)) + n X j=1 cij(t+ω)Zt+ω −∞ Dij(t+ω, s)gj(xj(s))ds  −Gi(t, t)xi(t)  n X j=1 aij(t)hj(xj(t)) + n X j=1 bij(t)fj(xj(t−τij (t)) + n X j=1 cij(t)Zt −∞ Dij(t, s)gj(xj(s))ds +ri(t)xi(t) =xi(t) ri(t)− n X j=1 aij(t)hj(xj(t)) − n X j=1 bij(t)fj(xj(t−τij (t)) − n X j=1 cij(t)Zt −∞ Dij(t, s)gj(xj(s))ds . For any t=tj, j ∈Z+, we have from (3.1) that xi(t+ j)−xi(tj) = Ztj+ω tjGit+ j, s−Gi(tj, s)xi(s) ×   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s))) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz  ds +X t+ j≤tk<tj+ω Git+ j, tkIik (tk, xi(tk)) −X tj≤tk<tj+ω Gi(tj, tk)Iik (tk, xi(tk)) =Iik (tk, xi(tk)) . Hence xiis a positive ω−periodic solution of (1.4). Thus, the proof of Lemma 3.1 is completed. 8 Define now PC (R,Rn) = nx= (x1, x2, ..., xn)T(3.5) :R→Rn|x∈C((tk, tk+1),Rn),such that xt− k, x t+ kexist and xt− k=x(tk), k ∈Z+. To apply Theorem 2.1, we need to define a Banach space Cω, a closed subset S of Cωand construct one mapping. Thus, we let (Cω,k.k)=(X, k.k) where Cω=nx= (x1, x2, ..., xn)T:x∈PC (R,Rn), x(t+ω) = x(t), t ∈Ro,(3.6) with the norm kxk= n X i=1 |xi|0,|xi|0= max t∈[0,ω]|xi(t)|, i = 1,2, ..., n, ∀x∈Cω.(3.7) Then, Cωwith the norm k.kis a Banach space. We denote θ= min (1, θ1, θ2, θ3), where θ1= min j=1,n Tj Tj,θ2= min j=1,n Fj Fj, θ3= min j=1,n min i=1,n Eij Eij Rj Rj, and σ= min ne−briω, i = 1,2, . . . , no. Let Kbe the cone in Cωdefined by K=nx(·)=(x1, x2, ..., xn)T∈Cω:xi(t)≥σ|xi|0, i = 1,2, ..., n, ∀t∈Ro. Use (3.1) to define the operator Φ : Cω→Cωby (Φx)(t) := [(Φ1x) (t),(Φ2x) (t), ..., (Φnx) (t)]T, where (Φix) (t) = Zt+ω t Gi(t, s)xi(s)   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s))) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz  ds +X t≤tk<t+ω Gi(t, tk)Iik (tk, xi(tk)) ,(3.8) 9 quently (Φix) (t)≥AiZω 0 xi(s)   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s))) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz  ds +X t≤tk<t+ω Gi(t, tk)Iik (tk, xi(tk)) ≥σ2Ai|xi|0 n X j=1 ω Z0 aij(s)Tjmin j=1,n Tj Tj|xj|0ds +σ2Ai|xi|0 n X j=1 ω Z0 Fjmin i=1,n Fj Fjbij(s)|xj|0ds +σ2Ai|xi|0 n X j=1 Zω 0 EijRjmin i=1,n Eij Eij ×Rj Rjcij(s)|xj|0ds +σ2Ai|xi|0X t≤tk<t+ω λik (tk)|xi(tk)|0 ≥θ1×σ2Aiωα2|xi|0 n X j=1baijx∗ j+θ2×σ2Aiωα2|xi|0 n X j=1b bijx∗ j +θ3×σ2Aiωα2|xi|0 n X j=1bcijx∗ j+ 1 ×σ2Aiωα2|xi|0b βikx∗ i ≥θ×σ2Aiωα2|xi|0 n X j=1baijx∗ j+θ×σ2Aiωα2|xi|0 n X j=1b bijx∗ j +θ×σ2Aiωα2|xi|0 n X j=1bcijx∗ j +θ×σ2Aiωα2|xi|0b βikx∗ i =θAiωσ2α2|xi|0  n X j=1 baij +b bij +bcijx∗ j+b βikx∗ i  = (Aibri)θωσ2α2|xi|0 ≥m0θωσ2α2|xi|0 ≥ |xi|0, i = 1,2, ..., n, 16 and thus kΦxk= n X i=1 |(Φixi)|0≥ n X i=1 |xi|0=kxk,∀x∈K∩∂Ω2. Obviously, Ω1and Ω2are open bounded subsets of Cωwith 0 ∈Ω1⊂ Ω1⊂Ω2.Hence, Φ : K∩(Ω2\Ω1)→Kis a completely continuous operator and satisfies condition (a) in Theorem 2.1. By Krasnoselskii’s Theorem, there exists a fixed point x∗(·) = (x∗ 1(·), x∗ 2(·), ..., x∗ n(·))T∈K∩(Ω2\Ω1) such that x∗(·) = (Φx∗)(·), i.e., x∗is a positive ω−periodic solution of system (1.4). The proof is completed. 3.2 The case of sublinear impulse functions In the case of sublinear impulses we can prove similar results for system (1.4). Theorem 3.2. Assume that (H1)−(H5)hold,assume further that: (H8)There exist nonnegative functions λik, λik ∈C(R+,R+)such that for all x∈R+, λik (t)x≤Iik (t, x)≤λik (t)x, i = 1,2, ..., n, k = 1,2, ... (H9)The linear system n X j=1 baij +b bij +bcijxj=bri, i = 1,2, ..., n, k = 1,2, ... possesses a unique positive solution. Then, system (1.4) possesses at least one positive ω−periodic solution. Proof. To prove that Φ : K→Kis completely continuous is similar to the corresponding proof in Lemma 3.2. We only need to prove (a) in Theorem 2.1. Let x∗= (x∗ 1, x∗ 2, ..., x∗ n)T with x∗ i>0, i = 1,2, ..., n, be a positive solution of (1.12). Set em0= min min 1≤i≤n{briAi},min 1≤i≤nnAib βiko, f M0= max max 1≤i≤n{briBi},max 1≤i≤nnBib βiko. Choose a constant f M≥f M0such that 0 <1−f Mω f Mω <1 where 0 <f Mω < 1. Let η1=1−f Mω f Mω and b Ω1=nx(t) = (x1(t), x2(t), ..., xn(t))T∈Cω:|xi|0< η1x∗ i, i = 1,2, ..., no. 17 If x∈K∩∂b Ω1,then σ|xi|0≤xi(t)≤ |xi|0=η1x∗ i, i = 1,2, ..., n, and (Φix) (t)≤BiZω 0    xi(s) n X j=1 aij(s)hj(xj(s)) +xi(s) n X j=1 bij(s)fj(xj(s−τij (s))) +xi(s) n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz ds +X t≤tk<t+ω Iik (tk, xi(tk))   ≤BiZω 0 |xi|0 n X j=1 Tjaij(s)|xj|0ds +BiZω 0 |xi|0 n X j=1 Fjbij(s)|xj|0ds +BiZω 0 |xi|0 n X j=1 RjEijcij(s)|xj|0ds +Bi|xi|0X t≤tk<t+ω λi(tk)|xi|0 ≤η1Biω|xi|0 n X j=1baijx∗ j+η1Biω|xi|0 n X j=1b bijx∗ j +η1Biω|xi|0 n X j=1bcijx∗ j+Biωb βik |xi|0 =η1Biω|xi|0   n X j=1 baij +b bij +bcijx∗ j  +Biωb βik |xi|0 =ω(Bibri)η1|xi|0+ωBib βik|xi|0 ≤η1f M0ω|xi|0+f