Discrete Dynamics in Nature and Society

Volume 2009 (2009), Article ID 239209, 18 pages

http://dx.doi.org/10.1155/2009/239209

## Multiple Positive Periodic Solutions for Delay Differential System

^{1}Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China^{2}Department of Mathematics, University of Science and Technology of China, Hefei 230026, China^{3}School of Mathematical Sciences, Fudan University, Shanghai 200433, China^{4}Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 11 September 2009; Accepted 6 December 2009

Academic Editor: Binggen Zhang

Copyright © 2009 Zhao-Cai Hao 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

We obtain some existence results for multiple positive periodic solutions of some delay differential systems. Examples are presented as applications. For a general positive integer , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our results are new also when .

#### 1. Introduction

It is known that multiple delay Logistic equations

are generalizations of many biological models, such as Logistic models of Single-species growth (see [1–3]), and red blood cell models (see [4–7]),

For biological models, positive periodic solutions are often important and many results have been achieved in this direction, for instance, [8–10].

To the best of our knowledge, few papers concerning the existence of multiple positive solutions of (1.1) can be found in literature. Furthermore, no papers have yet deal with the more general nonautonomous delay differential systems

where are all positive integer and

are given functions and signs , are given as follows:

The extension to systems is a natural one; for example, many occurrences in nature involve two or more populations coexisting in an environment, with the model being best described by a system of differential equations (see [11]).

The aim of this paper is to investigate systems (1.4) and (1.5). In what follows we only discuss the existence of positive periodic solutions of system (1.4); similar results can be obtained for system (1.5). By using multiple fixed-point theorems (see Lemmas 2.1 and 2.2), which are different from those used in [8–10], we obtain the existence of multiple positive periodic solutions of system (1.4) (see Theorems 3.1, 4.1, and 4.3). Some examples are given also to illustrate our main theorems. Main results of this paper are new also even if (see Remark 4.5).

This paper is organized as follows. In Section 2, we make some preliminaries. In Section 3, we derive existence result (see Theorem 3.1) for two positive periodic solutions of system (1.4). Example 3.2 is given below Theorem 3.1. The existence of three positive periodic solutions of system (1.4) is presented in Section 4 (see Theorems 4.1 and 4.3). Applications of Theorems 4.1 and 4.3 may be seen from Examples 4.2 and 4.4.

#### 2. Preliminaries

We make the basic assumption throughout this paper that

Let us now provide some preparations. Let be a real Banach space and let be a cone in . A map is said to be a nonnegative continuous concave functional on cone if is continuous and

For numbers , such that and a nonnegative continuous concave functional on cone , we define

Setting we define

Write

where

Then and are all Banach spaces and is a cone in . Set

It is easy to see that for any , functions , have properties

Now we define an operator as follows:

where

signs , are given in (1.7) and we often use them in the remainder of this paper. It is easy to say that a -periodic solution of operator equation

on , that is, a fixed point of operator , is a -positive periodic solution of system (1.4). So, our main results concerning multiple positive solutions of system (1.4) will arise as application of the following fixed-point theorem.

Lemma 2.1 (see [12]). *Let be a cone in a real Banach space . Let and be increasing, nonnegative, continuous functionals on , and let be a nonnegative continuous functional on with such that, for some and ,
**
Suppose 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 2.2 (see [13]). *Let be a cone in a real Banach space , let be completely continuous, and let be a nonnegative continuous concave functional on with for all Suppose that there exists such that*(i)* and for ;*(ii)* for all ;*(iii)* for with **Then has at least three fixed points , , satisfying
*

#### 3. Existence of Two Positive Solutions of System (1.4)

In this section, we apply Lemma 2.1 to establish Theorem 3.1, the existence result of two positive solutions of system (1.4). Example 3.2 will be given as an application of Theorem 3.1.

Theorem 3.1. *Assume that there exist numbers such that the following three assumptions are satisfied.**One has**
where is fixed and
**, , , and imply **
where
**, , , and imply **Then system (1.4) has at least two -positive periodic solutions.*

*Proof. *We begin by defining
Clearly, and are increasing, nonnegative, continuous functionals on , and is nonnegative a continuous functional on with . Moreover, we observe that
Now, we proceed to show that other conditions of Lemma 2.1 are also satisfied.

Firstly, we will show that
In fact, we have from (2.8), for any ,
which yields
Hence for all Furthermore, we know from the continuity of functions , , , that the operator is completely continuous. Hence, we conclude that (3.10) holds.

Secondly, let us prove
For any so that we get, in view of (1.7), (2.4) and (3.8),
Consequently, for any condition and (3.14) imply that
which gives (3.13).

Thirdly, we verify
As before, and (1.7), (2.4), and (3.8) also tell us that
Then condition , (3.17), and the fact that the function is concave imply
Thus (3.16) holds.

Finally, let us prove
Obviously,
In addition, for any we get
since and . So we have from condition that
Hence (3.19) holds.

