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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 980974, 12 pages
http://dx.doi.org/10.1155/2013/980974
Research Article

Global Positive Periodic Solutions of Generalized -Species Competition Systems with Multiple Delays and Impulses

1Department of Mathematics, National University of Defense Technology, Changsha 410073, China
2Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China

Received 20 April 2013; Accepted 18 June 2013

Academic Editor: M. Victoria Otero-Espinar

Copyright © 2013 Zhenguo Luo and Liping 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 in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized -species competition systems with multiple delays and impulses as follows: It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

1. Introduction

Let , , , , , , respectively, and let be a constant and , with the norm defined by ; , with the norm defined by   ; ; ; , with the norm defined by ; , with the norm defined by   ; , for any .

The theory of impulsive delay differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of nonimpulsive delay differential equations. Many evolution processes in nature are characterized by the fact that at certain moments of time they experience abrupt change of state. That was the reason for the development of the theory of impulsive differential equations and impulsive delay differential equations, see the monographs [14]. 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, which is referred to as [512]. However, 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 -species Lotka-Volterra competitive systems with multiple delays and impulses: where , , with . Moreover, is a constant, and . Furthermore, , (here represents the right limit of at the point ); ; that is, changes decreasingly suddenly at times . We assume that there exists an integer such that , , , where .

References [13, 14], G. Seifert investigated the following periodic single-species population growth models with periodic delay: They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , , for . The negative feedback term in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate. They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (3). But the discussion of global attractivity is only confined to the special case when the periodic delay is constant.

In [15], Freedman and Wu proposed the following periodic single-species population growth models with periodic delay: They had assumed that the net birth , the self-inhibition rate , and the delay are continuously differentiable -periodic functions, and , , , for . The positive feedback term in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation. They had established sufficient conditions which guarantee that system (4) has a positive periodic solution which is globally asymptotically stable.

References [16, 17] have studied the following two-species competitive system without delay: They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (5) by using differential inequalities and topological degree, respectively. In fact, in many practical situations the time delay occurs so often. A more realistic model should include some of the past states of the system. Motivated by the previous ideas, Liu et al. [18] considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays: where , , are -periodic functions. Here, the intrinsic growth rates are -periodic functions with , . They had derived the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the previous two competitive systems by using Gaines and Mawhin's continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional. However, to this day, no scholars had done works on the existence of positive periodic solution of (1) and (2). One could easily see that systems (3)–(6) are all special cases of system (1) and (2).

Throughout the paper, we make the following assumptions:, , with , ; satisfies and ; is a real sequence with , , ; is a -periodic function.

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 (1) and (2), if the following conditions are satisfied: is absolutely continuous on each ; for each , and exist and ; satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in and satisfies for , .
Under the previous hypotheses , we consider the following two classes of nonimpulsive Lotka-Volterra competitive systems: where By a solution of (7) and (8), it means an absolutely continuous function defined on that satisfies (7) and (8).

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 [5].

Lemma 2. Suppose that hold. Then(1)if are solutions of (7) and (8) on , then are solutions of (1) and (2) on ;(2)if are solutions of (1) and (2) on , then are solutions of (7) and (8) on .

Proof. (1) It is easy to see that are absolutely continuous on every interval , On the other hand, for any , , thus It follows from (10)–(12) that are solutions of (1). Similarly, if are solutions of (8), we can prove that are solutions of (2).
(2) Since is absolutely continuous on every interval , and in view of (12), it follows that for any , which implies that is continuous on . It is easy to prove that is absolutely continuous on . Similar to the proof of (1), we can check that are solutions of (7) on . Similarly, if , , is a solution of (2), we can prove that are solutions of (8). The proof of Lemma 2 is completed.

In the following section, we only discuss the existence of a periodic solution for (7) and (8).

Definition 3 (see [19]). Let be a real Banach space and a closed, nonempty subset of . is said to be a cone if(1) for all , and ,(2) imply .

Lemma 4 (see Krasnoselskii [20], Deimling [21], and Guo and Lakshmikantham [22]). Let be a cone in a real Banach space . Assume that and are open subsets of with , where . Let be a continuous and completely continuous operator satisfying(1), for any ;(2)there exists such that , for any and .
Then has fixed points in . The same conclusion remains valid if (1) holds for any   and (2) holds for any and .

The paper is organized as follows. In Section 2, firstly, we give some definitions and lemmas. Secondly, we derive a necessary and sufficient condition for at least one positive periodic solution of (1) which is established by using the fixed-point theorem in the cone of Banach space under some conditions. In the following section, we also get a necessary and sufficient condition for at least one positive periodic solution of (2) that is also established by applying the fixed-point theorem in the cone of Banach space under some conditions. Finally, as applications, we study some particular cases of system (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 Lemma 4. We will first make some preparations and list later a few preliminary results. Let with the norm , . It is easy to verify that is a Banach space.

