Discrete Dynamics in Nature and Society

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

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

Research Article

Existence of Positive Periodic Solutions for -Dimensional Nonautonomous System

Youjun Liu,¹ Huanhuan Zhao,¹ and Jurang Yan²

¹College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China

²School 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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