Abstract
By using a fixed point theorem of strict-set-contraction, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for -set contraction, we established some new criteria for the existence of positive periodic solution of the following generalized neutral delay functional differential equation with impulse: . As applications of our results, we also give some applications to several Lotka-Volterra models and new results are obtained.
1. Introduction
Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This complex dynamical behavior can be modeled by impulsive differential equations. Impulsive differential equations have become more important in recent years in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics; see [1–8]. There has been a significant development in impulse theory, in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs [9–11].
In this paper, we consider more general neutral delay functional differential equation with impulse: where , , are -periodic functions and is -periodic function with respect to its first argument. Moreover, , represents the right, left limit of at the point , respectively. In this paper, it is assumed that is left continuous at ; that is, changes decreasingly suddenly at times . , is a constant, , and , . We assume that there exists an integer such that , , where . For the ecological justification of (1) and the similar types refer to [8, 12–17].
In 1993, Kuang in [12] proposed an open problem (open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation: In [13], Fang and Li studied model (2) and gave an answer to the open problem 9.2 of [12]. But paper [13] required that , and , or , for , where . In [14], Yang and Cao studied a general neutral delay model of single-species population growth: They applied the theory of coincidence degree to obtain verifiable sufficient conditions of the existence of positive periodic solutions of system (3). In [15], Lu considered the following neutral functional differential equation: He obtained some sufficient conditions for the existence of positive periodic solutions of model (4) by using the theory of abstract continuous theorem of -set contractive operator and some analysis techniques. In [16], Yang and Cao used the theory of coincidence degree to investigate a complex neutral equation with several state-dependent delays as follows: They also got some verifiable sufficient conditions of the existence of positive periodic solutions of system (5). In [17], Li and Kuang considered the periodic Lotka-Volterra equation with state-dependent delays: They used the continuation theorem of coincidence degree theory to obtain some sufficient and realistic conditions for the existence of positive periodic solutions of system (6). In [8], Wang and Dai investigated the following periodic neutral population model with delays and impulse: They obtained some sufficient conditions for the existence of positive periodic solutions of model (7) by using the theory of abstract continuous theorem of -set contractive operator and some analysis techniques.
The main purpose of this paper is to establish new criteria to guarantee the existence of positive periodic solutions of the system (1) by using a fixed point theorem of strict-set-contraction [18–20].
For convenience, we introduce the notation where is a continuous -periodic function.
Throughout this paper, we assume the following. are -periodic functions. In addition, , , and . is -periodic function with , . There exist -periodic functions , , such that where , , , and , .We assume that .We assume that .We assume that .
The paper is organized as follows. In the next section, we give some definitions and lemmas to prove the main results of this paper. In Section 3, we established some criteria to guarantee the existence of at least one positive periodic solution of system (1) by using a fixed point theorem of strict-set-contraction. As applications in Section 4, we study some particular cases of system (1) which have been investigated extensively in the references mentioned previously.
2. Preliminaries
In order to obtain the existence of a periodic solution of system (1), we first introduce some definitions and lemmas.
Definition 1 (see [17]). 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 2 (see [18]). Let be a real Banach space and a closed, nonempty subset of . is a cone provided that(i) for all and all ;(ii) imply .
Definition 3 (see [18]). Let be a bounded subset in . Define : there is a finite number of subsets such that and , where denotes the diameter of the set ; obviously, . So is called the Kuratowski measure of noncompactness of .
Definition 4 (see [18]). Let , be two Banach spaces and ; a continuous and bounded map is called -set contractive if for any bounded set we have is called strict-set-contractive if it is -set contractive for some .
Definition 5 (see [19]). The set is said to be quasiequicontinuous in , if for any , there exists such that if , , , and , then .
Lemma 6 (see [19]). The set is relatively compact if and only if(1) is bounded, that is, , for each , and some ;(2) is quasiequicontinuous in .
Lemma 7. is an -periodic solution of (1) is equivalent to is an -periodic solution of the following equation: where
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 (11).
If is a periodic solution of (11), for any , from (11) we have
For any , , we have from (11) that
Hence is a positive -periodic solution of (1). Thus we complete the proof of Lemma 7.
Lemma 8 (see [18–20]). Let be a cone of the real Banach space and with . Assume that is strict-set-contractive such that one of the following two conditions is satisfied: (a), , and , , ; (b), , and , , .Then has at least one fixed point in .
In order to apply Lemma 8 to system (1), we set Define with the norm defined by and with the norm defined by . Then and are both Banach spaces. Define the cone in by Let the map be defined by where , , and is defined by (12). It is obvious to see that , , , and In what follows, we will give some lemmas concerning and defined by (22) and (23), respectively.