M0ω|xi|0 ≤ |xi|0, i = 1,2, ..., n. Hence for any x∈K∩∂b Ω1 kΦxk= n X i=1 |(Φix)|0≤ n X i=1 |xi|0=kxk. On the other hand, choose 0 <em≤em0such that 1−σemθω σ2emθω >1 where 18 0< σ emθω < 1.Let η2=1−σemθω σ2emθω and b Ω2={x∈Cω:|xi|0< η2x∗ i, i = 1,2, ..., n}. If x∈K∩∂b Ω2,then σ|xi|0≤xi(t)≤ |xi|0=η2x∗ i, i = 1,2, ..., n, and, consequently (Φix) (t)≥AiZω 0 xi(s)   n X j=1 aij(s)hj(xj(s)) + n X j=1 bij(s)fj(xj(s−τij (s))) + n X j=1 cij(s)Zs −∞ Dij(s, z)gj(xj(z)dz  ds +X t≤tk<t+ω Gi(t, tk)Iik (tk, xi(tk)) ≥σ2Ai|xi|0 n X j=1 ω Z0 aij(s)Tjmin j=1,n Tj Tj|xj|0ds +σ2Ai|xi|0 n X j=1 ω Z0 Fjmin i=1,n Fj Fjbij(s)|xj|0ds +σ2Ai|xi|0 n X j=1 Zω 0 EijRjmin i=1,n Eij Eij ×Rj Rjcij(s)|xj|0ds +σAiX t≤tk<t+ω λik (tk)|xi(tk)|0 ≥θ1×σ2Aiωη2|xi|0 n X j=1baijx∗ j+θ2×σ2Aiωη2|xi|0 n X j=1b bijx∗ j +θ3×σ2Aiωη2|xi|0 n X j=1bcijx∗ j+ 1 ×σAiω|xi|0b βik 19 ≥θ×σ2Aiωη2|xi|0 n X j=1baijx∗ j+θ×σ2Aiωη2|xi|0 n X j=1b bijx∗ j +θ×σ2Aiωη2|xi|0 n X j=1bcijx∗ j +θ×σAiω|xi|0b βik =θAiωσ2η2|xi|0 n X j=1 baij +b bij +bcijx∗ j +θ×σAib βikω|xi|0 = (Aibri)θωσ2η2|xi|0+Aib βikωθσ |xi|0 ≥em0θωσ2η2|xi|0+em0θωσ |xi|0 ≥ |xi|0, i = 1,2, ..., n, and therefore kΦxk= n X i=1 |(Φix)|0≥ n X i=1 |xi|0=kxk,∀x∈K∩∂b Ω2. Hence, Φ : K∩(b Ω2\b Ω1)→Kis a completely continuous operator and satisfies condition (a) in Theorem 2.1. Consequently, there exists a fixed point x∗(·) = (x∗ 1(·), x∗ 2(·), ..., x∗ n(·))T∈K∩(b Ω2\b Ω1) such that x∗(·) = (Φx∗)(·). Therefore, system (1.4) has a positive ω-periodic solution. The proof is completed. Remark 3.1: The method applied in this paper can be used to treat a more general nonlinear impulse function. For instance, assuming that Iik (·, x) satisfies (H10) There exist nonnegative functions λik, λik ∈C(R+,R+) and z∈ CRn +,Rn +such that for all x∈Rn +, λik (t)z(kxk)≤Iik (t, x)≤λik (t)z(kxk), i = 1,2, ..., n, k = 1,2, ... Note that (H6) and (H8) are special cases of condition (H10) which has been used in [6, 7, 8]. Remark 3.2: Notice that when aij = 0 in the second term on the right hand side of (1.4), Iik (tk, x (tk)) = 0, fj(xj)=0,and gj(xj) = xj, we can easily derive the corresponding results in [25]. Therefore, the results presented in this paper improve and extend the main results in Ref. [25]. 4 An example In this section, we analyze an example to show the effectiveness of our result. 