ISRN Mathematical AnalysisVolume 2013 (2013), Article ID 695891, 18 pageshttp://dx.doi.org/10.1155/2013/695891

Research Article

A New Existence Theory for Positive Periodic Solutions to a Class of Neutral Delay Model with Feedback Control and Impulse

Zhenguo Luo,1,2 Jianhua Huang,¹ and Binxiang Dai³

¹Department of Mathematics, National University of Defense Technology, Changsha 410073, China²Department of Mathematics, Hengyang Normal University, Hengyang 421008, China³School of Mathematical Sciences and Statistics, Central South University, Changsha 410075, China

Received 10 May 2013; Accepted 9 July 2013

Academic Editors: M. Ehrnström, S. U. Islam, G. Ólafsson, G. Schimperna, and L. Wang

Copyright © 2013 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 acquire some sufficient and realistic conditions for the existence of positive periodic solution of a general neutral impulsive -species competitive model with feedback control by applying some analysis techniques and a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for -set contraction. As applications, we also examine some special cases, which have been studied extensively in the literature, some known results are improved and generalized.

1. Introduction

In this paper, we consider the existence of the positive periodic solution of the following impulsive -species competition system with multiple delays and feedback control: with the following initial conditions: where , , , , , , , , , , , and are continuous -periodic functions; are continuous  -periodic functions with . The growth functions are not necessarily positive; since the environment fluctuates randomly, in some conditions, may be negative. Consider the following: ; and , , and . And and represent the birth rate and the harvesting (or stocking) rate of at time , respectively. When , it stands for harvesting, while means stocking. For the ecological justification of (1) and the similar types, refer to [1–14].

In 1991, in [1], Gopalsamy et al. have established the existence of a positive periodic solution for a periodic neutral delay logistic equation where , and are positive continuous -periodic functions with and is a positive integer. In 1993, in [2], Kuang proposed an open problem (Open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation: In [3], Li tried to give an affirmative answer to the previous open problem; however, there is a mistake in the proof of Theorem 2 in  [3]. With the aim of giving a right answer to the previous open problem,  [4–6] also have investigated the previous question. However, it is more complex to check the sufficient conditions of the system  [5, 6].  Moreover, in  [7], Li studied the existence of positive periodic solution of the neutral Lotka-Volterra equation with several delays where , and are positive continuous -periodic functions and are nonnegative constants. Recently, in [8], Lu and Ge investigated a neutral delay population model with multiple delays: They applied the theory of abstract continuous theorem of -set contractive operator and some analysis techniques to obtain some sufficient conditions for the existence of positive periodic solutions of the model (6).

It is of course very interesting to study the neutral delay population model for higher dimensional systems. In fact, in  [9], Li has studied the neutral Lotka-Volterra system with constant delays where , and , , and are positive continuous -periodic functions, and are nonnegative constants. He obtained sufficient conditions that guarantee the existence of positive periodic solution of the system (7), by applying a continuation theorem based on Gaines and Mawhin's coincidence degree. Noticing that delays arise frequently in practical applications, it is difficult to measure them precisely. In population dynamics, it is clear that a constant delay is only a special case. In most situations, delays are variable, and so in  [10], Liu and Chen investigated the following general neutral Lotka-Volterra system with unbounded delays: They introduced a new existence theorem to obtain a set of sufficient conditions for the existence of positive periodic solutions for the system (8), and their results improved and generalized some known results.

Moreover, in some situations,  people may wish to change the position of the existing periodic solution but keep its stability. This is of significance in the control of ecology balance. One of the methods for its realization is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control schemes or by harvesting procedure. In fact, during the last decade, the qualitative behaviors of the population dynamics with feedback control have been studied extensively; see [11, 15–23]. Recently, in [11], Chen considered the following neutral Lotka-Volterra competition model with feedback control of the form: With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, he established easily verifiable criteria for the global existence of positive periodic solutions of the system (9), and his results extended and improved existing results.

On the other hand, there are some other perturbations in the real world, such as fires and floods, that are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations  involving impulsive differential equations have attracted the interest of many researchers in the past twenty years, see [12–14, 24–30], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, and optimal control. For details, see [31–33].

In  [12], Huo studied the following neutral impulsive delay Lotka-Volterra system: By using some techniques of Mawhin’s coincidence degree theory, he obtained sufficient conditions for the existence of periodic positive solutions of the system (10).

In  [13], Wang and Dai investigated the following periodic neutral population model with delays and impulse: They obtained some sufficient conditions for the existence of positive periodic solutions of the model (11) by using the theory of abstract continuous theorem of -set contractive operator and some analysis techniques.

Recently, in  [14], Luo et al. studied the following -species competition system with general periodic neutral delay and impulse: They obtained some sufficient and realistic conditions for the existence of positive periodic solutions of the system (12), by using a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for -set contraction.

