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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 980935, 10 pages
The Existence and Uniqueness of Positive Periodic Solutions for a Class of Nicholson-Type Systems with Impulses and Delays
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
Received 27 March 2013; Accepted 9 July 2013
Academic Editor: Ming Mei
Copyright © 2013 Ruojun Zhang and Fuyun Lian. 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.
By establishing the equivalence, respectively, to the existence and uniqueness of positive periodic solutions for corresponding delay Nicholson-type systems without impulses, some criteria for the existence and uniqueness of positive periodic solutions for a class of Nicholson-type systems with impulses and delays are established. The results of this paper extend some earlier works reported in the literature.
As we know, in order to describe the dynamics of Nicholson’s blowflies, Gurney et al.  proposed a mathematical model where is the size of the population at time , is the maximum per capita daily egg production, is the size at which the population reproduces at its maximum rate, is the per capita daily adult death rate, and is the generation time. Nicholson’s blowflies model described by delay differential equation (1) belongs to a class of biological system, and more and more biology systems have attracted more attention because of their extensively realistic significance [2–6]. In particular, the effects of a periodically varying environment are important for evolutionary theory, as the selective forces on systems in a fluctuating environment differ from those in a stable environment. There have been some results in the literature of the problem of the existence of positive periodic solutions for Nicholson’s blowflies equation [7–11].
Recently, in order to describe the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics which are examples of Nicholson-type delay differential systems, Berezansky et al. , Wang et al. , and Liu  studied the following Nicholson-type delay systems: where , , , , and , , . For coefficients and delays that are constants, Berezansky et al.  presented several results for the permanence and globally asymptotic stability of system (2). Supposing that , , , , and are almost periodic functions, Wang et al.  obtained some criteria to ensure that the solutions of system (2) converge locally exponentially to a positive almost periodic solution. Furthermore, Liu  established some criteria for existence and uniqueness of positive periodic solutions of system (2) by applying the method of the Lyapunov functional.
However, besides delay effects, impulsive effects likewise exist widely in many evolution processes in which states are changed abruptly at certain moment of time, involving such fields as medicine and biology, economics, mechanics, electronics, and telecommunications. That was the reason for the development of the theory of impulsive differential equations. We refer the reader to the monographs [15–17]. In practical, Yan  provided the method for studying a class of impulsive differential equations by changing impulsive equations into corresponding equations without impulses.
Therefore, it is necessary and reasonable to consider impulsive effects on the existence and uniqueness of positive periodic solutions for Nicholson-type delay systems (2). However, to the best of our knowledge, there are few results of this problem. Thus, techniques and methods of existence and uniqueness of positive periodic solutions for system (2) with impulsive effects should be developed and explored.
In this paper, we consider a class of Nicholson-type systems with impulses and delays where , , , , , , . are the impulses at moments .
In (3), we will use the following hypotheses: are fixed impulsive points with ; is a real sequence and ;, , , , , and are periodic functions with common periodic , , , and .
Here and in the sequel we assume that a product equals unit if the number of factor is equal to zero. We will only consider the solutions of (3) with initial values given by where , and , , , .
The organization of this paper is as follows. In Section 2, we introduce some notations, definitions, and lemmas. In Section 3, some new sufficient conditions ensuring the existence and uniqueness of positive periodic solutions of (3) are derived by changing (3) into a corresponding equation without impulses. We study the existence of positive periodic solutions for the corresponding equation without impulses by coincidence degree theory, and we study the uniqueness of positive periodic solutions for the corresponding equation without impulses by the Lyapunov function. Finally, some conclusions are drawn in Section 4.
For the sake of convenience, throughout this paper, we adopt the following notations:
Definition 1. A map is said to be a solution of (3) on satisfying the initial value condition (4), if(i) is absolutely continuous on each interval and ;(ii)for any , , and exist, and ;(iii) satisfies (3), .
Under the previous hypotheses , we consider the following delay differential equation without impulses: with initial condition (4) where
By a solution of (6) on , we mean that an absolutely continuous function on satisfies (6) for and initial condition (4). Similar to the method of , we have the following.
Lemma 2. Assume that hold. Then(i)if is a solution (or positive -periodic solution) of (6) on , then is a solution (or positive -periodic solution) of (3) on ;(ii)if is a solution (or positive -periodic solution) of (3) on , then is a solution (or positive -periodic solution) of (6) on .
Proof. First, we prove (i). If is a solution (or positive -periodic solution) of (6) on , then it is easy to see that is absolutely continuous on each interval and , , and, for any ,
Similarly, we have
On the other hand, for every , ,
Thus, for every ,
It follows from (9) and (12) that is the solution (or positive -periodic solution) of (3) corresponding to initial condition (4).
