A New Existence Theory for Positive Periodic Solutions to a Class of Neutral Delay Model with Feedback Control and Impulse
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.
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 , 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 , 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 , Li tried to give an affirmative answer to the previous open problem; however, there is a mistake in the proof of Theorem 2 in . 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 , 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 , 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 , 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 , 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 , 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 , 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 , 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 , 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.
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 , 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 .
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 ).
We will also need the following lemmas.
Lemma 4 (see ). Suppose that is a differently continuous -periodic function on with . Then, to any .
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 .
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 , ,
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.
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
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