Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 268418, 7 pages

http://dx.doi.org/10.1155/2014/268418

## Existence of Positive Periodic Solutions for -Dimensional Nonautonomous System

^{1}College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China^{2}School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 25 February 2014; Revised 8 May 2014; Accepted 13 June 2014; Published 17 August 2014

Academic Editor: Guang Zhang

Copyright © 2014 Youjun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper we consider the existence, multiplicity, and nonexistence of positive periodic solutions for *n*-dimensional nonautonomous functional differential system , where are -periodic in and there exist -periodic functions such that for all with and , exist for ; are -periodic functions and , for ; is an -periodic function. We show that the system has multiple or no positive -periodic solutions for sufficiently large or small , respectively.

#### 1. Introduction

In this paper, we consider the first-order -dimensional nonautonomous functional differential system where is a parameter;

Let and for any , the norm of is defined as .

Throughout this paper, we use , unless otherwise stated.

For the system (1), we assume that is an -periodic function, are -periodic functions, and , for ; are -periodic in and there exist -periodic functions such that In addition, exist for .

We note that in (1) may have a singularity near ; that is,

As we well know, the system (1) is sufficiently general to include particular mathematical models which describe multiple population dynamics. Recently, due to the theoretical and practical significance, the existence of positive periodic solution of some particular cases of periodic system (1) has been extensively studied; see, for example, [1–15]. Cheng and Zhang [1], Kang and Cheng [2], Kang et al. [3], Kang and Zhang [4], and Liu et al. [5] studied the existence, multiplicity, and nonexistence of positive periodic solutions. The existence of positive periodic solutions of the scalar functional differential equation
has been studied by Wang [6]. By employing behaviours of the quotient as and , several interesting results on the existence and nonexistence of positive periodic solutions of (7) have been obtained. In [7], Weng and Sun studied more general scalar periodic functional differential equation
where the existence theorems of positive periodic solutions of (8) are obtained by employing the behaviours of at any point and . The result in [7] generalized and improved those in [6]. O’Regan and Wang [8] investigated the* n*-dimensional periodic system
By employing behaviours of as and , under quite general conditions, several existence theorems of positive periodic solutions are proved.

A solution of (1) is said to be positive if its all components are positive; is said to be -periodic if .

#### 2. Preliminary

Lemma 1 (see [9]). *Let be a Banach space and a cone in . For , define . Assume that is completely continuous such that for . *(i)*If for , then
*(ii)*If for , then
*

Lemma 2 (see [9, 10]). *Let be a Banach space and a cone in . Assume are open subsets of with . Let
**
be a completely continuous operator such that one of the following conditions is satisfied: *(a)* for and for ;*(b)* for and for .**Then has at least one fixed point in .*

In order to apply Lemmas 1 and 2 to system (1), we take endowed with the norm , where ; then is a Banach space.

Define the operator by where Let and , clearly; Define a set by

We use the following notations.

Let be a constant, and , defining

Lemma 3. *Assume that - hold; then and is continuous and completely continuous.*

*Proof. *In view of the definition of , for , we have
It is easy to see that is a constant because of the periodicity of .

Notice that, for and ,
Thus and it is easy to show that is continuous and completely continuous.

Lemma 4. *Assume that - hold; then a function is a positive -periodic solution of (1) if and only if , .*

*Proof. *If and , then
Thus is a positive -periodic solution of (1). On the other hand, if is a positive -periodic solution of (1), then and
Thus, ; furthermore, in view of the proof of Lemma 3, we also have for . That is, is a fixed point of in .

Lemma 5. *Assume that - hold; for any and , if there exists a component of such that , then
*

*Proof. *Since and , we have
Thus . The proof is complete.

For each , let be the function given by Let and .

Lemma 6 (see [11]). *Assume holds. Then and .*

Lemma 7. *Assume that - hold and let . If and there exists an , such that for , then
*

*Proof. *From the definition of , for , we have
That is, . The proof is complete.

The following two lemmas are weak forms of Lemmas 5 and 7.

Lemma 8. *Assume that - hold. If , then
**
where .*

Lemma 9. *Assume that - hold. If , then
**
where .*

#### 3. The Main Results

Theorem 10. *Assume that - hold and there exist positive constants and with such that
**
then for
**
the system (1) has at least a positive -periodic solution satisfying .*

*Proof. *From (32) for satisfying (33) we have that
or
Let and ; by (16) and (34), we have
That is, . This implies that for .

If and , by (16) and (35), we have
This implies that for . By Lemma 2(a), has a fixed point in . It follows from Lemma 2 that (1) has an -periodic solution with . The proof is complete.

Theorem 11. *Assume that - hold and there exist positive constants and with such that
**
then for
**
the system (1) has at least a positive -periodic solution satisfying .*

*Proof. *The proof of Theorem 11 is similar to that of Theorem 10, so we omit it. The proof is complete.

Theorem 12. *Assume that - hold.*(a)*If or , then (1) has positive -periodic solution(s) for .*(b)*If or , then (1) has positive -periodic solution(s) for .*(c)*If or , then (1) has no positive -periodic solution for sufficiently large or small , respectively.*

*Proof. *(a) Choose a number . By Lemma 8 we infer that there exists such that

If , this implies that . It follows from Lemma 6 that ; therefore, we choose so that , where the constant satisfies . By Lemma 7, it follows that
it follows from Lemma 1 that
Thus , which implies has a fixed point in , which is positive -periodic solution of (1) for .

If , then . It follows from Lemma 6 that . Therefore, we choose such that , where the constant satisfies . By Lemma 7, it follows that
it follows from Lemma 1 that
thus , which implies has a fixed point in , which is positive -periodic solution of (1) for .

If , it is easy to see from the above proof that has fixed points in and in such that
Consequently, (1) has two positive -periodic solutions for .

(b) Choose a number ; by Lemma 9 we infer that there exists a such that

If , there exists a component such that . Therefore there is a positive number such that
where the constant satisfies . Lemma 5 implies that
It follows from Lemma 1 that
Thus which implies has a fixed point in , which is positive -periodic solution of (1) for .

If , there exists a component such that . Therefore there is a positive number such that
where the constant satisfies . Let ; if , then
Hence,
Again, it follows from Lemma 5 that
It follows from Lemma 1 that
Thus , which implies has a fixed point in , which is positive -periodic solution of (1) for .

If , it is easy to see from the above proof that has fixed points in and in such that
Consequently, (1) has two positive -periodic solutions for .

(c) If , then and ; there exist two components and such that
It is easy to show (see [11]) that positive numbers exist such that

Assume is a positive -periodic solution of (1), we will show that this leads to a contradiction for . In fact, if , (57) implies that
On the other hand, if , then
which together with (58), implies that
Since , for , it follows from Lemma 5 that, for ,
which is a contradiction.

If , then and ; there exist two components and such that and . It is easy to show (see [11]) that positive numbers exist such that
Assume that is a positive -periodic solution of (1); we will show that this leads to a contradiction for . In fact, for , since , we find
which is a contradiction. The proof is complete.

Theorem 13 is a direct consequence of the proof of Theorem 12(c). Under the conditions of Theorem 13, we are able to give explicit intervals of such that (1) has no positive -periodic solution.

Theorem 13. *Assume that - hold.*(a)*If there is a such that , , then there exists a such that, for all , (1) has no positive -periodic solution.*(b)*If there is a such that , , then there exists a such that, for all , (1) has no positive -periodic solution.*

Theorem 14. *Assume that - hold and . If
**
then (1) has a positive -periodic solution.*

*Proof. *(a) If , then there exist two components and such that . It is easy to see that there exists an such that
Now, turning to and , there is an such that
Thus
We have by Lemma 7 that
On the other hand, there is an such that
Let . It follows that
Thus , for and . In view of Lemma 7, we have
It follows from Lemma 1 that
Thus . Hence, has a fixed point in . Consequently, (1) has a positive -periodic solution.

(b) If . The remaining part of the proof is similar to that of Theorem 14(a); therefore it is omitted. The proof is complete.

#### 4. Remarks

*Remark 15. *Based on the condition of [8], we may obtain the inequality sequence of
Comparatively, in terms of condition of presented in this paper, we have the inequality sequence of
it is clear that result of this paper can be applied to even more wider domain. Additionally, an extra requirement of
included in of [8], is not demanded here.

For example, when letting
and then
we can obtain the formula of
According to condition , the formulas would be further achieved as
However, there do not exist such that
Therefore, function of cannot satisfy the conditions proposed in [8].

*Remark 16. *The technique of proof of theorems in this paper differs from that of theorems in [8]. What is more, the condition of Theorems 10 and 11 is obviously weaker here than proposed in [8].

*Remark 17. *In this paper the application scope of Theorems 12–14 is relatively wider than that of Theorems in [6], respectively.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are very grateful to the referee for their valuable comments and suggestions. They appreciate the generous aid from Professor Jianwen Zhang in the process of revision.

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