Multiple Periodic Solutions for Discrete Nicholson ’ s Blowflies Type System

and Applied Analysis 3 Proof. Sufficiency. Assume that y : Z → Rn is a N-periodic function satisfying (9); that is,


Introduction and Preliminaries
In 1954 Nicholson [1] and later in 1980 Gurney et al. [2] proposed the following delay differential equation model: where () is the size of the population at time ,  is the maximum per capita daily egg production, 1/ is the size at which the population reproduces at its maximum rate,  is the per capita daily adult death rate, and  is the generation time.Now, Nicholson's blowflies model and its various analogous equations have attracted more and more attention.There is large literature on this topic.Recently, the study on Nicholson's blowflies type systems has attracted much attention (cf.[3][4][5][6][7][8] and references therein).In particular, several authors have made contribution on the existence of periodic solutions for Nicholson's blowflies type systems (see, e.g., [6,7]).In addition, discrete Nicholson's blowflies type models have been studied by several authors (see, e.g., [9][10][11][12] and references therein).Stimulated by the above works, in this paper, we consider the following discrete Nicholson's blowflies type system:  1 ( + 1) =  11 ()  1 () +  12 ()  2 () +  () where  > 1 is a constant,  is a nonnegative integer, and   , ,  = 1, 2, , and  are all -periodic functions from Z to R.
In fact, there are seldom results concerning the existence of multiple periodic solutions for Nicholson's blowflies type equations.It seems that the only results on this topic are due to Padhi et al. [13][14][15], where they established several existence theorems about multiple periodic solutions of 2 Abstract and Applied Analysis Nicholson's blowflies type equations.In addition, recently, several authors have investigated the existence of almost periodic solutions for Nicholson's blowflies type equations (see, e.g., [11,16,17] and references therein).However, to the best our knowledge, there are few results concerning the existence of multiple periodic solutions for Nicholson's blowflies type systems.That is the main motivation of this paper.
Next, let us recall the Leggett-Williams fixed point theorem, which will be used in the proof of our main results.
Let  be a Banach space.A closed convex set  in  is called a cone if the following conditions are satisfied: (i) if  ∈ , then  ∈  for any  ≥ 0; (ii) if  ∈  and − ∈ , then  = 0.
A nonnegative continuous functional  is said to be concave on  if  is continuous and Letting  1 ,  2 , and  3 be three positive constants and letting  be a nonnegative continuous functional on , we denote In addition, we call that  is increasing on  if () ≥ () for all ,  ∈  with  −  ∈ .
Then Φ has at least three fixed points  1 ,  2 , and Throughout the rest of this paper, we denote by Z the set of all integers, by R the set of all real numbers, and by  ∞  (Z, R  ) the space of all -periodic functions  : Z → R  , where  is a fixed positive integer.It is easy to see that where  = ( 1 ,  2 , . . .,   )  .In addition, we denote

Main Results
To study the existence of multiple periodic solutions for system (2), we first consider the following more general dimensional functional difference system: where, for every  ∈ Z, () is -periodic and nonsingular  ×  matrix, and  = ( 1 , . . .,   )  : Z × R  → R  is periodic in the first argument and continuous in the second argument.
To note that the existence of periodic solutions for system (7) and its variants had been of great interest for many authors (see, e.g., [19][20][21][22][23][24][25] and references therein) is needed.However, in some earlier works (see, e.g., [21]) on the existence of periodic solutions for system (7), the matrix () is assumed to be diagonal.In this paper, we will remove this restrictive condition by utilizing an idea in [22], where the authors studied the existence of periodic solutions for a class of nonlinear neutral systems of differential equations. Let We first present some basic results about Φ() and (, ).
Lemma 2. For all ,  ∈ Z with  ≤  ≤  +  − 1, the following assertions hold: Proof.One can show (i) and (ii) by some direct calculations and noting that ( + ) = ().So we omit the details.In addition, the assertion (iii) follows from the assertion (i) and the assertion (iv) follows from the assertion (ii).
By using Lemma 2, we can get the following result.
Lemma 3. A function  : Z → R  is a -periodic solution of system (7) if and only if  is a -periodic function satisfying Proof.Sufficiency.Assume that  : Z → R  is a -periodic function satisfying (9); that is, where Thus, we conclude that  is a -periodic solution of system (7).
Necessity.Let  : Z → R  be a -periodic solution of system (7).Then, we have For all  ≥ , we have which yields Letting  =  +  − 1 and noting that  is -periodic, we get Noting that we conclude That is, (9) holds.This completes the proof.(H1) There exist two constants (H2) There exists a constant  2 ∈ ( 1 ,  4 ) such that  2 ≤  4 , and Then system (7) has at least three nonnegative -periodic solutions.
Proof.Firstly, by (H0) and noting that ( + ,  + ) = (, ), Φ is an operator from  to .Secondly, noting that  is continuous for the second argument, by similar proof to [21, Lemma 2.5], one can show that Φ :  →  is completely continuous.Let It is easy to see that  is a concave nonnegative continuous functional on  and () ≤ ‖‖.Now, we show that Φ maps   4 into   4 .For every  ∈   4 , we have ( − ) ∈ R  + and ‖( − )‖ ≤  4 for all  ∈ Z.Then, by (H1), we have Similarly, for every  ∈   1 , it follows from (H1) that That is, condition (ii) of Lemma 1 holds.
Let  3 = (/) 2 .Next, let us verify condition (i) of Lemma 1.It is easy to see that the set In addition, for every  ∈ (,  2 ,  3 ), we have Then, by Lemma 1, we know that Φ has at least three fixed points in   4 .Then, it follows from Lemma 3 that system (7) has at least three nonnegative -periodic solutions.
Next, we give a concrete example for Nicholson's blowflies type system (2).(42) So, by Corollary 5, we know that system (2) has at least three nonnegative 2-periodic solutions.