Research Article | Open Access
Existence of Positive Periodic Solutions for a Class of Higher-Dimension Functional Differential Equations with Impulses
By employing the Krasnoselskii fixed point theorem, we establish some criteria for the existence of positive periodic solutions of a class of -dimension periodic functional differential equations with impulses, which improve the results of the literature.
Some evolution processes are distinguished by the circumstance that the evolutions change very rapidly at certain instants. In mathematical simulations, impulsive delay differential equations may express several simulation processes in real world which depend on their prehistory and are subject to short time disturbances. Such processes occur in the theory of optional control, population dynamics, biotechnologies, economics, and so forth. 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; we refer the reader to [1–4]. For other related works on studying for impulsive delay differential equations, we refer the reader to [5–7].
Zhang et al.  investigated the following form:
In this paper, we will consider the -dimension differential equation with impulses as follows: where is a parameter, , is -periodic, and is an operator defined on (here denotes the Banach space of bounded continuous operator with the norm where . For and , is defined by for (see , Zheng). Consider that and (here represents the right limit of at the point ), , that is, changes decreasingly suddenly at , is a constant, and are the sets of all nonnegative and nonpositive real numbers, respectively. We assume that there exists an integer such that , where .
2. Some Preliminaries
, is continuous everywhere except at a finite number of points at which and exist and , . For each , the norm of is defined as .
Throughout the paper, we make the following assumptions: for all ; is a continuous function of for each , ; for any and , there exists such that for , , , and imply that
To conclude this section, we summarize in the following a few concepts and results that will be needed in our arguments.
Definition 1. Let be a Banach space, and let be a closed, nonempty subset of ; is a cone if(i) for all and all ;(ii) imply .
Let with the norm , where ; then is a Banach space.
If is a solution of (3), then where See , Zhang et al.
It is clear that , for all , and by , for and .
Define for , Let where . It is not difficult to verify that is a cone in . We define an operator as follows: where Then, it can be immediately obtained from the assumptions and that the operator is completely continuous. On the other hand, it is not difficult to check that is a positive -periodic solution of (3) if and only if is a fixed point of the operator .
Before stating the main results, we shall give some important lemmas.
Lemma 2. The mapping maps into , that is, .
Proof. For any , it is easy to see that . From (11), we have Noting that , we can also obtain Hence, . The proof is complete.
Lemma 3. Let be a Banach space, and let be a cone in . Suppose that and are open subsets of such that . Suppose that
is a completely continuous operator and satisfies either (1) or (2)Then, has a fixed point in .
The proof of Lemma 3 can be found in , Guo et al.
Lemma 4. Assume that – hold and there exists , such that Then,
Proof. If , then Thus, we have
Lemma 5. Assume that – hold and let , if there exists a sufficiently small such Then,
Proof. For any ,
3. Main Results
For the sake of convenience, we introduce the following notations: where denotes either 0 or .
Theorem 6. Assume that – hold and then (3) has at least one positive -periodic solution.
Proof. By , for any there exists , such that Choose , satisfying , by Lemma 5, we have Next, by , there exists , such that where is chosen, so that . It follows from Lemma 4 that It follows from Lemma 3 that (3) has a positive -periodic solution satisfying .
Theorem 7. Assume that – hold and then (3) has at least one positive -periodic solution.
Proof. Since , one can find an , such that
where is chosen so that . It follows from Lemma 4 that
By , we know that there exists and such that
Choose , satisfying ; then
where , , and , .
This implies that , for any .
Therefore, (3) has at least one positive -periodic solution.
Theorem 8. Suppose that there exists , such that , for , there exists , such that, , for hold; then (3) has at least one positive -periodic solution.
Proof. Without loss of generality, we may assume that . If , then by , one can get
in particular, , for all.
On the other hand, by, one has Therefore, (3) has at least one positive -periodic solution.
Theorem 9. If , ; hold, then (3) has at least one positive -periodic solution.
Proof. By assumption , for , there exists a sufficiently small such that
So, is satisfied.
By assumption , for , there exists a sufficiently large such that that is therefore, holds. By Theorem 8, we complete the proof.
Theorem 10. If ; , hold, then (3) has at least one positive -periodic solution.
Proof. By , for , there exists a sufficiently small , such that that is Again, By , for , there exists a sufficiently small such that that is In the following, we consider two cases to prove to be satisfied:(i);(ii), .The bounded case is clear. If case (ii) is valid, then there exists , such that Since , then we have which implies that condition holds. By Theorem 8, we complete the proof.
Corollary 11. If one of the following pairs and ; and ; and ; and ; and ; and ; and ; and ; and is valid, then system (3) has at least one positive -periodic solution.
This research was supported by the National Nature Science Foundation of China (no. 11371027 and no. 11071001); Special Research Fund for the Doctoral Program of the Ministry of Education of China (no. 20123401120001); National Science Foundation of Educational Committee of Anhui Province of China (no. KJ2011B052); Program of Natural Science of Colleges of Anhui Province (no. KJ2013A032 and no. KJ2012A019).
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