Journal of Applied Mathematics

Volume 2014 (2014), Article ID 751612, 17 pages

http://dx.doi.org/10.1155/2014/751612

Research Article

Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters

Zhenguo Luo,1,2 Liping Luo,¹ and Yunhui Zeng¹

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

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

Received 29 June 2013; Accepted 2 October 2013; Published 5 January 2014

Academic Editor: Meng Fan

Copyright © 2014 Zhenguo Luo 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 apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature.

1. Introduction

First, we give the following definitions. 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 .

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; numerous examples are given in [1–3]. In recent years, in [4–11], many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability, and existence of solutions. However, to this day, still no scholars investigate the existence of multiple positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with two parameters:

where

and , are two parameters, ,  ,    are -periodic, that is, , , , is an operator on (here denotes the Banach space of bounded continuous operator with the norm , where ,  , (here represents the right limit of at the point ), , that is, changes decreasingly suddenly at times . is a constant, , , , and . We assume that there exists an integer such that , , , where .

Models of forms (1) and (2) have been proposed for population dynamics (single species growth models), physiological processes (such as production of blood cells, respiration, and cardiac arrhythmias), and other practical problems. Equations (1) and (2) are very general and incorporate many famous mathematical models extensively studied in the literature [12–21]. 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, and are locally Lebesgue measurable in for each fixed and are continuous in for each fixed are -periodic functions in . Moreover, there exist -periodic functions , which are locally bounded Lebesgue measurable so that ,   and , .   , are two parameters. is -periodic with respect to the first variable, that is, such that , .The delay kernel is integrable and is normalized such that , such that , ., satisfies , and ; , satisfy Caratheodory conditions and are -periodic functions in . 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.

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

Definition 1 (see [22]). 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 ;(2) imply .

Lemma 2 (see Krasnoselskii’s fixed point theorem [23–26]). 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 satisfies either

   , for any and , for any , or   , for any and , for any .

Then, has a fixed point in .

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

where denotes either or ,  , and .

The paper is organized as follows. In Section 2, firstly, we give some definitions and lemmas. Secondly, we derive some existence theorems for one or two positive periodic solutions of (1) which are established by using Krasnoselskii’s fixed point theorem under some conditions. In Section 3, existence theorems for one or two positive periodic solutions of (2) are also established by using Krasnoselskii’s fixed point theorem under some conditions. As applications in Section 4, we study some particular cases of systems (1) and (2) which have been investigated extensively in the references mentioned earlier.

2. Existence of Periodic Solution of (1)

We establish the existence of positive periodic solutions of (1) by applying the Krasnoselskii’s fixed point theorem on cones. We will first make some preparations and list below a few preliminary results. For , , we define

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

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 Krasnoselskii’s fixed point theorem in cones. To make use of the fixed point theorem in cone, firstly, we need to introduce some definitions and lemmas.

Definition 3 (see [1]). 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 4 (see [22]). 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 .

Lemma 5 (see [27]). Assume that and are continuous nonnegative functions defined on the interval ; then there exists such that

Lemma 6. 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 , , , .

Therefore,

which can be transformed into

Thus, is a periodic solution for (9).

If and with , then for any , derivative the two sides of (9) about ,

For any , , we have from (9) that

Hence is a positive -periodic solution of (1). Thus we complete the proof of Lemma 6.

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

Proof. From (9), it is easy to verify that is continuous in , and exist, and for each . Moreover, for any

Therefore, . From (9), we have

On the other hand, we obtain

Therefore, . This completes the proof of Lemma 7.

Lemma 8. 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 (9), we have

Since is bounded, in view of the continuity of , it follows from (21) that is bounded and is uniformly bounded. Finally, we show that the family of functions is equicontinuous on . Let with . From (9), for any , we have

Since for , , , , and are uniformly bounded in , in view of (23), 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 Arzela-Ascoli theorem, the operator is completely continuous. The proof of Lemma 8 is complete.

Our main results of this paper are as follows.

Theorem 9. Assume that and

there exists a such that , , for ;

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

Proof. First, we define ; then is an open subset of . Then, for any , we have . Consequently,

then

and from the definition of , we know . From (9), , and Lemma 5, we get

This yields

On the other hand, if holds, then we can choose , so that ; from the definition of , we know . Thus, we have , for , , , where constant satisfies . By (9) and Lemma 5, we can obtain

This yields

In view of (27) and (29), by Lemma 5, it follows that has a fixed point with , which is a positive -periodic solution of (1).

Likewise, in view of , for any , there is such that

Let , where

Then for any , from (9), (30), and (31), we have

where , . This yields

In view of (27) and (33), by Lemma 5, 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 9.