Lemma 9. Assume that hold.(i)If , then is well defined.(ii)If holds and , then is well defined.
Proof. For any , it is clear that . From (23), for , we have
That is, , . So . In view of , for , , we have
Therefore, for , we find
Furthermore, for , we have
Now, we show that , .
On the other hand, from (23), we obtain
It follows from (29) and (30) that if , then
On the other hand, from (30) and , if , then
It follows from (31) and (32) that . So . By (29) we have . Hence, . This completes the proof of (i).
(ii) In view of the proof of (i), we only need to prove that implies . From (23), (26), , and , we have
The proof of (ii) is complete. Thus we complete the proof of Lemma 9.
Lemma 10. Assume that hold and .(i)If , then is strict-set-contractive.(ii)If holds and , then is strict-set-contractive, where .
Proof. We only need to prove (i), since the proof of (ii) is similar. It is easy to see that is continuous and bounded. Now we prove that a for any bounded set . Let ; then, for any positive number , there is a finite family of subsets satisfying with . Therefore,
As and are precompact in , it follows that there is a finite family of subsets of such that and
In addition, for any and , we have
Hence,
Applying the Arzela-Ascoli theorem, we know that is precompact in . Then, there is a finite family of subsets of such that and
From (34)–(39) and , for any , we have
where
From (40) we obtain
As is arbitrary small, it follows that
Therefore, is strict-set-contractive. The proof of Lemma 10 is complete.
Lemma 11. Assume that hold.(i)If , then is a positive -periodic solution of model (1), where is a nonzero fixed point of the operator on .(ii)If holds and , then is a positive -periodic solution of model (1), where is a nonzero fixed point of the operator on .
3. Main Results
In this section, we will study the existence of positive -periodic solutions of system (1).
Theorem 12. Assume that , and hold.(i)If , then system (1) has at least one positive -periodic solution.(ii)If holds and , then system (1) has at least one positive -periodic solution.
Proof. We only need to prove (i), since the proof of (ii) is similar. Let
Then it is easy to see that . From Lemmas 9 and 10, we know that is strict-set-contractive on . By Lemma 11, we see that if there exists such that , then is one positive -periodic solution of system (1). Now, we will prove that condition (b) of Lemma 8 holds.
First, we prove that , , . Otherwise, there exist , , such that . So and , which implies that
Moreover, for , we have
In view of (45) and (46), we obtain
which is a contradiction.
Finally, we prove that , , . For this case, for the sake of contradiction, suppose that there exist , such that . Furthermore, for any , we have
In addition, for any , we find
which is a contradiction. Therefore, condition (b) of Lemma 8 holds. By Lemma 8, we see that has at least one nonzero fixed point in . Thus, the system (11) has at least one positive -periodic solution. Therefore, it follows from Lemma 7 that system (1) has a positive -periodic solution. The proof of Theorem 12 is complete.
4. Applications
In this section, we apply the result obtained in the previous section to some periodic population models with impulses which are mentioned in the first section.
First, we consider a general neutral delay model of single-species population growth with impulse: and we investigate a complex neutral equation with several state-dependent delays and impulse: For convenience, we list several assumptions: , , , and are the same as , , , and , respectively; are -periodic functions and
Theorem 13. Assume hold.(i)If , then systems (50) and (51) have at least one positive -periodic solution.(ii)If holds and , then systems (50) and (51) have at least one positive -periodic solution.
Proof. The proof is similar to that of Theorem 12; we omit the details here.
Second, we consider a general neutral delay model of single-species population growth with impulse:
and we investigate a periodic Lotka-Volterra equation with state-dependent delays and impulse:
For convenience, we list several assumptions: , , , and are the same as , , , and , respectively; are -periodic functions and
Then we can obtain the following theorem.
Theorem 14. Assume hold.(i)If , then systems (53) and (54) have at least one positive -periodic solution.(ii)If holds and , then systems (53) and (54) have at least one positive -periodic solution.
Proof. The proof is similar to that of Theorem 12; we omit the details here.
Remark 15. We apply the main result obtained in the previous section to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of a positive periodic solution which plays the role played by the equilibrium of the autonomous models and means that the species is in an equilibrium state. From Theorems 13 and 14, we see that, under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems. Therefore, our result generalizes and improves the corresponding results in [12–17].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is supported by NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (nos. 2012M512162 and 2013T60934), NSF of Hunan province (nos. 11JJ900, 12JJ9001, and 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, 12C0361, and 13C084), and the construct program of the key discipline in Hunan province.