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Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System
This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for the -dimensional functional difference system , where is not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.
1. Introduction and Preliminaries
In 1954 Nicholson  and later in 1980 Gurney et al.  proposed the following delay differential equation model: where is the size of the population at time , is the maximum per capita daily egg production, 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. [3–8] 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., [9–12] and references therein).
Stimulated by the above works, in this paper, we consider the following discrete Nicholson’s blowflies type system: where is a constant, is a nonnegative integer, and , , , and are all -periodic functions from to .
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. [13–15], where they established several existence theorems about multiple periodic solutions of 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 ; (ii) if and , then .
A nonnegative continuous functional is said to be concave on if is continuous and
Letting , , and 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 .
Lemma 1 (see ). Let be a cone in a Banach space , let be a positive constant, let be a completely continuous mapping, and let be a concave nonnegative continuous functional on with for all . Suppose that there exist three constants , , and with such that (i), and for all ;(ii) for all ;(iii) for all with .Then has at least three fixed points , , and in . Furthermore, , and .
Throughout the rest of this paper, we denote by the set of all integers, by the set of all real numbers, and by the space of all -periodic functions , where is a fixed positive integer. It is easy to see that is a Banach space under the norm where . In addition, we denote
2. 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 , is -periodic and nonsingular matrix, and 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., [19–25] and references therein) is needed. However, in some earlier works (see, e.g., ) 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 , where the authors studied the existence of periodic solutions for a class of nonlinear neutral systems of differential equations.
We first present some basic results about and .
Lemma 2. For all with , the following assertions hold: (i),(ii),(iii),(iv).
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 is a -periodic solution of system (7) if and only if is a -periodic function satisfying
Proof. Sufficiency. Assume that is a -periodic function satisfying (9); that is,
Then, we have
Thus, we conclude that is a -periodic solution of system (7).
Necessity. Let be a -periodic solution of system (7). Then, we have which yields For all , we have which yields Letting and noting that is -periodic, we get Noting that we conclude That is, (9) holds. This completes the proof.
Now, we introduce a set It is not difficult to verify that is a cone in . Finally, we define an operator on by
Theorem 4. Assume that and the following assumptions hold. (H0), for all and , and for all , , and .(H1)There exist two constants such that (H2)There exists a constant such that , and Then system (7) has at least three nonnegative -periodic solutions.
Proof. Firstly, by (H0) and noting that , 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 is completely continuous.
Let It is easy to see that is a concave nonnegative continuous functional on and .
Now, we show that maps into . For every , we have and for all . Then, by (H1), we have Similarly, for every , it follows from (H1) that That is, condition (ii) of Lemma 1 holds.
Let . Next, let us verify condition (i) of Lemma 1. It is easy to see that the set
In addition, for every , we have , , and for all . Then, by (H2), we get which means that condition (i) of Lemma 1 holds.
It remains to verify that condition (iii) of Lemma 1 holds. Let with ; we have and which yields Then, we have
Then, by Lemma 1, we know that has at least three fixed points in . Then, it follows from Lemma 3 that system (7) has at least three nonnegative -periodic solutions.
Corollary 5. Assume that , and (), , and are all nonnegative for and . Then the system (2) has at least three nonnegative -periodic solutions provided that , and
Proof. We only need to verify that all the assumptions of Theorem 4 are satisfied. Firstly, it is easy to see that (H0) holds. Let
Secondly, let us check (H1). In fact, one can choose sufficiently small such that, for all with , there holds
In addition, for all , we have
we conclude that (H1) holds.
It remains to verify (H2). For all with and , by using (34), we have This completes the proof.
Next, we give a concrete example for Nicholson’s blowflies type system (2).
Example 6. Let , , , , and By a direct calculation, we can get Then, we have and . In addition, we have and So, by Corollary 5, we know that system (2) has at least three nonnegative -periodic solutions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the referees for valuable suggestions and comments. Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province.
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