Multiple Positive Periodic Solutions for a Functional Difference System

Abstract

We obtain two existence results about multiple positive periodic solutions for a class of functional difference system. Two examples are given to illustrate our results.

1. Introduction and Preliminaries

Throughout this paper, we denote by the set of all integers, by the set of all real numbers, and by a real Banach space. Moreover, let and let () be the space of all -periodic functions (), where is fixed positive integer. It is well known that is a Banach space under the norm where .

The aim of this paper is to investigate the existence of multiple positive periodic solutions to the following functional difference system: where is an -dimensional vector function , , are -periodic functions from to , is a function from to , and is defined by for all .

The existence of periodic solutions has been an important topic in the qualitative theory of functional differential equations and functional difference equations. There is a large body of literature on this interesting topic. We refer the reader to [1–17] and references therein for some recent contributions. Especially, the existence of periodic solutions for system (3) and its variants has been of great interest for many authors (see, e.g., [5, 6, 8, 9, 17] and references therein).

It is needed to note that Raffoul [8] and Raffoul and Tisdell [9] have made an important contribution to this topic. In fact, Raffoul constructed Green function for system (3) and transformed system (3) into an equivalent system. This enables us to use some suitable fixed point theorems to investigate the existence of periodic solutions for system (3). In addition, we would like to draw the reader’s attention to [6], where Dix et al. initiated the study on the multiple periodic solutions for a variant of system (3) in a 1-dimensional case.

Stimulated by [6, 8, 9], in this paper, we will make further study on this topic for an -dimensional case. Next, we recall two fixed point theorems, which will be used in the proof of our main results. We first recall some definitions and notations.

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 a concave on if is continuous and Letting 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 [18]). Let be a cone in , let and be increasing, nonnegative, continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and , for all . Suppose that there exists a completely continuous operator and such that and (i), for all ;(ii), for all ;(iii) and , for all .
Then has at least two fixed points and belonging to such that

Lemma 2 (see [19]). Let be a cone in , 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 with such that (i) and for all ;(ii) for all ;(iii) for all with .
Then has at least three fixed points in . Furthermore, and .

2. Main Results

Throughout the rest of this paper, we assume that the following assumptions for system (3) hold.(H0)For every for all and (H1) belongs to whenever .(H2)For every and , there exists a such that  for all with , and .

Now, we define for with .

Then, by a proof similar to [8], we can transform (3) into the following equivalent equation: where

It is easy to see that for all with . In addition, it follows from (H0)–(H2) that, for every has a positive denominator, while the numerator is a positive and increasing function of . Thus, for with , we have Letting we have

Next, we introduce a set where . It is not difficult to verify that is a cone in . Finally, we define an operator on by

Lemma 3. is an operator from to .

Proof. Let . By (H1) and , we get for all . So .
In addition, for , we have where is the th component of . Then, we obtain for all . Thus, . This completes the proof.

2.1. Existence of Two Positive Periodic Solutions of System (3)

In this section, we apply Lemma 1 to establish an existence result about two positive periodic solutions of system (3). For convenience, we list some assumptions.(H3)There exists a constant such that (H4)There exists a constant such that (H5)There exists a constant such that

Theorem 4. Assume that there exist three constants with such that (H0)–(H5) hold. Then system (3) has at least two positive -periodic solutions.

Proof. Firstly, by Lemma 3, is an operator from to . Secondly, by a proof similar to [9, Lemma 2.5], one can show that is completely continuous.
Now, we begin to verify that all the assumptions of Lemma 1 hold. Let It is clear that , , and are increasing, nonnegative, continuous functionals on with . Moreover, we have for all and .
Next, we proceed to show that conditions (i)–(iii) of Lemma 1 are also satisfied. For every , noting that , by (H3), we conclude that that is, condition (i) of Lemma 1 holds. For every , since , by (H4), we get that is, condition (ii) of Lemma 1 holds. Finally, it is easy to see that and for every , it follows from (H5) that Thus, condition (iii) of Lemma 1 holds.
Now, by applying Lemma 1, there exist two fixed points , which are just two -periodic solutions to system (3). This completes the proof.

Remark 5. In Theorem 4, the two -periodic solutions do not equal zero. In fact, according to Lemma 1, we have

Corollary 6. Assume that (H0)–(H2) and (H4) hold. Moreover, Then system (3) has at least two positive -periodic solutions.

Proof. By (34), there exists a constant such that (H3) holds. By (35), there exists a constant such that (H5) holds. Then, by applying Theorem 4, we complete the proof.

Next, we present a simple example, which does not aim at generality but illustrates how to use our existence theorem.

Example 7. Consider the following system: where We have , It is easy to verify that conditions (H0)–(H2) hold. Since, for , we conclude that (34) and (35) are satisfied. It remains to verify (H4). Letting , for all with , we have which means that (H4) holds. Therefore, by Corollary 6, we know that system (36) has at least two positive -periodic solutions.

Remark 8. In the above example, is obviously a trivial periodic solution for system (36). But by Remark 5, we know that the two positive -periodic solutions do not equal zero.

2.2. Existence of Three Nonnegative Periodic Solutions of System (3)

In [6], Dix et al. investigated the existence of multiple nonnegative periodic solutions for a first order functional difference equation by the Leggett-Williams fixed point theorem. In this section, we will investigate the existence of multiple nonnegative periodic solutions for system (3) by using an idea similar to that of [6]. For convenience, we also list some assumptions.(H6)There exists a constant such that (H7)There exists a constant such that (H8)There exists a constant such that

Theorem 9. Assume that (H0)–(H2) and (H6)–(H8) hold. Then system (3) has at least three nonnegative -periodic solutions.

Proof. By the proof of Theorem 4, we know that is an operator from to and completely continuous. Let It is easy to see that is a concave nonnegative continuous functional on and .
Firstly, we show that maps into . For every , we have . Combining this with (H8), we get
Secondly, let us verify condition (i) of Lemma 2. Since , , then, it is easy to see that the set In addition, for every , we have . Then, by (H7), we have which means that condition (i) of Lemma 2 holds.
Thirdly, for every , since , it follows from (H6) that that is, condition (ii) of Lemma 2 holds.
Finally, for every with , we have and which yields that Then, we have that is, condition (iii) of Lemma 2 holds.
Now, by Lemma 2, we know that has at least three fixed points in , and thus system (3) has at least three nonnegative -periodic solutions.

Corollary 10. Assume that (H0)–(H2) hold and there exists a constant such that Moreover, there hold Then system (3) has at least three nonnegative -periodic solutions.

Proof. We only need to verify that (H6) and (H8) hold. Let Then . There exists a constant such that, for all with , there holds Taking , (H6) holds. In addition, there exists a constant such that, for all with , there holds Taking (H8) holds.

Next, we also provide a simple example to illustrate our existence theorem.

Example 11. Let be the same as in Example 7, and By Example 7, we have , and (H0)–(H2) hold.
By a direct calculation, we get Then, it is easy to see that (53) holds.
Let . Then, for all we have Thus, all the assumptions of Corollary 10 hold. Then, we know that the considered functional difference system 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.

Acknowledgments

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.