Journal of Mathematics

Volume 2014 (2014), Article ID 214093, 13 pages

http://dx.doi.org/10.1155/2014/214093

Research Article

Multiple Positive Periodic Solutions for Two Kinds of Higher-Dimension Impulsive Differential Equations with Multiple Delays and Two Parameters

Zhenguo Luo1,2

¹Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

²Department of Mathematics, National University of Defense Technology, Changsha 410073, China

Received 8 September 2013; Revised 13 February 2014; Accepted 13 February 2014; Published 6 April 2014

Academic Editor: Nan-Jing Huang

Copyright © 2014 Zhenguo Luo. 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 applying the fixed point theorem, we derive some new criteria for the existence of multiple positive periodic solutions for two kinds of  -dimension periodic impulsive functional differential equations with multiple delays and two parameters: ), a.e., , , , , , and ), a.e., , , , , , As an application, we study some special cases of the previous systems, which have been studied extensively in the literature.

1. Introduction

Let , , , , , and , respectively. Denote by the set of operators which are continuous for , and have discontinuities of the first kind at the points but are continuous from the left at these points. For each , the norm of is defined as . Let denote the Banach space of bounded continuous functions with the norm , where . The matrix means that each pair of corresponding elements of and satisfies the inequality . In particular, is called a positive matrix if .

Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences, and numerous examples are given in [1–4]. 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, and we refer the reader to [5–17]. Recently, in [5], Jiang and Wei studied the following nonimpulsive delay differential equation:

They obtained sufficient conditions for the existence of the positive periodic solutions of (1). Motivated by [5], in [6], Zhao et al. investigated the following impulsive delay differential equation:

They derived some sufficient conditions for the existence of the positive periodic solutions of (2). In [7], Huo et al. considered the following impulsive delay differential equation:

They got sufficient conditions for the existence and attractivity of the positive periodic solutions of (3). Motivated by [5–7], in [8], Zhang et al. studied the following impulsive delay differential equation:

They obtained some sufficient conditions for the existence of the positive periodic solutions of (4). However, to this day, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of impulsive functional differential equations with two parameters:

with initial conditions:

where

and , , are -periodic; that is, , , with , (here representing the right limit of at the point ). ; that is, changes decreasingly suddenly at times , and is a constant. We assume that there exists an integer such that , , where .

Throughout the paper, we make the following assumptions:

are locally summable -periodic functions; that is, , , and for all , , and are two parameters; and for all , such that , ;, satisfies and . satisfy Caratheodory conditions and are -periodic functions in . Moreover, there exists a positive constant such that , . Without loss of generality, we can assume that and ; is a real sequence such that , , and satisfies for all .

In addition, the parameters in this paper are assumed to be not identically equal to zero.

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

Definition 1 (see [1]). A function is said to be a positive solution of (5) and (6) if the following conditions are satisfied:

(a) is absolutely continuous on each ;(b)for each and exist, and ;(c) satisfies the first equation of (5) and (6) for almost everywhere (for short a.e.) in and satisfies for , .

Under the previous hypotheses , we consider the neutral nonimpulsive system:

with initial conditions:

where

By a solution of (9) and (10), it means an absolutely continuous function defined on that satisfies (9) and (10), that is, for , and , on .

The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem  1 in [18].

Lemma 2. Suppose that hold. Then

(i)if are solutions of (9) and (10) on , then are solutions of (5) and (6) on ;(ii)if are solutions of (5) and (6) on , then are solutions of (9) and (10) on .

Proof. It is easy to see that are absolutely continuous on every interval , , ,

On the other hand, for any , ,

and thus

It follows from (13)–(15) that are solutions of (5). Similarly, if are solutions of (10), we can prove that are solutions of (6).

Since is absolutely continuous on every interval , , , and in view of (14), it follows that, for any ,

which implies that are continuous on . It is easy to prove that are absolutely continuous on . Similar to the proof of , we can check that are solutions of (9) on . Similarly, if are solutions of (6), we can prove that are solutions of (10). The proof of Lemma 2 is completed.

In the following section, we only discuss the existence of a periodic solution for (9) and (10).

Definition 3 (see [19]). Let be a real Banach space, and let be a closed, nonempty subset of . is said to be a cone if

(1) for all , and , and(2) imply .

