Multiple Positive Periodic Solutions for Two Kinds of Higher-Dimension Impulsive Differential Equations with Multiple Delays and Two Parameters
By applying the fixed point theorem, we derive some new criteria for the existence of multiple positive periodic solutions for two kinds of -dimension periodic impulsive functional differential equations with multiple delays and two parameters: ), a.e., , , , , , and ), a.e., , , , , , As an application, we study some special cases of the previous systems, which have been studied extensively in the literature.
Let , , , , , and , respectively. Denote by the set of operators which are continuous for , and have discontinuities of the first kind at the points but are continuous from the left at these points. For each , the norm of is defined as . Let denote the Banach space of bounded continuous functions with the norm , where . The matrix means that each pair of corresponding elements of and satisfies the inequality . In particular, is called a positive matrix if .
Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences, and numerous examples are given in [1–4]. 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, and we refer the reader to [5–17]. Recently, in , Jiang and Wei studied the following nonimpulsive delay differential equation: They obtained sufficient conditions for the existence of the positive periodic solutions of (1). Motivated by , in , Zhao et al. investigated the following impulsive delay differential equation: They derived some sufficient conditions for the existence of the positive periodic solutions of (2). In , Huo et al. considered the following impulsive delay differential equation: They got sufficient conditions for the existence and attractivity of the positive periodic solutions of (3). Motivated by [5–7], in , Zhang et al. studied the following impulsive delay differential equation: They obtained some sufficient conditions for the existence of the positive periodic solutions of (4). However, to this day, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of impulsive functional differential equations with two parameters: with initial conditions: where and , , are -periodic; that is, , , with , (here representing the right limit of at the point ). ; that is, changes decreasingly suddenly at times , and is a constant. We assume that there exists an integer such that , , where .
Throughout the paper, we make the following assumptions: are locally summable -periodic functions; that is, , , and for all , , and are two parameters; and for all , such that , ;, satisfies and . satisfy Caratheodory conditions and are -periodic functions in . Moreover, there exists a positive constant such that , . Without loss of generality, we can assume that and ; is a real sequence such that , , and satisfies for all .
In addition, the parameters in this paper are assumed to be not identically equal to zero.
To conclude this section, we summarize in the following a few definitions and lemmas that will be needed in our arguments.
Definition 1 (see ). A function is said to be a positive solution of (5) and (6) if the following conditions are satisfied:(a) is absolutely continuous on each ;(b)for each and exist, and ;(c) satisfies the first equation of (5) and (6) for almost everywhere (for short a.e.) in and satisfies for , .Under the previous hypotheses , we consider the neutral nonimpulsive system:
with initial conditions:
By a solution of (9) and (10), it means an absolutely continuous function defined on that satisfies (9) and (10), that is, for , and , on .
The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem 1 in .
Lemma 2. Suppose that hold. Then(i)if are solutions of (9) and (10) on , then are solutions of (5) and (6) on ;(ii)if are solutions of (5) and (6) on , then are solutions of (9) and (10) on .
Proof. It is easy to see that are absolutely continuous on every interval , , ,
On the other hand, for any , ,
It follows from (13)–(15) that are solutions of (5). Similarly, if are solutions of (10), we can prove that are solutions of (6).
Since is absolutely continuous on every interval , , , and in view of (14), it follows that, for any , which implies that are continuous on . It is easy to prove that are absolutely continuous on . Similar to the proof of , we can check that are solutions of (9) on . Similarly, if are solutions of (6), we can prove that are solutions of (10). The proof of Lemma 2 is completed.
In the following section, we only discuss the existence of a periodic solution for (9) and (10).
Definition 3 (see ). Let be a real Banach space, and let be a closed, nonempty subset of . is said to be a cone if(1) for all , and , and(2) imply .
Lemma 4 (Krasnoselskii fixed point theorem see [20–22]). Let be a cone in a real Banach space . Assume that and are open subsets of with , where . Let be a completely continuous operator and satisfy either(1), for any and , for any or(3), for any and , for any .Then has a fixed points in .
The paper is organized as follows. In next section, firstly, we give some definitions and lemmas. Secondly, we derive some existence theorems for one or two positive periodic solutions of system (5) by using Krasnoselskii fixed point theorem under some conditions. In Section 3, we also get some existence theorems for one or two positive periodic solutions of system (6) that are also established by applying Krasnoselskii fixed point theorem under some conditions. Finally, as an application, we give two examples to show our results.
2. Existence of Periodic Solutions of System (5)
We establish the existence of positive periodic solutions of (5) by applying the Krasnoselskii fixed point theorem on cones. We will first make some preparations and list a few preliminary results. For , , we define It is clear that , , . In view of , we also define for , Let with the norm , where , and it is easy to verify that is a Banach space. Define to be a cone in by and we easily verify that is a cone in .
We define an operator as follows: where For convenience in the following discussion, we introduce the following notations: where denotes either or , . Moreover, we list several assumptions:: ;: ;: ;: ;: ;: ;: ;: ;: there exists , such that , for any ;: there exists , such that , for any .The proofs of the main results in this paper are based on an application of Krasnoselskii fixed point theorem in cones. To make use of fixed point theorem in cones, firstly, we need to introduce some definitions and lemmas.
Lemma 5. Assume that hold. The existence of positive -periodic solution of (9) is equivalent to that of nonzero fixed point of in .
Proof. Assume that is a periodic solution of (9). Then, we have
Integrating the above equation over , we can have
Therefore, we have
which can be transformed into
Thus, are periodic solutions for (21).
If and with , then for any , derivative the two sides of (21) about , Hence is a positive -periodic solution of (9). Thus we complete the proof of Lemma 5.
Lemma 6. Assume that hold. Then the solutions of (5) are defined on and are positive.
Proof. By Lemma 2, we only need to prove that the solutions of (9) are defined on and are positive on . From (9), we have that, for any and , Therefore, are defined on and are positive on . The proof of Lemma 6 is complete.
Lemma 7. Assume that hold. Then is well defined.
Proof. From (21), for any , Therefore, . From (21), we have On the other hand, we obtain Therefore, . The proof of Lemma 7 is complete.
Lemma 8. Assume that hold, and there exists such that and then
Proof. For any , then Thus, we have The proof of Lemma 8 is complete.
Lemma 9. Assume that hold, and let . If there exists a sufficiently small such that and then
Proof. For any , we have The proof of Lemma 9 is complete.
Our main results of this paper are as follows.
Theorem 10. In addition to , if and hold, then system (5) has at least one positive -periodic solution.
Proof. By , there exists such that where the constant satisfies . Then by Lemma 8, we have On the other hand, by , for any , there exists such that We choose If , then where This implies that