However, to this day, no scholars had done works on the existence of positive periodic solution of the system (1). One could easily see that systems (3)–(12) are all special cases of the system (1). Therefore, we propose and study the system (1) in this paper.

For the sake of generality and convenience, we make the following notations and assumptions: 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 .Then, the previous spaces are all Banach spaces. We also denote that and make the following assumptions: (A), , , , , , , , , , and are continuous -periodic functions, and , ,  and  ;(B) satisfies and ;(C) is a real sequence such that , are -periodic functions.

The organization of this paper is as follows. In the following section, we introduce some lemmas and an important existence theorem developed in [34, 35]. In the third section, we derive some sufficient conditions, which ensure the existence of positive periodic solution of system (1) by applying this theorem and some other techniques. Finally, we study some special cases of system (1), which have been studied extensively in the literature. These examples show that our sufficient conditions are new, and some known results can be improved and generalized.

2. Preliminaries

In this section, in order to obtain the existence of a periodic solution for system (1) and (2), we will give some concepts and results from [35], and we will state an existence theorem and some lemmas.

For a fixed , let , if for some and , then for is defined by for . The supremum norm in is denoted by , that is, for , where denotes the norm in and for . Consider the following neutral functional differential equation: where is completely continuous, and is continuous. Moreover, we assume the following: (1) there exists such that for every , we have and ;(2) there exists a constant such that , for and .

By using the continuation theorem for composite coincidence degree, in [34], Erbe et al. proved the following existence theorem (see also Theorem 4.7.1 in [35]).

Lemma 1. Assume that there exists a constant such that (i) for any and any -periodic solution of the system  One has that for ;(ii) for , where , and denotes the constant mapping from to with the value ;(iii). Then, there exists at least one -periodic solution of the system (14) that satisfies .

The following remark is introduced by Fang (see Remark 1 in [36]).

Remark 2. Lemma 1 remains valid if the assumption (ii) is replaced by the following:(ii*) there exists a constant such that for and with as given in condition (i) of Lemma 1.

We will also need the following lemmas.

Lemma 3 (see [8, 13]). Suppose that and . Then, the function has a unique inverse satisfying with , , and if , and , then .

Lemma 4 (see [27]). Suppose that is a differently continuous -periodic function on with . Then, to any .

Lemma 5. Consider that   is the positive invariable region of the system (1) and (2).

Proof. In view of biological population, we obtain . By the system (1) and (2), we have where Then, the solution of the systems (1) and (2) is positive.

Definition 6. A function is said to be a positive solution of (1) and (2) on , 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) and (2) for almost everywhere (for short a.e.) in and satisfies for .

Consider the following nonimpulsive delay differential equation: with the following initial conditions: where The following lemmas will be used in the proofs of our results. The proof of Lemma 7 is similar to that of Theorem 1 in [24].

Lemma 7. Suppose that (A)–(C) hold, then (i) if is a solution of (18) and (19) on , then is a solution of (1) and (2) on , where ; (ii) if is a solution of (1) and (2) on , then is a solution of (18) and (19) on , where .

Proof. (i) It is easy to see that is absolutely continuous on every interval , On the other hand, for any , , thus, which implies that is a solution of the system (1) and (2). Therefore, if is a solution of the system (18) and (19) on , we can prove that are solutions of the system (1) and (2) on .(ii) Since is absolutely continuous on every interval , and in view of (23), 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 (i), we can check that is a solution of the system (18) and (19) on . The proof of Lemma 7 is completed.

Lemma 8. Consider that is a -periodic solution of (18) and (19) if and only if is a -periodic solution of the following system: where and is defined by (17).

Proof. The proof of Lemma 8 is similar to that of Lemma  2.2 in [11], and we omit the details here.

From Lemmas 7 and  8, if we want to discuss the existence of positive periodic solutions of systems (1) and (2), we only discuss the existence of positive periodic solutions of systems (25) and (26).

3. The Main Result

Since , we see that all have their inverse function. Throughout the following part, we set to that represent the inverse function of , respectively. We denote that

Remark 9. From Lemma 3, we get that , then Similarly, Thus,

Here, we have the following notations: where , and are defined by (27), and .

Theorem 10. Suppose that the following conditions hold: (1) the system of algebraic equations  has a unique positive solution ;(2), , , and ;(3). Then the system (1) and (2) has at least one positive -periodic solution.

To prove the previous theorem, we make the change of variables Then, the system (25) can be rewritten in the following form: Let denote the linear space of real value continuous -periodic functions on . The linear space is a Banach space with the usual norm for a given .

We define the following maps: Clearly, and are complete continuation functions, and system (34) takes the form In the proof of our main result below, we will use the following two important lemmas.

Lemma 11. If the assumptions of Theorem 10 are satisfied and if , where such that , then , for and .