Corollary 10. Assume that and

There exists a such that , , for ;, or

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

Theorem 11. Assume that and

there exists a such that , , for ;

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

Proof. We define , where R satisfied ; then is an open subset of and . For any , by the definition of , we get . Furthermore, by (9), , and Lemma 5, we have

This implies for any

On the one hand, since , then for any there exists such that

Letting , then for any , one has , , . Consequently,

where, together with (42) and Lemma 5, we have

This implies for any

In view of (35) and (39), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). On the other hand, if , then for any there exists such that

Letting , then for any , one has , , . Consequently,

where, together with (41) and Lemma 5, we have

This yields

In view of (35) and (43), by Lemma 2, 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.

Corollary 12. Assume that and

there exists a such that , , for ;, or ;

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

Theorem 13. Assume that , and hold. Then, (1) has a positive -periodic solution y with lying between R and r, which are defined in and , respectively.

Proof. Without loss of generality, we may assume that ; then for any , by the definition of , we get . Furthermore, by (9), , and Lemma 5, we have

This implies for any

Now, we let ; then is an open subset of . Then, for any , we have . Consequently,

and from the definition of , we know . From (9), , and Lemma 5, we get

This yields

In view of (45) and (48), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). This proves Theorem 13.

Theorem 14. In addition to , suppose the following conditions hold:

, ;, ;

then, (1) has a positive -periodic solution.

Proof. In view of , , there exists a sufficiently small such that

which yields

Therefore, condition is satisfied. On the other hand, since , , there exists a sufficiently large such that

which yields

Thus, condition is satisfied. By Theorem 13, we complete the proof.

Theorem 15. Assume that and

, ;,

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

Proof. In view of , , for , there exists a sufficiently small such that

which yields

Therefore, condition is satisfied. On the other hand, since , , for , there exists a sufficiently large such that

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

Since , then we get

Thus, condition is satisfied. By Theorem 14, we complete the proof.

Theorem 16. Assume that , , and hold. Then, (1) has at least two positive -periodic solutions , satisfying , where R is defined in .

Proof. From and the proof of Theorem 9, we know that there exists a sufficiently large such that

On the one hand, from and the proof of Theorem 14, we know that there exists a sufficiently small such that

In view of (58) and (59), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1).

On the other hand, from and the proof of Theorem 15, we know that there exists a sufficiently large such that

In view of (58) and (60), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). Thus, (1) has at least two positive -periodic solutions , satisfying . The proof is completed.

Corollary 17. In addition to , suppose or holds. Then, (1) has at least one positive -periodic solution.

Theorem 18. Assume that , , , and hold. Then, (1) has at least two positive -periodic solutions , satisfying , where r is defined in .

Proof. From and the proof of Theorem 9, we know that there exists a sufficiently small such that

On the one hand, from and the proof of Theorem 14, we know that there exists a sufficiently large such that

In view of (61) and (62), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1).

On the other hand, from and the proof of Theorem 15, we know that there exists a sufficiently small such that

In view of (61) and (63), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). Thus, (1) has at least two positive -periodic solutions , satisfying . The proof is completed.

Corollary 19. In addition to , and , suppose or holds. Then, (1) has at least one positive -periodic solution.

Theorem 20. Assume that , , hold. Then, (1) has at least two positive -periodic solutions , satisfying .

Proof. From and the proof of Theorem 9, if , we know that there exists a sufficiently small such that

In view of , we know that there exists a sufficiently large such that

On the one hand, from and the proof of Theorem 14, we know that there exists a sufficiently large such that

In view of (64) and (66), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). In view of (65) and (66), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). Thus, (1) has at least two positive -periodic solutions , satisfying . The proof is completed.

Corollary 21. Assume that , , and hold. Then, (1) has at least one positive -periodic solution.

Theorem 22. Assume that , , and hold. Then, (1) has at least two positive -periodic solutions , satisfying .

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

From and the proof of Theorem 9, on the one hand, if , we know that there exists a sufficiently small such that

In view of (67) and (68), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1).

On the other hand, if , we know that there exists a sufficiently large such that

In view of (67) and (69), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). Thus, (1) has at least two positive -periodic solutions , satisfying . The proof is completed.

Corollary 23. Assume that , , and hold. Then, (1) has at least one positive -periodic solution.

Similarly, one can prove the following theorems and corollaries.

Theorem 24. Assume that , , and hold. Then, (1) has at least two positive -periodic solutions , satisfying .

Corollary 25. Assume that , , and hold. Then, (1) has at least one positive -periodic solution.

Theorem 26. Assume that , , and hold. Then, (1) has at least two positive -periodic solutions , satisfying .

Corollary 27. Assume that , , and hold. Then, (1) has at least one positive -periodic solution.

3. Existence of Periodic Solution of (2)

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

For , , we define

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

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

Lemma 28. Assume that hold. The existence of positive -periodic solution of (2) is equivalent to that of nonzero fixed point of in .

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

Lemma 30. Assume that hold. Then is completely continuous.

