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International Journal of Engineering Mathematics
Volume 2013 (2013), Article ID 912858, 12 pages
Existence and Global Attractivity of Positive Periodic Solutions for The Neutral Multidelay Logarithmic Population Model with Impulse
1Department of Mathematics, National University of Defense Technology, Changsha 410073, China
2Department of Mathematics, Hengyang Normal University, Hengyang, Hunan 421008, China
3School of Mathematical Sciences and Statistics, Central South University, Changsha, Hunan 410075, China
Received 28 January 2013; Accepted 15 April 2013
Academic Editor: Shouming Zhong
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.
Suffiicient and realistic conditions are established in this paper for the existence and global attractivity of a positive periodic solution to the neutral multidelay logarithmic population model with impulse by using the theory of abstract continuous theorem of k-set contractive operator and some inequality techniques. The results improve and generalize the known ones in Li 1999, Lu and Ge 2004, Y. Luo and Z. G. Luo 2010, and Wang et al. 2009. As an application, we also give an example to illustrate the feasibility of our main results.
In this paper, we investigate the existence and uniqueness of the positive periodic solution of the following neutral population system with multiple delays and impulse: with the following initial conditions: where are positive continuous -periodic functions with . Furthermore,, for all . For the ecological justification of (1) and similar types refer to [1–7]. In recent years, Gopalsamy  and Kirlinger  had proposed the following single species logarithmic model:
In , Li considered the following nonautonomous single species logarithmic model: He used the continuation theorem of the coincidence degree theory to establish sufficient conditions for the existence and attractivity of positive periodic solutions of the system (4).
For more works on the periodic solution of the neutral type logistic model or the Lotka-Volterra model, see [8–12] for details. Only little scholars considered the neutral logarithmic model (see [4–7]). Li  had studied the following single species neutral logarithmic model: Lu and Ge  and Y. Luo and Z. G. Luo  employed an abstract continuous theorem of -set contractive operator to investigate the following equation: They established some criteria to guarantee the existence of positive periodic solutions of the system (6), respectively.
In , Wang et al. had investigated the existence and uniqueness of the positive periodic solution of the following neutral multispecies logarithmic population model: By using an abstract continuous theorem of a -set contractive operator, the criteria are established for the existence and global attractivity of positive periodic solutions for model (7).
On the other hand, there are some other perturbations in the real world such as fires and floods, which 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 [13–19], 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 [20–22].
However, to this day, no scholars had done works on the existence, uniqueness, and global stability of the positive periodic solution of (1). One could easily see that the systems (5)–(7) are all special cases of the system (1). Therefore, we propose and study the system (1) in this paper.
Throughout this paper, we make the following notations.
Let be a constant, , with the norm defined by , , with the norm defined by .
Then, those spaces are both Banach spaces. We also denote that
For the sake of generality and convenience, we always make the following fundamental assumptions: are all positive periodic continuous functions with period ; are fixed impulsive points with ; is a real sequence such that , and is an -periodic function.
In the following section, some definitions and some useful lemmas are listed. In the third section, by using an abstract continuous theorem of -set contractive operator and some inequality techniques, we acquired some sufficient conditions which ensure the existence and uniqueness of the positive periodic solution of the systems (1) and (2). Finally, we give an example to show our results.
Definition 1 (see ). A function is said to be a positive solution of (1) and (2), 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 .
We can easily get the following Lemma 3.
Lemma 3. The region is invariant with respect to (1).
Proof. In view of biological population, we obtain . By the system (1), we have
Then, the solution of (1) and (2) is positive.
Under the above hypotheses , we consider the following neutral nonimpulsive system: with the following initial conditions: where By a solution of (10) and (11), it means an absolutely continuous function defined on that satisfies (10) a.e., for , and on .
The following lemmas will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in .
Proof. It is easy to see that is absolutely continuous on every interval ,
On the other hand, for any ,
It follows from (13)–(15) that is a solution of (1) and (2).
Since is absolutely continuous on every interval , , and in view of (15), 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), one can check that are solutions of (10) and (11) on . The proof of Lemma 3 is completed.
Definition 5 (see ). Let be a bounded subset in . Define that where denotes the diameter of the set , obviously, . So, is called the (Kuratowski) measure of noncompactness of .
Definition 6 (see ). Let be two Banach spaces and ; a continuous and bounded map is called -set contractive if for any bounded set one has
is called strict-set-contractive if it is -set contractive for some .