Proof. For and , we have for some . Then, we get Hence, The proof of Lemma 11 is thus completed.

Lemma 12. If the assumptions of Theorem 10 are satisfied, then every solution of the system satisfies .

Proof. Let , for , that is, which yields, after integrating from 0 to , that where is defined by (27). From (41), we derive It follows from (41)–(43) that By amplification, it follows from (42) that In view of Remark 9 and by a similar analysis, we obtain As , it follows that . So we find from (45) that That is, By the mean value theorem, we see that there exist points such that which implies that By (44) and (50), we can see that For , one can find that Besides, from (41), we have Notice that for all , one has Then, by (52) and (54), we get Furthermore, we have By the assumption (3) of Theorem 10, we see that Then, we have Since , there exists a such that It follows from (42) that It follows from (30) and (60) that From (61), we obtain As , it follows from the previous formula that On the other hand, there also exists a such that It follows from (42), (54), and (64) that From (41), (54), and (64), we can have That is, Now, from (60) and (63) we know that there exist such that From (54), (64), and Lemma 3, we have Then, Obviously, is independent of ; the proof of Lemma 12 is completed.

Based on the previous results, we can now apply Lemma 1 and Remark 2 to (34) and obtain a proof of Theorem 10.

Proof. Obviously, for as given in Lemma 11, condition (i) in Lemma 1 is satisfied. Let . Since and , we have for any . That is, condition (ii) in Lemma 1 holds. At last, we verify that condition (iii) of Lemma 1 also holds. By assumption (1) of Theorem 10 and the formula for the Brouwer degree (see Theorem 2.2.3 in [35, 36]), a straightforward calculation shows that By now, all the assumptions required in Lemma 1 hold. It follows from Lemma 1 and Remark 2 that system (34) has an -periodic solution. Returning to , we infer that systems (18) and (19) have at least one positive -periodic solution. By Lemmas 7 and 8, is the unique positive periodic solution of the system (1) and (2), where . The proof of Theorem 10 is complete.

Consider the following: which is a special case of system (1) without impulse. We get easily the following result. Here, we have the following notations: and represent the inverse function of , respectively.

Corollary 13. Suppose that the following conditions hold; (1) the system of algebraic equations  has a unique positive solution ;(2), , , and ;(3). Then, (73) has at least one positive -periodic solution.

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Similarly, we can get the following results.

Theorem 14. Assume that conditions of Theorem 10 hold, and then, the conclusion of Theorem 10 holds for the following system:

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Corollary 15. Assume that conditions of Corollary 13 hold, and then, the conclusion of Corollary 13 holds for the following system:

Proof. Its proof is similar to the proof of Theorem 10. Here, we omit it.

Remark 16. When , we can derive some immediate corollaries from Theorems 10 and 14; thus, our results generalize the corresponding results in [14].

4. Applications

In order to illustrate some features of our main result, in the following, we will apply Theorem 10 to some special cases, which have been studied extensively in the literature.

Application 17. We consider an -species neutral delay competition system in a periodic environment with impulse: where , , , , , , , , , and are continuous -periodic functions. And are continuous -periodic functions with . Here, we give the following notations: where represent the inverse function of , respectively. Applying Theorem 10 to (78), we can obtain the following theorem.

Theorem 18. Assume that the following conditions are satisfied: (1) the system of algebraic equations  has a unique positive solution ;(2), , , , and ;(3). Then, (78) has at least one positive -periodic solution.

Remark 19. When , , we can derive some immediate corollaries of Theorem 18, then, Theorem 18 generalizes the corresponding results in [9, 10]. On the other hand, when , we can derive an immediate corollary of Theorem 18, then, Theorem 18 generalizes the corresponding results in  [11].

Application 20. We consider the single specie neutral delay logistic equation with impulse: where , , , , , , , , , , and are continuous -periodic functions. And are continuous -periodic functions with . Here, we have the following notations:

and represent the inverse function of , respectively. Applying Theorem 10 to (81), we can obtain the following theorem.

Theorem 21. Assume that the following conditions are satisfied: (1), , and ;(2). Then, system (81) has at least one positive -periodic solution.

Remark 22. When , , , , and , we can derive an immediate corollary of Theorem 21, which is also an answer to the open problem 9.2 due to Kuang  [2]. On the other hand, when , , and , we can derive some immediate corollaries of Theorem 21 (that is the corresponding results in  [7, 8]), therefore, our result improves and generalizes the corresponding result in  [7, 8]. Moreover, when , , we can see that our Theorem 21 can hold without the assumption . When , Wang's main result (see Theorem 3.1 in  [13]) cannot be applied. Therefore, in comparison with [13], our result improves and generalizes the result in  [13].

Acknowledgments

This research is supported by NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (no. 2012M512162), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, and 13C084), and the construct program of the key discipline in Hunan province.

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