EXISTENCE OF POSITIVE PERIODIC SOLUTION OF A PERIODIC COOPERATIVE MODEL WITH DELAYS AND IMPULSES

where ai, bi, and ci (i= 1,2) are nonnegative ω-periodic continuous functions. It is well known that more realistic and interesting species models should take into account both the seasonality of the changing environment and time delays [4, 8, 9], and that the birth of many species is not continuous but occurs at fixed time intervals (some wild animals have seasonal births), in the long run; the birth of these species can be considered as an impulse to the system [1, 2, 5, 7]. To describe this phenomenon exactly, we proposed


Introduction
In 1974 May [10] suggested the following cooperative species model [3]: where a i , b i , and c i , i = 1,2, are positive constants.Recently, paper [11] has studied the existence of positive periodic solutions of the following system: where a i , b i , and c i (i = 1,2) are nonnegative ω-periodic continuous functions.It is well known that more realistic and interesting species models should take into account both the seasonality of the changing environment and time delays [4,8,9], and that the birth of many species is not continuous but occurs at fixed time intervals (some wild animals have seasonal births), in the long run; the birth of these species can be considered as an impulse to the system [1,2,5,7].To describe this phenomenon exactly, we proposed the following periodic cooperative species model with delays and impulses, which is a generalization of (1.1) where Δx(t k ) = x(t + k ) − x(t − k ) are the impulses at moment t k and t 1 < t 2 < ••• is a strictly increasing sequence such that lim k→∞ t k = +∞ and there exists a positive integer q such that t k+q = t k + ω, γ i(k+q As usual in the theory of impulsive differential equations, at the points of discontinuity t k of the solution t → x i (t) we assume that x i (t k ) ≡ x i (t − k ).It is clear that, in general, the derivatives x i (t k ) do not exist.On the other hand, according to the first equality (1.3) there exist the limits x i (t ∓ k ).According to the above convention, we assume x i (t k ) ≡ x i (t − k ).Throughout this paper, we assume that are ω-periodic functions.The organization of this paper is as follows.In Section 2, we introduce some notations and definitions, and state some preliminary results needed in later sections.We then study, in Section 3, the existence of periodic solutions of system (1.3) by using the continuation theorem of coincidence degree theory proposed by Gaines and Mawhin [6].

Preliminaries
In order to obtain the existence of a positive periodic solution of system (1.3), we first make the following preparations.Consider the impulsive system Y. Li and W. Xing 3 where x ∈ R n , f : R × R n+1 → R n is continuous, and f is ω-periodic with respect to its first argument; I k : R n → R n are continuous, and there exists a positive integer q such that t k+q = t k + ω, I k+q (x Consider the following nonimpulsive delay differential system with initial condition y i (t) = φ i (t), t ≤ 0, where φ i (t) is defined as above.
In the following, we will establish a fundamental theorem that enables us to reduce the existence of solution of system (1.3) to the corresponding problem for the nonimpulsive delay differential system (2.4).Theorem 2.2.Assume that (1.4) holds.Then (i) if y = (y 1 , y 2 ) T is a solution of (2.4), then T is a solution of (1.3), then is a solution of (2.4).
Proof.First, we prove (i).It is easy to see that , are absolutely continuous on the interval (t k ,t k+1 ] and that for any satisfies system (1.3).On the other hand, for every t k ∈ {t k }, (2.8) Thus, for every k = 1,2,..., (2.9) The proof is complete.Next, we prove (ii).Since x i (t) is absolutely continuous on each interval (t k ,t k+1 ] and, in view of (2.9), it follows that, for any k = 1,2,..., which implies that y i (t), i = 1,2, are continuous on [0,∞).It is easy to prove that y i (t) are absolutely continuous on [0, ∞).Now, one can easily check that is a solution of (2.9).The proof is complete.
Y. Li and W. Xing 5

Existence of periodic solutions
In this section, based on Mawhin's continuation theorem, we will study the existence of at least one periodic solution of (1.3).To do so, we will make some preparations.Let X and Y be real Banach spaces, L : DomL ⊂ X → Y a linear mapping, and N : X → Y a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dimKer L = co dim Im L < +∞ and ImL is closed in Y.If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that Im P = Ker L and Ker Q = Im(I − Q), it follows that mapping L| DomL∩Ker P : (I − P)X → Im L is invertible.We denote the inverse of that mapping by K P .If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if QN(Ω) is bounded and In what follows, we will use the following notations: where h(t) is a periodic continuous function with period ω, Before we proceed to state and prove our main result, we introduce a lemma which is useful in the proof of our main result. Let 6 Positive periodic solution of a cooperative model then (2.4) is transformed into . (3.4) One can easily check that if system (3.4) has an ω-periodic solution (z * 1 (t),z * 2 (t)) T , then (e z * 1 (t) ,e z * 2 (t) ) T is a positive ω-periodic solution of system (2.4). and , are the same as those in system (1.3) and Y. Li and W. Xing 7 then it is easy to see that, for ( (3.9) Therefore, From the property of invariance under a homotopy, we have By a straightforward computation, we find This completes the proof.
We are now in a position to state and prove the existence of periodic solutions of (1.3).
Proof.According to the discussion made in Section 2, we need only to prove that the nonimpulsive delay differential system (3.4) has an ω-periodic solution.In order to use Y. Li and W. Xing 9 the continuation theorem of coincidence degree theory to establish the existence of ωperiodic solutions of (3.4), we take where , . (3.14) So, ImL is closed in X and L is a Fredholm mapping of index zero.It is easy to show that P and Q are continuous projectors such that Furthermore, the generalized inverse (to L) K P : ImL → DomL ∩ Ker P is given by Clearly, QN and K P (I − Q)N are continuous.Using the Arzela-Ascoli theorem, it is not difficult to show that Then isomorphism J of ImQ onto KerL can be the identity mapping since ImQ = Ker L. Now we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem.Corresponding to the operator equation Lx = λNx, λ ∈ (0,1), we have . Suppose that z(t) = (z 1 (t),z 2 (t)) T ∈ X is a solution of system (3.20) for some λ ∈ (0,1).Integrating (3.20) over the interval [0,ω], we obtain dt.