For a Fredholm operator with index zero, according to [9, 24], we define
Lemma 7 (see [9, 24]). Let be a Fredholm operator with zero index and be a fixed point. Suppose that is called a -set contractive with , where is bounded, open, and symmetric about . Further, one also assumes that(1), for ; (2), for ;
where is a bilinear form on and Q is the projection of Y onto Coker(), where Coker() is the cokernel of the operator . Then, there is an such that .
In order to use Lemma 7 to study (17), we set , and defined by It is easy to see from  that is a Fredholm operator with index zero. Thus, (17) has a positive -periodic solution if and only if has a solution , where .
Lemma 8 (see ). The differential operator is a Fredholm operator with index zero and satisfies .
Lemma 9. Let be two positive constants and . If , then is a k-set contractive map.
3. Main Theorem
Since , , we see that all have its inverse function. Throughout the following part, we set that represent the inverse function of , , respectively. We also denote
Remark 12. From Lemma 9, we get that ; ; then, Similarly, Thus,
Theorem 13. In addition to , suppose that the following conditions hold: there exists a constant such that , for all , where is defined by (23); if , and .
Then, (1) has at least one positive -periodic solution.
Proof. Let be an arbitrary -periodic solution of the operator equation as follows: where , defined by (21) and (22), respectively. Then, satisfies the following operator equation: Integrating both sides of (28) over , we have Let , then and By Lemma 10, we have Similarly, we have Substituting (31) and (32) into (29), we have Considering assumption , we know that , and then, it follows from the integro mean value theorem that there exists a satisfying By Lemma 11, we can get that Multiplying both sides of (28) by and integrating them over , we have By using the Cauchy-Schwarz inequality, we get that Meanwhile, we see that Substituting (38) and (35) into (37), we can find that which gives that From , we get that . Then, there exists a constant such that From (35) and the inequality, we obtain that Again from (28), we get that From condition , it is easy to see that Now, we take and . Then, . So by (42) and (44), we can find that all the conditions of Lemma 4 except (2) hold. In what follows, we will prove that condition (2) of Lemma 4 is also satisfied. In order to do this, we defined a bounded bilinear form on by as the following . Also, we defined as . It is obvious that Without loss of generality, suppose that ; then, since , then By (46), we get that Therefore, by using Lemma 4, we obtain that (1) has at least one positive -periodic solution; the proof of Theorem 13 is completed.
Since , , it follows that . So from Theorem 13, we have the following result.
On the other hand, if , for all , it follows that . So from Theorem 13, we also have the following result.
Our next theorem is concerned with the global attractivity of periodic solution of the system (1).
Theorem 16. Suppose that and the following conditions hold: there is a positive constant such that
, as .
Then, the positive -periodic solution of (1) is globally attractive, where .
Proof. Suppose that is a positive -periodic solution of (10), is another positive solution of (10). Similar to (17), we have Let ; then, Multiply both sides of (51) with and then integrate from 0 to to obtain that Then, Meanwhile, we see that where Substituting (55) into (54), we get that Therefore, we have From , we have From , we have thus, , as ; that is, the positive -periodic solution of (10) is globally attractive; by Definition 2, the positive -periodic solution of (1) is globally attractive. The proof is completed.
Consider the following equation: which is a special case of the system (1) without impulse. Similarly, we can get the following conclusions.
Corollary 17. Suppose that the following conditions hold:,; ; are all positive -periodic continuous functions with , , , , for all , for all ; furthermore, , , for all , for all ; there exists a constant such that , for all , where is defined by the following:
if , and .
Then, (61) has at least one positive -periodic solution.
Corollary 18. Suppose that and the following conditions hold: there is a positive constant such that
, as .
Then, the positive -periodic solution of (61) is globally attractive, where .
Remark 19. One could easily see that the systems (5)–(7) are all special cases of the system (61); we can get the similar results, and we omit them here. Hence, our results improve and generalize the corresponding results in [4–7].
4. An Example
Consider the following impulsive model: where are all positive -periodic continuous functions with ; furthermore, , , , .
Corollary 20. Suppose that and the following conditions hold: there exists a constant such that , for all , where is defined by the following:
if , and .
Then, (64) has at least one positive -periodic solution, where
Corollary 21. Suppose that and the following conditions hold: there is a positive constant such that