Abstract and Applied Analysis

Volume 2014 (2014), Article ID 812867, 13 pages

http://dx.doi.org/10.1155/2014/812867

Research Article

Multiple Positive Periodic Solutions for Functional Differential Equations with Impulses and a Parameter

Zhenguo Luo1,2

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

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

Received 19 November 2013; Revised 24 December 2013; Accepted 7 January 2014; Published 26 March 2014

Academic Editor: Yun Kang

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.

We apply the Krasnoselskii fixed-point theorem to investigate the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with a parameter; some verifiable sufficient results are established easily. In particular, our results extend and improve some previous results.

1. Introduction

It is well known that impulsive differential equations arise naturally from a wide variety of applications such as aircraft control, the inspection processes in operations research, drug administration, and threshold theory in biology. Therefore, the impulsive differential equations represent a more natural framework for the mathematical model of many real world phenomena than differential equations (see [1–7]). In recent years, many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability and existence of solutions (see [8–16]). However, there are a little work discussing the existence of multiple positive periodic solutions for the high-dimensional functional differential equations with impulse and parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with a parameter:

where is a parameter, , , are -periodic. , is an operator on (here denoting the Banach space of bounded continuous operator with the norm , where ); . If , then for any , where is defined by for and (here representing the right limit of at the point ). Consider that ; that is, changes decreasingly suddenly at times . is a constant, , , , and . We assume that there exists an integer such that , , where .

It is well known that the functional differential system (1) includes many mathematical ecological models for example:

 in the case (a), , in [17], Zeng et al. studied the existence of multiple positive periodic solutions of (1) by applying the Krasnoselskii fixed-point theorem. in the case (b), , ; in [18], Zhang et al. established the existence of positive periodic solutions of (1) by using the fixed-point theorem in cones. in the case (c), , , and ; in [19], Jiang et al. investigated the existence, multiplicity, and nonexistence of positive periodic solutions of (1).

In this paper, we will study the existence of positive periodic solutions in more cases than the previously mentioned papers and obtain some easily verifiable sufficient criteria.

Throughout the paper, we make the following assumptions:

satisfy Caratheodory conditions; that is, , are locally Lebesgue measurable in for each fixed , are continuous in for each fixed , and are -periodic functions in . Moreover, there exist -periodic functions , , , which are locally bounded Lebesgue measurable such that , and , ; for all , and is a continuous function of for each , ;for any and , there exists such that for , , , and imply that , ;, satisfies and ; , , satisfy Caratheodory conditions and are -periodic functions in and, moreover, for all . There exists a positive constant such that , , . Without loss of generality, we can assume that and .

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

Furthermore, we will use the following notation. Let 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 . The matrix means that each pair of corresponding elements of and satisfies the inequality “” (“”). In particular, is called a positive matrix if .

The paper is organized as follows. In Section 2, we give some definitions and lemmas to prove the main results of this paper. In Section 3, existence theorems for one or two positive periodic solutions of (1) are established by using the Krasnoselskii fixed-point theorem under some conditions.

2. Preliminaries

In this section, we make some preparations for the following sections. For , , we define

It is clear that , , . For all and by , we have

In view of , we also define for the following:

Let with the norm , . It is easy to verify that is a Banach space. Define as a cone in by

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

where

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

Definition 1 (see [20]). A function is said to be a positive solution of (1), if the following conditions are satisfied:

(a) is absolutely continuous on each ;(b)for each and exist, and ;(c) satisfies the first equation of (1) for almost everywhere in and satisfies the second equation of (1) at impulsive point .

Definition 2 (see [21]). Let be a real Banach space; is a cone of . The semiorder induced by the cone is denoted by “”; that is, if and only if for any .

Secondly, let us introduce the Krasnoselskii point theorem in cones which will be used in this paper.

Lemma 3 (for the Krasnoselskii fixed-point theorem; see [22–24]). 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(2), for any and , for any .

Then has a fixed point in .

Lemma 4 (see [25]). Assume that and are continuous nonnegative functions defined on the interval ; then there exists such that

Lemma 5. Assume that hold. The existence of positive -periodic solution of (1) is equivalent to that of nonzero fixed point of in .

Proof. Assume that is a periodic solution of (1). Then, we have

Integrating the above equation over , we can have

where , , , and . Therefore,

which can be transformed into

Thus, is a periodic solution for (7).

If and with , then, for any we can get the derivation of (7) about ,

For any , , we have from (7) that

Hence is a positive -periodic solution of (1). Thus we complete the proof of Lemma 5.

Lemma 6. Assume that hold. Then is well defined.

Proof. From (7), it is easy to verify that is continuous in , and exist, and for each . Moreover, for any ,

Therefore, . From (7), we have

Noticing that , we obtain

Therefore, . This completes the proof of Lemma 6.

Lemma 7. Assume that hold. Then is completely continuous.

Proof. We first show that is continuous. By -, and are continuous in ; it follows that, for any , let be small enough to satisfy that, if , with ,

Therefore,

which implies that is continuous on .

Next we show that maps a bounded set into a bounded set. Indeed, let be a bounded set. For any and , by (7), we have

Since is bounded, in view of the continuity of , it follows from (19) that is bounded and is uniformly bounded. Finally, we show that the family of functions is equicontinuous on . Let with . From (7), for any , we have

Since for , , , , , and are uniformly bounded in ; in view of (21), it is easy to see that when tends to zero, tends uniformly to zero in . Hence, is a family of uniformly bounded and equicontinuous functions on . By Ascoli-Arzelà theorem, the operator is completely continuous. The proof of Lemma 7 is complete.

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

where denotes either or , denotes a positive number, and .

3. Main Results

Our main results of this paper are as follows.

Theorem 8. Assume that and the following conditions:

, ;

hold. Then (1) has two positive -periodic solutions.

Proof. First, we define ; then is an open subset of . From (7), , and Lemma 4, for any , we have

This yields

On the other hand, if holds, then we can choose , such that and for , , and , where constant satisfies . By (7) and Lemma 4, we can obtain

This yields

In view of (24) and (26), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1).

Likewise, if holds, then there is such that and for , , and , where constant satisfies . Let and it follows that for , , and . Thus

By (7) and Lemma 4, we have

this yields

In view of (24) and (29), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, . This proves Theorem 8.

Remark 9. Assume that and the following conditions:

;

hold. Then (1) has a positive -periodic solution.

Corollary 10. Assume that and hold.

is satisfied; then (1) has two positive -periodic solutions;is satisfied; then (1) has a positive -periodic solution.

Theorem 11. Assume that and the following conditions:

;

hold. Then (1) has two positive -periodic solutions.

Proof. We define , for a positive number . Then is an open subset of and . By (7), , and Lemma 4, for any , we have

This implies that for any

On the one hand, since , there exists and small enough satisfies such that, for any with ,

Define ; then is an open subset of . For any , by (7) and Lemma 4, we have

This yields

In view of (31) and (34), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). On the other hand, if , we can find small enough that satisfies and large enough , such that ,

Define and ; then is an open subset of . For any , from (7) and Lemma 3, we have

This yields

In view of (31) and (37), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore, (1) has at least two positive periodic solutions; that is, . This proves Theorem 11.

Remark 12. Assume that and the following conditions:

;, or

hold. Then (1) has a positive -periodic solution.

Corollary 13. Assume that and hold.

is satisfied; then (1) has two positive -periodic solutions;is satisfied; then (1) has a positive -periodic solution.

Theorem 14. Assume that and

;

hold. Then (1) has a positive -periodic solution, where , , , and are positive constants.

Proof. From , we can choose such that . Thus there exists such that, for , and ,

by (7), , and Lemma 4, we have

This implies that for any

On the other hand, choose such that and , and from , we can obtain

It is easy to see that there exists large enough , such that ,

Define and ; then is an open subset of . From (7), , and Lemma 4, for any , we have

This yields

In view of (40) and (44), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). This proves Theorem 14.

Corollary 15. Assume that and the following condition:

hold. Then (1) has a positive -periodic solution.

Similarly, we can prove the following theorem and corollary.

Theorem 16. Assume that and the following conditions:

;

hold. Then (1) has a positive -periodic solution, where , , , and are positive constants.

Corollary 17. Assume that and the following condition:

hold. Then (1) has a positive -periodic solution.

Theorem 18. Assume that , and the following condition:

, and , ,

hold. Then (1) has two positive -periodic solutions.

Proof. First, we define ; then is an open subset of . From (7), , and Lemma 4, for any , we have

This yields

On the one hand, since and , there exists such that for

Set ; then is an open subset of . From (7), , and Lemma 4, for any , , and , we have

This yields

In view of (46) and (49), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1).

On the other hand, if , and hold, then there is such that

for , , . Let and it follows that for , , and . Thus

By (7), , and Lemma 4, we have

This yields

In view of (46) and (53), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, . This proves Theorem 18.

Remark 19. Assume that , and the following condition:

, or ,

hold. Then (1) has a positive -periodic solution.