Lemma 4 (Krasnoselskii fixed point theorem see [20–22]). Let be a cone in a real Banach space . Assume that and are open subsets of with , where . Let be a completely continuous operator and satisfy either

(1), for any and , for any  or(3), for any and , for any .

Then has a fixed points in .

The paper is organized as follows. In next section, firstly, we give some definitions and lemmas. Secondly, we derive some existence theorems for one or two positive periodic solutions of system (5) by using Krasnoselskii fixed point theorem under some conditions. In Section 3, we also get some existence theorems for one or two positive periodic solutions of system (6) that are also established by applying Krasnoselskii fixed point theorem under some conditions. Finally, as an application, we give two examples to show our results.

2. Existence of Periodic Solutions of System (5)

We establish the existence of positive periodic solutions of (5) by applying the Krasnoselskii fixed point theorem on cones. We will first make some preparations and list a few preliminary results. For , , we define

It is clear that , , . In view of , we also define for ,

Let with the norm , where , and it is easy to verify that is a Banach space. Define to be a cone in by

and we easily verify that is a cone in .

We define an operator as follows:

where

For convenience in the following discussion, we introduce the following notations:

where denotes either or , . Moreover, we list several assumptions:

: ;: ;: ;: ;: ;: ;: ;: ;: there exists , such that , for any ;: there exists , such that , for any .

The proofs of the main results in this paper are based on an application of Krasnoselskii fixed point theorem in cones. To make use of fixed point theorem in cones, firstly, we need to introduce some definitions and lemmas.

Lemma 5. Assume that hold. The existence of positive -periodic solution of (9) is equivalent to that of nonzero fixed point of in .

Proof. Assume that is a periodic solution of (9). Then, we have

Integrating the above equation over , we can have

Therefore, we have

which can be transformed into

Thus, are periodic solutions for (21).

If and with , then for any , derivative the two sides of (21) about ,

Hence is a positive -periodic solution of (9). Thus we complete the proof of Lemma 5.

Lemma 6. Assume that hold. Then the solutions of (5) are defined on and are positive.

Proof. By Lemma 2, we only need to prove that the solutions of (9) are defined on and are positive on . From (9), we have that, for any and ,

Therefore, are defined on and are positive on . The proof of Lemma 6 is complete.

Lemma 7. Assume that hold. Then is well defined.

Proof. From (21), for any ,

Therefore, . From (21), we have

On the other hand, we obtain

Therefore, . The proof of Lemma 7 is complete.

Lemma 8. Assume that hold, and there exists such that

and then

Proof. For any , then

Thus, we have

The proof of Lemma 8 is complete.

Lemma 9. Assume that hold, and let . If there exists a sufficiently small such that

and then

Proof. For any , we have

The proof of Lemma 9 is complete.

Our main results of this paper are as follows.

Theorem 10. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Proof. By , there exists such that

where the constant satisfies . Then by Lemma 8, we have

On the other hand, by , for any , there exists such that

We choose

If , then

where

This implies that

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, and then has a fixed point in . By Lemma 5, system (5) has at least one positive -periodic solution. The proof of Theorem 10 is complete.

Theorem 11. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Proof. By , for any , there exists such that

Then by Lemma 9, we have

On the other hand, by , there exists such that

where the constant satisfies . Then by Lemma 8, we have

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, then has a fixed point in . By Lemma 4, the system (5) has at least one positive -periodic solution. The proof of Theorem 11 is complete.

Theorem 12. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where is defined in .

Proof. By , there exists such that

where the constant satisfies . Then by Lemma 8, we have

Likewise, from , there exists such that

where the constant satisfies . Then by Lemma 8, we have

Define . Then from , for any , , we obtain

which yields

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, and then has a fixed point . Likewise, under the assumptions and , satisfies all the conditions in Lemma 4, and then has a fixed point . By Lemma 5, the system (5) has at two positive -periodic solutions and satisfying . The proof of Theorem 12 is complete.

Theorem 13. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where is defined in .

Proof. By , for any , there exists such that

Then by Lemma 9, we have

Likewise, by , for any , there exists such that

We choose

If , then

where

This implies that

Define . Then from , for any , we obtain

which yields

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, and then has a fixed point . Likewise, under the assumptions and , satisfies all the conditions in Lemma 4, and then has a fixed point . By Lemma 4, the system (5) has at two positive -periodic solutions and satisfying . The proof of Theorem 13 is complete.