We define an operator as follows: where where It is clear that , , . In view of , we also define for Define to be a cone in by We easily verify that is a cone in . For convenience of expressions, we define an operator by The proof of the main result in this paper is based on an application of Krasnoselskii fixed-point theorem in cones. To make use of fixed point theorem in the cone, firstly, we need to introduce some definitions and lemmas.

Lemma 5. Assume that hold. Then the solutions of (1) are defined on and are positive.

Proof. By Lemma 2, we only need to prove that the solutions of (7) are defined on and are positive on . From (7), we have that for any , , and , Therefore, are defined on and are positive on . The proof of Lemma 5 is complete.

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

Proof. In view of the definitions of and , for any , we have Therefore, . Furthermore, for any , it follows from (15) that On the other hand, we obtain Therefore, . The proof of Lemma 6 is complete.

Lemma 7. The operator is continuous and completely continuous.

Proof. By using a standard argument one can show that is continuous on . Now, we show that is completely continuous. Let be any positive constant and a bounded set. For any , by (15) we have Therefore, for any , we obtain which implies that is a uniformly bounded set. On the other hand, in view of the definitions of and , we have Again, from (15), we obtain which implies that , for any , is also uniformly bounded. Hence, is a family of uniformly bounded and equicontinuous functions. By the well-known Ascoli-Arzela theorem, we know that the operator is completely continuous. The proof of Lemma 7 is complete.

Lemma 8. Assume that hold. The existence of positive -periodic solution of (7) is equivalent to that of nonzero fixed point of in .

Proof. Assume that is a periodic solution of (7). Then, we have Integrating the previous equation over , we can have Therefore, we have which can be transformed into Thus, is a periodic solution for (15).
If and with , then for any derivative the two sides of (15) about , Hence, is a positive -periodic solution of (7). Thus we complete the proof of Lemma 8.

Our main result of this paper is as follows.

Theorem 9. Assume . Then condition is necessary and sufficient for system (1) to have at least one positive -periodic solution.

Proof. (Sufficiency) Let by condition (33), we know that . Choose a constant such that . Let and For any , , from (15), we obtain Hence, for any , , we have which implies that condition (1) in Lemma 4 is satisfied.
On the other hand, we choose such that . Let and suppose that . We show that for any and , . Otherwise, there exist and , such that . Let , since , it follows that which is a contradiction. This proves that condition (2) in Lemma 4 is also satisfied. By Lemmas 4 and 8, system (7) has at least one positive -periodic solution. From Lemma 2, system (1) has at least one positive -periodic solution.
(Necessity) Suppose that (33) does not hold. Then there exists at least an such that If system (7) has a positive -periodic solution , then we have Integrating the previos equation over , we can have which is a contradiction. The proof of Theorem 9 is complete.

3. Existence of Periodic Solution of (2)

Now, we are at the position to study the existence of positive periodic solutions of (2). 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 (2). For simplicity, we prefer to list later 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 to be a cone in by We easily verify that is a cone in . We define an operator as follows: where where The proof of the following lemmas and theorems is similar to those in the Section 2, so we all omit the details here.

Lemma 10. Assume that hold. Then the solutions of (2) are defined on and are positive.

Lemma 11. Assume that hold. Then is well defined.

Lemma 12. The operator is continuous and completely continuous.

Lemma 13. Assume that hold. The existence of positive -periodic solution of (8) is equivalent to that of nonzero fixed point of in .

Theorem 14. Assume . Moreover, if the condition holds, then the system (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. We consider the following periodic single-species population growth models with periodic delay and impulse: which is a special case of system (1), and where are -periodic. Thus from Theorem 9 we have the following.

Theorem 15. Assume that hold. Then condition: is necessary and sufficient for system (49) to have at least one positive -periodic solution, where

Application 2. We consider the following periodic single-species population growth models with periodic delay and impulse: which is a special case of system (2), and where are -periodic. Thus from Theorem 14 we have the following.

Theorem 16. Assume that hold. Moreover, if the condition holds, the system (52) has at least one positive -periodic solution, where

Application 3. We study the following two-species competitive system with impulses: which is a special case of system (1), and where are -periodic. Thus from Theorem 9 we have the following.

Theorem 17. Assume that hold. Then condition is necessary and sufficient for system (55) to have at least one positive -periodic solution, where

Application 4. We study the following two-species competitive system with impulses: which is a special case of system (1), and where