Corollary 20. Assume that and hold.

is satisfied; then (1) has two positive -periodic solutions.is satisfied; then (1) has a positive -periodic solution.

From the arguments in the previous proof, we have the following consequences immediately.

Theorem 21. Assume that , and the following condition:

, and ,

hold. Then (1) has two positive -periodic solutions.

Remark 22. Assume that , and the following condition:

, or ,

hold. Then (1) has a positive -periodic solution.

Corollary 23. Assume that and hold.

is satisfied; then (1) has two positive -periodic solutions.is satisfied; then (1) has a positive -periodic solution.

Theorem 24. Assume that , and one of the following conditions

and , ,, and

hold. Then (1) has two positive -periodic solutions.

Proof. We only consider the case . When the case holds, the conclusion remains true by a similar proof and we will omit it. We define ; then is an open subset of . From (7), , and Lemma 4, for any , we have

This yields

On the one hand, if holds, then we can choose , such that and for , , and , where constant satisfies . By (7), , and Lemma 4, we can obtain

This yields

In view of (55) and (57), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1).

On the other hand, if and hold, then there is such that

for , , and ; let and it follows that for , , and . Thus

By (7), , and Lemma 4, we have

This yields

In view of (55) and (61), by Lemma 3, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, . This proves Theorem 24.

Remark 25. Assume that , and one of the following conditions:

or , ,, or

hold. Then (1) has a positive -periodic solution.

Corollary 26. Assume that and hold.

 Either or is satisfied; then (1) has two positive -periodic solutions. Either or is satisfied; then (1) has a positive -periodic solution.

From the arguments in the previous proof, we have the following consequences immediately.

Theorem 27. Assume that , and one of the following conditions:

and , ,, and

hold. Then (1) has two positive -periodic solutions.

Remark 28. Assume that , and one of the following conditions

or , ,, or

hold. Then (1) has a positive -periodic solution.

Corollary 29. Assume that and hold.

 Either or is satisfied; then (1) has two positive -periodic solutions. Either or is satisfied; then (1) has a positive -periodic solution.

Remark 30. Suppose that and , under some conditions; we can obtain the corresponding results of [17]. Hence, our results generalize and improve the corresponding results of [17].

Remark 31. Assume that , , under some conditions; we can obtain the corresponding results of [18]. Hence, our results generalize and improve the corresponding results of [18].

Conflict of Interests

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

Acknowledgments

This 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 Science Foundation of Hengyang Normal University (no. 11B36), and the Construct Program of the Key Discipline in Hunan Province.

References

D. Y. Xu and Z. C. Yang, “Impulsive delay differential inequality and stability of neural networks,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 107–120, 2005.

A. M. Zhao and J. R. Yan, “Asymptotic behavior of solutions of impulsive delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 201, no. 3, pp. 943–954, 1996.

Y. P. Xing and M. A. Han, “A new approach to stability of impulsive functional differential equations,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 835–847, 2004.

Z. G. Luo and L. P. Luo, “Global positive periodic solutions of generalized -species competition systems with multiple delays and impulses,” Abstract and Applied Analysis, vol. 2013, Article ID 980974, 12 pages, 2013.

W. T. Li and H. F. Huo, “Existence and global attractivity of positive periodic solutions of functional differential equations with impulses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 6, pp. 857–877, 2004.

J. S. Yu, “Stability for nonlinear delay differential equations of unstable type under impulsive perturbations,” Applied Mathematics Letters, vol. 14, no. 7, pp. 849–857, 2001.

X. Y. Li, X. N. Lin, D. Q. Jiang, and X. Y. Zhang, “Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 683–701, 2005.

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.

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.

M. Liu and K. Wang, “Asymptotic behavior of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 909–925, 2013.

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

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.

Y. F. Shao and B. X. Dai, “The dynamics of an impulsive delay predator-prey model with stage structure and Beddington-type functional response,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3567–3576, 2010.

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, 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.

Z. J. Zeng, B. Li, 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.

N. Zhang, B. X. Dai, and X. Z. Qian, “Periodic solutions for a class of higher-dimension functional differential equations with impulses,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 3, pp. 629–638, 2008.

D. Q. Jiang, D. O'Regan, R. P. Agarwal, and X. J. Xu, “On the number of positive periodic solutions of functional differential equations and population models,” Mathematical Models & Methods in Applied Sciences, vol. 15, no. 4, pp. 555–573, 2005.

D. J. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Shandong, China, 2001, (in Chinese).

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

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

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

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, Orlando, Fla, USA, 1988.

Y. H. Fan, W. T. Li, and L. L. Wang, “Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 5, no. 2, pp. 247–263, 2004.