20 Example 4.1. Let us consider the following system: x0 i(t) = xi(t)  ri(t)− 2 X j=1 bij(t)fj(xj(t−τij (t))) − 2 X j=1 cij(t)Zt −∞ Dij(t, s)gj(xj(s)ds  ,(4.1) t6=tk=1 2kπ, k ∈Z+, t > 0, xi(t+ k)−xi(t− k) = Iik (tk, xi(tk)) = x2 i(tk) cos (2tk)(|sin xi(tk)|+ 1) ,(4.2) for i= 1,2.This model corresponds to system (1.4) when n= 2, ω = 2π. Let r1(t) = 1 2(1 + sin 2t), r2(t) = 1 3(1 + cos 2t), and τij ∈(R+,R+) be arbitrary continuous functions which satisfy τij(t+ω) = τij (t), i = 1,2. We then have br1=1 ωZω 0 r1(t)dt =1 4πZ2π 0 (1 + sin 2t)dt =1 2, br2=1 ωZω 0 r2(t)dt =1 6πZ2π 0 (1 + cos 2t)dt =1 3, and it is straightforward to check that Ai≤Gi(t, s)≤Bi,for i= 1,2,where Gi(t, s) = 1 1−e−briωexp −Zs t ri(ξ)dξ, i = 1,2, and A1:= e−br1ω 1−e−br1ω=e−π 1−e−π, A2:= e−br2ω 1−e−br2ω=e−2π 3 1−e−2π 3 , B1:= 1 1−e−br1ω=1 1−e−π, B2:= 1 1−e−br2ω=1 1−e−2π 3 . Let fj(x) = x 2e(sin x)+1, gj(x) = x 3e|cos x|+ 1, Ii(t, x) = πx2 cos 2t(|sin x|+ 1) , i, j = 1,2, and Dij (t, s) = es−t(cos t+ 2) ×3 2, i 6=j= 1,2, Dij (t, s) = es−t(sin t+ 4) ×1 3, i =j= 1,2. 21 Since |sin x| ≤ 1 and |cos x| ≤ 1 for x∈R+, we have Fjx≤fj(x)≤Fjx, Rjx≤gj(x)≤Rjx, and 3 2=Eij ≤Zt −∞ Dij (t, s)ds ≤Eij =9 2, i 6=j, 1 = Eij ≤Zt −∞ Dij (t, s)ds ≤Eij =5 3, i =j, and λi(t)x2≤Ii(t, x)≤λi(t)x2, where Fj=e2 2, Fj=1 2, Rj=e+ 1 3, Rj=2 3, λi(t) = π cos 2t, λi(t) = 2π cos 2t, i, j = 1,2. We can choose b11(t) = (1 + cos 2t) 3F1 , b12(t) = (1 + sin 2t) 2F2 , b21(t)=0, b22(t) = cos (4t), which implies b b11 =F1 ωZω 0 b11(s)ds =1 3,b b12 =F2 ωZω 0 b12(s)ds =1 2, b b21 =F1 ωZω 0 b21(s)ds = 0,b b22 =F2 ωZω 0 b22(s)ds = 0, and also choose c11(t)=0, c12(t) = 2 (1 + cos 4t) E1R12 , c21(t) = (1 + sin 2t) E2R21 , c22(t) = (1 + sin 2t) 2E2R22 ,obtaining bc11 =E1R11 ωZω 0 c11(s)ds = 0,bc12 =E1R12 ωZω 0 c12(s)ds = 2, bc21 =E2R21 ωZω 0 c21(s)ds = 1,bc22 =E2R22 ωZω 0 c22(s)ds =1 2. Choosing q= 8,we have b βik =1 ωX 0≤tk<ω λik (tk) = 1 2π q X k=1 2π cos 2tk =1 2 8 X k=1 2 cos 2 1 2kπ= 1, i = 1,2. 22 Moreover, it is easy to verify that the corresponding system of nonlinear equation (4.1) ,           2 X j=1 b b1j+bc1jxj+b β1kx1=br1 2 X j=1 b b2j+bc2jxj+b β2kx2=br2, has a unique positive solution x= (x1, x2) = 1 6,1 9. It is straightforward to show that all conditions of Theorem 3.1 are fulfilled. 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