Theorem 14. In addition to , if and hold, then system (5) has at least one positive -periodic solution satisfying , where and are defined in and , respectively.

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

which yields

Likewise, for , then from , we can get

which yields

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, and then has a fixed point . By Lemma 4, the system (5) has at least one positive -periodic solution satisfying , where and are defined in and , respectively. The proof of Theorem 14 is complete.

Theorem 15. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

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

that is

which implies that is satisfied.

Likewise, by , for any , there exists a sufficiently large such that

that is

which implies that is satisfied. Therefore, by Theorem 14, we complete the proof.

Theorem 16. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

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

that is

which implies that is satisfied. On the other hand, by , for any , there exists a sufficiently large such that

In the following, we consider two cases to prove to be satisfied: are bounded and unbounded. The bounded case is clear. If are unbounded, then there exist and such that

Since , then we get

which implies that the condition holds. By Theorem 14, we complete the proof.

Theorem 17. In addition to , if , , and hold, then system (5) has at two positive -periodic solutions and satisfying , where is defined in .

Proof. By and the proof of Theorem 15, there exists a sufficiently small such that

On the other hand, from and the proof of Theorem 16, there exists a sufficiently large such that

Therefore, from the proof of Theorem 14, there exist two positive solutions and satisfying , where is defined in ; the proof of Theorem 17 is complete.

Theorem 18. In addition to , if , , and hold, then system (5) has at least two positive -periodic solutions and satisfying , where R is defined in .

Proof. The proof is similar to that of Theorem 17, and we omit the details here.

Theorem 19. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Proof. Let . By and the proof of Theorem 10, there exists a sufficiently small such that

Likewise, by and the proof of Theorem 16, there exists a sufficiently large such that

In conclusion, under the assumptions and , satisfies the conditions in Lemma 4, and then has a fixed point in . By Lemma 4, the system (5) has at least one positive -periodic solution. The proof of Theorem 19 is complete.

Similar to Theorem 19, we can get the following consequences.

Theorem 20. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Theorem 21. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Theorem 22. In addition to , if and hold, then system (5) has at least one positive -periodic solution.

Theorem 23. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where is defined in .

Proof. Let . By and the proof of Theorem 10, there exists a sufficiently small such that

Likewise, by and the proof of Theorem 15, there exists a sufficiently large such that

Incorporating and the proof of Theorem 14, we know that there exist two positive -periodic solutions and satisfying , where is defined in . The proof of Theorem 23 is complete.

Similar to Theorem 23, one immediately has the following consequences.

Theorem 24. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where is defined in .

Theorem 25. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where R is defined in .

Theorem 26. In addition to , if , , and hold, then system (5) has two positive -periodic solutions and satisfying , where R is defined in .

3. Existence of Periodic Solutions of System (6)

Now, we are at the position to study the existence of positive periodic solutions of system (6). By carrying out similar arguments as in Section 2, it is not difficult to derive sufficient criteria for the existence of positive periodic solutions of system (6). For simplicity, we prefer to list below the corresponding criteria for system (6) without proof, since the proofs are very similar to those in Section 2.

For , , we define

and it is clear that , , . In view of , we also define for

Let with the norm , , and it is easy to verify that is a Banach space. Define to be a cone in by

We easily verify that is a cone in . We define an operator as follows:

where

The proof of the following lemmas and theorems is similar to those in Section 2, and we all omit the details here.

Lemma 27. Assume that hold. The existence of positive -periodic solution of system (6) is equivalent to that of nonzero fixed point of in .

Lemma 28. Assume that hold. Then the solutions of system (6) are defined on and are positive.

Lemma 29. Assume that hold. Then is well defined.

Lemma 30. Assume that hold, and there exists such that

and then

Lemma 31. Assume that hold, and let . If there exists a sufficiently small such that

and then

Theorem 32. Assume that and hold. Moreover, if one of the following conditions holds:

 ; ; ; ,

then system (6) has two positive -periodic solutions and satisfying , where is defined in .

Theorem 33. Assume that , and hold. Moreover, if one of the following conditions holds:

 ; ; ; ,

then system (6) has two positive -periodic solutions and satisfying , where r is defined in .