To sum up, (3.6)–(3.10), (3.13), (3.16), and (3.19) tell us that conditions of Lemma 2.1 all hold here. Consequently, system (1.4) has at least two -positive periodic solutions and belonging to such that

As an application of Theorem 3.1, we provide the following example. For convenience, all examples in this paper are given when

*Example 3.2. *Assume that is a fixed constant. Consider the following system:
where
We set
Then
We may verify that conditions , , and are all satisfied. Hence, Theorem 3.1 tells us that system (3.24) has at least two -positive periodic solutions and such that

#### 4. Existence of Three Positive Solutions of System (1.4)

For the sake of convenience we list the assumptions to be used in this section as follows.

There exists a number such thatwhere and , are given in .

There exist numbers and such thatwhere

One has

There exists a number such that,

Let us now state the first existence result of three positive solutions of system (1.4).

Theorem 4.1. *Assume that conditions , , and hold. Then system (1.4) has at least three -positive periodic solutions.*

*Proof. *Firstly, we set
Obviously, is a nonnegative continuous concave functional on and

Secondly, condition implies that there exists a number such that
In fact, we know from condition that there exist numbers
satisfying
So
Set
Then
Let us choose
Then for any , we have
which implies for all . Moreover, we know from the proof of (3.10) that is completely continuous.

Thirdly, let us show that numbers
satisfy conditions (i), (ii), and (iii) of Lemma 2.2.*Step 1. *We prove that
Clearly, Moreover, for any we have
Then condition , (4.6), and (4.18) imply that
which gives for And then we arrive at (4.17).*Step 2. *Condition implies
In fact, for any , that is, , from
and condition we have
which yields (4.20).*Step 3. * for with This is the case because implies

At present, we may say that hypotheses of Lemma 2.2 (the Leggett-Willaims theorem) are satisfied. Hence system (1.4) has at least three -positive periodic solutions:
such that

We give the following example to illustrate Theorem 4.1.

*Example 4.2. *Consider the following system:
where is a fixed constant and

We set
Then
We may verify also that conditions , , and hold. Hence, Theorem 4.1 tells us that system (4.26) has at least three -positive periodic solutions:
such that

The second existence result of three positive solutions of system (1.4) is as follows.

Theorem 4.3. *Assume that conditions , , and hold. Then system (1.4) has at least three -positive periodic solutions.*

*Proof. *If we can get (4.8) with replaced by in this case, then the proof is complete. In fact, for any , condition implies
Then
as desired. This ends the proof.

The following example is an application of Theorem 4.3.

*Example 4.4. *Consider system
where is a fixed constant and

If we set
then
We choose
Then assumptions , , and hold. Hence we know from Theorem 4.3 that system (4.34) has at least three -positive periodic solutions:
such that

We end this paper by the following remark.

*Remark 4.5. *For a general positive integer , main results of this paper do not appear in former literatures as we know. Comparing with the existing results, our Theorems 3.1, 4.1, and 4.3 are new also when .

#### Acknowledgments

Z.-C. Hao acknowledges support from NSFC (10771117), and PH.D. Programs Foundation of Ministry of Education of China(20093705120002), and NSF of Shandong Province of China (Y2008A24), China Postdoctoral Science Foundation (20090451290), Shandong Province Postdoctoral Foundation (200801001), Foundation of Qufu Normal University (BSQD07026).

#### References

- K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at MathSciNet - Y. Kuang,
*Delay Differential Equations with Applications in Population Dynamics*, vol. 191 of*Mathematics in Science and Engineering*, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet - S. M. Lenhart and C. C. Travis, “Global stability of a biological model with time delay,”
*Proceedings of the American Mathematical Society*, vol. 96, no. 1, pp. 75–78, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. N. Chow, “Existence of periodic solutions of autonomous functional differential equations,”
*Journal of Differential Equations*, vol. 15, pp. 350–378, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,”
*Science*, vol. 197, no. 4300, pp. 287–289, 1977. View at Google Scholar · View at Scopus - J. Mallet-Paret and R. D. Nussbaum, “Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,”
*Annali di Matematica Pura ed Applicata*, vol. 145, pp. 33–128, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wazewska-Czyzwska and A. Lasota, “Mathematical problems of the dynamics of the red blood cells system,”
*Annals of Polish Mathematical Society, Series III*, vol. 17, pp. 23–40, 1988. View at Google Scholar - K. Gopalsamy, X. Z. He, and L. Z. Wen, “On a periodic neutral logistic equation,”
*Glasgow Mathematical Journal*, vol. 33, no. 3, pp. 281–286, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gopalsamy and B. G. Zhang, “On a neutral delay logistic equation,”
*Dynamics and Stability of Systems*, vol. 2, no. 3-4, pp. 183–195, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Liu, “Positive periodic solution for a nonautonomous delay differential equation,”
*Acta Mathematicae Applicatae Sinica*, vol. 19, no. 2, pp. 307–316, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Beltrami,
*Mathematics for Dynamic Modeling*, Academic Press, Boston, Mass, USA, 1987. View at MathSciNet - R. I. Avery and J. Henderson, “Two positive fixed points of nonlinear operators on ordered Banach spaces,”
*Communications on Applied Nonlinear Analysis*, vol. 8, no. 1, pp. 27–36, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,”
*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673–688, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet