Abstract and Applied Analysis

Volume 2013 (2013), Article ID 396509, 6 pages

http://dx.doi.org/10.1155/2013/396509

Research Article

Existence of Positive Periodic Solutions for a Class of Higher-Dimension Functional Differential Equations with Impulses

Zhang Suping1,2 and Jiang Wei¹

¹School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China

²Department of Mathematics and Physics, Anhui Jianzhu University, Hefei, Anhui 230601, China

Received 3 August 2013; Accepted 1 October 2013

Academic Editor: Youyu Wang

Copyright © 2013 Zhang Suping and Jiang Wei. 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.

By employing the Krasnoselskii fixed point theorem, we establish some criteria for the existence of positive periodic solutions of a class of -dimension periodic functional differential equations with impulses, which improve the results of the literature.

1. Introduction

Some evolution processes are distinguished by the circumstance that the evolutions change very rapidly at certain instants. In mathematical simulations, impulsive delay differential equations may express several simulation processes in real world which depend on their prehistory and are subject to short time disturbances. Such processes occur in the theory of optional control, population dynamics, biotechnologies, economics, and so forth. In recent years, the existence theory of positive periodic solutions of delay differential equations with impulsive effects or without impulsive effects has been an object of active research; we refer the reader to [1–4]. For other related works on studying for impulsive delay differential equations, we refer the reader to [5–7].

In [8], Zeng et al. studied the following functional differential equations without impulses:

and obtained sufficient conditions for the existence of positive periodic solutions of (1).

Zhang et al. [9] investigated the following form:

In this paper, we will consider the -dimension differential equation with impulses as follows:

where is a parameter, , is -periodic, and is an operator defined on (here denotes the Banach space of bounded continuous operator with the norm where . For and , is defined by for (see [10], Zheng). Consider that and (here represents the right limit of at the point ), , that is, changes decreasingly suddenly at , is a constant, and are the sets of all nonnegative and nonpositive real numbers, respectively. We assume that there exists an integer such that , where .

2. Some Preliminaries

, is continuous everywhere except at a finite number of points at which and exist and , . For each , the norm of is defined as .

Throughout the paper, we make the following assumptions:

  for all ;  is a continuous function of for each , ;  for any and , there exists such that for , , , and imply that

To conclude this section, we summarize in the following a few concepts and results that will be needed in our arguments.

Definition 1. Let be a Banach space, and let be a closed, nonempty subset of ; is a cone if

(i) for all and all ;(ii) imply .

Let with the norm , where ; then is a Banach space.

If is a solution of (3), then

where

See [9], Zhang et al.

It is clear that , for all , and by ,

for and .

Define for ,

Let

where . It is not difficult to verify that is a cone in . We define an operator as follows:

where

Then, it can be immediately obtained from the assumptions and that the operator is completely continuous. On the other hand, it is not difficult to check that is a positive -periodic solution of (3) if and only if is a fixed point of the operator .

Before stating the main results, we shall give some important lemmas.

Lemma 2. The mapping maps into , that is, .

Proof. For any , it is easy to see that . From (11), we have

Noting that , we can also obtain

Hence, . The proof is complete.

Lemma 3. Let be a Banach space, and let be a cone in . Suppose that and are open subsets of such that . Suppose that

is a completely continuous operator and satisfies either

(1)

 or (2)

Then, has a fixed point in .

The proof of Lemma 3 can be found in [11], Guo et al.

Lemma 4. Assume that – hold and there exists , such that

Then,

Proof. If , then

Thus, we have

Lemma 5. Assume that – hold and let , if there exists a sufficiently small such

Then,

Proof. For any ,

3. Main Results

For the sake of convenience, we introduce the following notations:

where denotes either 0 or .

Theorem 6. Assume that – hold and

then (3) has at least one positive -periodic solution.

Proof. By , for any there exists , such that

Choose , satisfying , by Lemma 5, we have

Next, by , there exists , such that

where is chosen, so that . It follows from Lemma 4 that

It follows from Lemma 3 that (3) has a positive -periodic solution satisfying .

Theorem 7. Assume that – hold and

then (3) has at least one positive -periodic solution.

Proof. Since , one can find an , such that

where is chosen so that . It follows from Lemma 4 that

By , we know that there exists and such that

Choose , satisfying ; then

Take

where , , and , .

This implies that , for any .

Therefore, (3) has at least one positive -periodic solution.

Theorem 8. Suppose that

  there exists , such that , for ,  there  exists , such that, , for

hold; then (3) has at least one positive -periodic solution.

Proof. Without loss of generality, we may assume that . If , then by , one can get

in particular, , for all.

On the other hand, by, one has

Therefore, (3) has at least one positive -periodic solution.

Theorem 9. If

 , ; 

hold, then (3) has at least one positive -periodic solution.

Proof. By assumption , for , there exists a sufficiently small such that

that is

So, is satisfied.

By assumption , for , there exists a sufficiently large such that

that is

therefore, holds. By Theorem 8, we complete the proof.

Theorem 10. If

 ; ,

hold, then (3) has at least one positive -periodic solution.

Proof. By , for , there exists a sufficiently small , such that

that is

Again, By , for , there exists a sufficiently small such that

that is

In the following, we consider two cases to prove to be satisfied:

(i);(ii), .

The bounded case is clear. If case (ii) is valid, then there exists , such that

Since , then we have

which implies that condition holds. By Theorem 8, we complete the proof.

Corollary 11. If one of the following pairs

  and ; and ; and ; and ; and ;  and ; and ; and ; and

is valid, then system (3) has at least one positive -periodic solution.

Acknowledgments

This research was supported by the National Nature Science Foundation of China (no. 11371027 and no. 11071001); Special Research Fund for the Doctoral Program of the Ministry of Education of China (no. 20123401120001); National Science Foundation of Educational Committee of Anhui Province of China (no. KJ2011B052); Program of Natural Science of Colleges of Anhui Province (no. KJ2013A032 and no. KJ2012A019).

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