Theorem 34. Assume that hold. Moreover, if one of the following conditions holds:

 ; ; ; ; ; ; ; ; ,

then system (6) has at least one positive -periodic solution.

4. Examples

In order to illustrate our results, we take the following examples.

Example 1. We consider the following generalized so-called Michaelis-Menton type single species growth model with impulse:

which is a special case of system (5), and where are -periodic, and , are two parameters.

Theorem 35. Assume that hold. Moreover, if the following condition holds:

then system (93) has at least one positive -periodic solution.

Proof. Note that

We can construct the same Banach space and cone as in Section 2. Then for any , we have

This can lead to

Then we can have

On the other hand, we have

This can lead to

That is

By Theorem 21, it follows that system (93) has at least one positive -periodic solution. The proof of Theorem 35 is complete.

Example 2. We consider the following generalized hematopoiesis model with impulse:

which is a special case of system (6), and where is the number of red blood cells at time , , , and are -periodic and , are two parameters.

Theorem 36. Assume that hold. Moreover, if the following condition holds:

then system (102) has at least one positive -periodic solution.

Proof. Note that

We can construct the same Banach space and cone as in Section 2. Then for any , we have

This can lead to

By Theorem 34, it follows that system (102) has at least one positive -periodic solution. The proof of Theorem 36 is complete.

Remark 37. We apply the main results obtained in the previous sections to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of positive periodic solutions which play the role played by the equilibrium of the autonomous models and means that the species is in an equilibrium state. From Theorems 35 and 36, we see that under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of the systems.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by NSF of China (nos. 11161015, 11371367, and 11361012), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, 12C0361, and 13C084), and the Construct Program of the Key Discipline in Hunan Province.

References

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, London, UK, 1993.

A. M. Samoikleno and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

D. Q. Jiang and J. J. Wei, “Existence of positive periodic solutions for nonautonomous delay differential equations,” Chinese Annals of Mathematics A, vol. 20, no. 6, pp. 715–720, 1999 (Chinese).

A. M. Zhao, W. P. Yan, and J. R. Yan, “Existence of positive periodic solution for an impulsive delay differential equation,” in Topological Methods, Variational Methods and Their Applications, pp. 269–274, World Scientific, London, UK, 2002.

H.-F. Huo, W.-T. Li, and X. Liu, “Existence and global attractivity of positive periodic solution of an impulsive delay differential equation,” Applicable Analysis, vol. 83, no. 12, pp. 1279–1290, 2004.

X. Y. Zhang, J. R. Yan, and A. P. Zhao, “Existence of positive periodic solutions for an impulsive differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 3209–3216, 2008.

J. R. Yan, “Existence of positive periodic solutions of impulsive functional differential equations with two parameters,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 854–868, 2007.

Z. J. Zeng, L. Bi, and M. Fan, “Existence of multiple positive periodic solutions for functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1378–1389, 2007.

J. R. Yan, “Global attractivity for impulsive population dynamics with delay arguments,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5417–5426, 2009.

G. Ballinger and X. Z. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000.

D. Lin, X. D. Li, and D. O'Regan, “Stability analysis of generalized impulsive functional differential equations,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1682–1690, 2012.

Z. G. Luo, L. P. Luo, and Y. H. Zeng, “Positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters,” Journal of Applied Mathematics, vol. 2014, Article ID 751612, 17 pages, 2014.

X. D. Li and X. L. Fu, “On the global exponential stability of impulsive functional differential equations with infinite delays or finite delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 3, pp. 442–447, 2014.

Z. G. Luo, B. X. Dai, and Q. H. Zhang, “Existence of positive periodic solutions for an impulsive semi-ratio-dependent predator-prey model with dispersion and time delays,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3390–3398, 2010.

X. Q. Ding, C. W. Wang, and P. Chen, “Permanence for a two-species Gause-type ratio-dependent predator-prey system with time delay in a two-patch environment,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 9099–9105, 2013.

J. R. Yan and A. M. Zhao, “Oscillation and stability of linear impulsive delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 187–194, 1998.

D. J. Guo, Nonlinear Functional Analysis, ShanDong Science and Technology Press, Ji'nan, China, 2001.

M. A. Krasnoselskii, Positive Solution of Operator Equation, P. Noordhoff, Groningen, The Netherlands, 1964.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, Fla, USA, 1988.