Theorem 31. Assume and hold. Moreover, if one of the following conditions holds:

then, (2) has two positive -periodic solutions and satisfying , where R is defined in .

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

then, (2) has two positive -periodic solutions and satisfying , where r is defined in .

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

then, (2) has two positive -periodic solutions and satisfying .

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

then, (2) has at least one positive -periodic solution.

Theorem 35. Assume and hold. Moreover, if one of the following conditions holds:

then, (2) has at least one positive -periodic solution.

Theorem 36. Assume hold. Moreover, if one of the following conditions holds:

then, (2) has at least one positive -periodic solution.

4. Applications

In this section, as some applications of our main results, we will consider some special cases of systems (1) and (2), which have been investigated extensively in the literature.

Application 1. Consider the following equations:

where , which are special cases of systems (1) and (2) without impulse, respectively. First, we list several assumptions:

;;;there exists a such that for ;there exists a such that for ;; ; ; ; ; ; ; .

By applying theorems in Sections 2 and 3 and to systems (81) and (82), respectively, we obtain the following theorems.

Theorem 37. Assume hold. Moreover, if one of the following conditions holds:

then, (81) and (82) have at least two positive -periodic solutions.

Theorem 38. Assume and hold. Moreover, if one of the following conditions holds:

then, (81) and (82) have at least two positive -periodic solutions.

Theorem 39. Assume hold. Moreover, if one of the following conditions holds:

then, (81) and (82) have at least two positive -periodic solutions.

Theorem 40. Assume hold. Moreover, if one of the following conditions holds:

then, (81) and (82) have at least one positive -periodic solution.

Theorem 41. Assume and hold. Moreover, if one of the following conditions holds:

then (81) and (82) have at least one positive -periodic solution.

Theorem 42. Assume , hold. Moreover, if one of the following conditions holds:

then, (81) and (82) have at least one positive -periodic solution.

Application 2. Consider the following equations:

where

which are special cases of systems (1) and (2), respectively. For convenience in the following discussion, on the one hand, we introduce the following notations:

where denotes either or , . On the other hand, we list several assumptions:

 ; ; ; ;there exists a such that , , for ;there exists a such that , , for ;; ; ; ; , ;, ;, ;, .

By applying theorems in Sections 2 and 3 and to systems (89) and (90), respectively, we obtain the following theorems.

Theorem 43. Assume hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least two positive -periodic solutions.

Theorem 44. Assume and hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least two positive -periodic solutions.

Theorem 45. Assume hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least two positive -periodic solutions.

Theorem 46. Assume hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least one positive -periodic solution.

Theorem 47. Assume and hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least one positive -periodic solution.

Theorem 48. Assume hold. Moreover, if one of the following conditions holds:

then, (89) and (90) have at least one positive -periodic solution.

Application 3. Consider the following equations:

where

which are special cases of systems (89) and (90) without impulse, respectively, that is, . For convenience in the following discussion, we list several assumptions:

;;;there exists a such that , for ;there exists a such that , for ;; ;; ;;;;.

We obtain the following theorems.

Corollary 49. Assume hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least two positive -periodic solutions.

Corollary 50. Assume and hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least two positive -periodic solutions.

Corollary 51. Assume hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least two positive -periodic solutions.

Corollary 52. Assume hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least one positive -periodic solution.

Corollary 53. Assume , hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least one positive -periodic solution.

Corollary 54. Assume hold. Moreover, if one of the following conditions holds:

then, (99) and (100) have at least one positive -periodic solution.

Hence, our results generalize and improve the corresponding results of [16].

Application 4. Consider the generalized logistic model of single species [16, 21] with impulse and two parameters:

where are -periodic, , are two parameters, and is integrable such that .

Theorem 55. Assume that and the following conditions hold:

(1), ;(2);

then, (108) has at least one positive -periodic solution, where

Proof. We let

Then, (108) can be seen as a special form of (2) satisfying . We can construct the same Banach space and cones E as in Section 2. Then for any ,

so we have

that is, . On the other hand,

which can lead to

that is, . Then in view of (2), (112), and (114), we can obtain that (108) has at least one positive -periodic solution. The proof is complete.

Application 5. Consider the generalized so-called Nicholson’s Blowflies model [19, 20] with impulse and two parameters:

where are -periodic and , are two parameters.

Theorem 56. Assume that and the following conditions hold:

(3), ;

then, (115) has at least one positive -periodic solution, where

Proof. We let . Then, (115) can be seen as a special form of (1) satisfying . We can construct the same Banach space and cones as in Section 2. Then for any ,

So we have

that is, . On the other hand,

which can lead to

that is, . Then in view of (1), (93), and (120), we can obtain that (115) has at least one positive -periodic solution. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by NSF of China (nos. 10971229, 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 and 13C084), the Science Foundation of Hengyang Normal University (no. 11B36), and the construct program of the key discipline in Hunan Province.

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