Research Article | Open Access
Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters
We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature.
First, we give the following definitions. Let 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 . 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; numerous examples are given in [1–3]. In recent years, in [4–11], many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability, and existence of solutions. However, to this day, still no scholars investigate the existence of multiple positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with two parameters: where and , are two parameters, , , are -periodic, that is, , , , is an operator on (here denotes the Banach space of bounded continuous operator with the norm , where , , (here represents the right limit of at the point ), , that is, changes decreasingly suddenly at times . is a constant, , , , and . We assume that there exists an integer such that , , , where .
Models of forms (1) and (2) have been proposed for population dynamics (single species growth models), physiological processes (such as production of blood cells, respiration, and cardiac arrhythmias), and other practical problems. Equations (1) and (2) are very general and incorporate many famous mathematical models extensively studied in the literature [12–21]. In this paper, we will study the existence of positive periodic solutions in more cases than the previously mentioned papers and obtain some easily verifiable sufficient criteria.
Throughout the paper, we make the following assumptions. satisfy Caratheodory conditions; that is, and are locally Lebesgue measurable in for each fixed and are continuous in for each fixed are -periodic functions in . Moreover, there exist -periodic functions , which are locally bounded Lebesgue measurable so that , and , . , are two parameters. is -periodic with respect to the first variable, that is, such that , .The delay kernel is integrable and is normalized such that , such that , ., satisfies , and ; , satisfy Caratheodory conditions and are -periodic functions in . Moreover, for all . There exists a positive constant such that , , . Without loss of generality, we can assume that and .
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 concepts and results that will be needed in our arguments.
Definition 1 (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 ;(2) imply .
Lemma 2 (see Krasnoselskii’s fixed point theorem [23–26]). 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 satisfies either , for any and , for any , or , for any and , for any .
Then, has a fixed point in .
For convenience in the following discussion, we introduce the following notations: where denotes either or , , and .
The paper is organized as follows. In Section 2, firstly, we give some definitions and lemmas. Secondly, we derive some existence theorems for one or two positive periodic solutions of (1) which are established by using Krasnoselskii’s fixed point theorem under some conditions. In Section 3, existence theorems for one or two positive periodic solutions of (2) are also established by using Krasnoselskii’s fixed point theorem under some conditions. As applications in Section 4, we study some particular cases of systems (1) and (2) which have been investigated extensively in the references mentioned earlier.
2. Existence of Periodic Solution of (1)
We establish the existence of positive periodic solutions of (1) by applying the Krasnoselskii’s fixed point theorem on cones. We will first make some preparations and list below a few preliminary results. For , , we define It is clear that , , . In view of , we also define for Let with the norm , . It is easy to verify that is a Banach space. Define as a cone in by We easily verify that is a cone in . We define an operator as follows: where The proofs of the main results in this paper are based on an application of Krasnoselskii’s fixed point theorem in cones. To make use of the fixed point theorem in cone, firstly, we need to introduce some definitions and lemmas.
Definition 3 (see ). A function is said to be a positive solution of (1) 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) for almost everywhere in and satisfies the second equation of (1) at impulsive point , .
Definition 4 (see ). Let be a real Banach space; is a cone of . The semiorder induced by the cone is denoted by “”. That is, if and only if for any .
Lemma 5 (see ). Assume that and are continuous nonnegative functions defined on the interval ; then there exists such that
Lemma 6. Assume that hold. The existence of positive -periodic solution of (1) is equivalent to that of nonzero fixed point of in .
Proof. Assume that is a periodic solution of (1). Then, we have
Integrating the above equation over , we can have
where , , , .
Therefore, which can be transformed into Thus, is a periodic solution for (9).
If and with , then for any , derivative the two sides of (9) about , For any , , we have from (9) that Hence is a positive -periodic solution of (1). Thus we complete the proof of Lemma 6.
Lemma 7. Assume that hold. Then is well defined.
Proof. From (9), it is easy to verify that is continuous in , and exist, and for each . Moreover, for any Therefore, . From (9), we have On the other hand, we obtain Therefore, . This completes the proof of Lemma 7.
Lemma 8. Assume that hold. Then is completely continuous.
Proof. We first show that is continuous. By , and are continuous in ; it follows that for any , let be small enough to satisfy that if , with ,
which implies that is continuous on .
Next we show that maps a bounded set into a bounded set. Indeed, let be a bounded set. For any and , by (9), we have Since is bounded, in view of the continuity of , it follows from (21) that is bounded and is uniformly bounded. Finally, we show that the family of functions is equicontinuous on . Let with . From (9), for any , we have Since for , , , , and are uniformly bounded in , in view of (23), it is easy to see that when tends to zero, tends uniformly to zero in . Hence, is a family of uniformly bounded and equicontinuous functions on . By Arzela-Ascoli theorem, the operator is completely continuous. The proof of Lemma 8 is complete.
Our main results of this paper are as follows.
Theorem 9. Assume that andthere exists a such that , , for ;hold. Then, (1) has two positive -periodic solutions.
Proof. First, we define ; then is an open subset of . Then, for any , we have . Consequently,
and from the definition of , we know . From (9), , and Lemma 5, we get
On the other hand, if holds, then we can choose , so that ; from the definition of , we know . Thus, we have , for , , , where constant satisfies . By (9) and Lemma 5, we can obtain
In view of (27) and (29), by Lemma 5, it follows that has a fixed point with , which is a positive -periodic solution of (1).
Likewise, in view of , for any , there is such that Let , where Then for any , from (9), (30), and (31), we have where , . This yields In view of (27) and (33), by Lemma 5, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore (1) has at least two positive periodic solutions; that is, . This proves Theorem 9.
Corollary 10. Assume that andThere exists a such that , , for ;, or hold. Then, (1) has a positive -periodic solution.
Theorem 11. Assume that andthere exists a such that , , for ;hold. Then, (1) has two positive -periodic solutions.
Proof. We define , where R satisfied ; then is an open subset of and . For any , by the definition of , we get . Furthermore, by (9), , and Lemma 5, we have This implies for any On the one hand, since , then for any there exists such that Letting , then for any , one has , , . Consequently, where, together with (42) and Lemma 5, we have This implies for any In view of (35) and (39), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). On the other hand, if , then for any there exists such that Letting , then for any , one has , , . Consequently, where, together with (41) and Lemma 5, we have This yields In view of (35) and (43), by Lemma 2, it follows that has a fixed point with , which is a positive -periodic solution of (1). Therefore, (1) has at least two positive periodic solutions; that is . This proves Theorem 11.
Corollary 12. Assume that andthere exists a such that , , for ;, or ;hold. Then, (1) has a positive -periodic solution.
Theorem 13. Assume that , and hold. Then, (1) has a positive -periodic solution y with lying between R and r, which are defined in and , respectively.