Next, we prove (ii). If is a solution (or positive -periodic solution) of (3) on , then is absolutely continuous on each interval and , and in view of (12), it follows that, for any , , Equations in (13) imply that is continuous on . It is easy to prove that is also absolutely continuous on . Now, one can easily check that is the solution (or positive -periodic solution) of (6) corresponding to initial condition (4). This completes the proof.
Lemma 3. Assume that hold. Then every solution of (3) is defined and positive on .
In the next section, we will study the existence of positive periodic solution of system (3). The method to be used in this paper involves the applications of the Mawhin’s continuous theorem of the coincidence degree theory. We introduce some concepts and results concerning the coincidence degree as follows. Let and be real Banach spaces, let be a linear mapping, and let be a continuous mapping. The mapping is called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, there exist continuous projectors and such that ; and .
Lemma 4 (see ). Let be a Banach space. Suppose that is a Fredholm operator with index zero and is L-compact on with open bounded in . Moreover, assume that all the following conditions are satisfied:(i), for all ;(ii), for all ;(iii)the Brouwer degree Then equation has at least one solution in .
For ease of exposition, let We denote as the set of all continuously positive -periodic functions defined on and denote Then, is a Banach space when it is endowed with the norm .
Again from (17) and (18), it is not difficult to show that , is closed in , and . From the definitions of continuous projectors and , we can easily to get It follows that the operator is a Fredholm operator with index zero. Furthermore, the generalized inverse (of ) reads as Therefore, it is easy to see from (17) and (23) that is -compact on , where is any open bounded set in .
3. Main Results
Theorem 5. Assume that hold. Moreover, the following condition is satisfied:
Then (3) has at least one positive -periodic solution.
Proof. Based on Lemma 4, what we need to do is just to search for an appropriate open bounded subset for applying Mawhin’s continuous theorem. To do this, it suffices to prove that the set of all possible positive -periodic solution of (6) is bounded.
Let be an arbitrary positive -periodic solution of (6). Corresponding to the operator equation (21), we have Multiplying and the first formula of (24), and then integrating from to , we obtain hence, Furthermore, from the Hölder inequality and (or ) for , we have that is, Similarly, we have Combining (28) and (29), we get Since , (30) implies that and are bounded. Therefore, according to the previous proof, there exists a positive constant (independent of ) such that From (24), together with the Hölder inequality and for , we can obtain (31)–(32) imply that there exist two positive constants , , such that Equations in (31) imply that there exist two points , , and two positive constants , , such that Since, for , from (33), it follows that there exists a positive constant such that Clearly, is independent of . Let , where is taken sufficiently large so that Now, we take . This satisfied condition (i) of Lemma 4.
When , is a constant vector in with , thenIn view of (38), we have Equations in (39), together with (37), imply that Consequently, condition (ii) of Lemma 4 is satisfied.
Furthermore, we define a continuous function by When and , is a constant vector in with , then, from (37), we obtain It follows that Hence, using the homotopy invariance theorem, we obtain Condition (iii) of Lemma 4 is also satisfied. Thus, by Lemma 4, we conclude that ; that is, (6) has at least one solution in . Then, by Lemma 2, we immediately obtain that (3) has at least one positive -periodic solution. This completes the proof.
Theorem 6. Let hold; furthermore, assume that the following conditions are satisfied:,
and , , .
Then (3) has a unique positive -periodic solution.
Proof. By Lemma 2, it suffices to prove the uniqueness of positive -periodic solutions for system (6). According to Theorem 5, we know that (6) has at least a positive -periodic solution with initial condition (4). Suppose that is an arbitrary positive -periodic solution of (6) with initial condition (4). Then it follows from (6) that Calculating the upper-right derivative, we have For , is a real number, , and by mean value theorem, we have where . Thus, for any fixed , we also have Hence, Similarly, we have We define a Lyapunov function by for , and by virtue of (49), (50), and assumption , we get According to , and it follows that Hence, we obtain So In view of (56) and periodicity of , we have Then by Lemma 2, we conclude that (3) has a unique positive -periodic solution. This completes the proof.
In this paper, a class of Nicholson-type systems with impulses and delays are investigated. We conquer the difficulty of coexistence of impulsive and delay factors in a dynamic system and give some results of the existence and uniqueness of positive periodic solutions. The results in this paper extend some earlier works reported in the literature. Moreover, our results are easy to test and important in applications of periodic oscillatory delayed Nicholson-type systems with impulsive control.
This work is supported by the National Natural Science Foundation of China (Grant no. 11171374) and the Scientific Research Fund of Shandong Provincial of China (Grant no. ZR2011AZ001).
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