Positive periodic solution for Nicholsontype delay systems with impulsive effects
 Ruojun Zhang^{1}Email author,
 Yanping Huang^{1} and
 Tengda Wei^{1}
https://doi.org/10.1186/s1366201507052
© Zhang et al. 2015
Received: 4 June 2015
Accepted: 19 November 2015
Published: 4 December 2015
Abstract
In this paper, a class of Nicholsontype delay systems with impulsive effects is considered. First, an equivalence relation between the solution (or positive periodic solution) of a Nicholsontype delay system with impulsive effects and that of the corresponding Nicholsontype delay system without impulsive effects is established. Then, by applying the cone fixed point theorem, some criteria are established for the existence and uniqueness of positive periodic solutions of the given systems. Finally, an example and its simulation are provided to illustrate the main results. Our results extend and improve greatly some earlier works reported in the literature.
Keywords
Nicholsontype systems positive periodic solutions delay impulsive effect cone fixed point theorem1 Introduction
However, species living in certain medium might undergo abrupt change of state at certain moments, and this occurs due to some seasonal effects such as weather change, food supply, and mating habits. That is to say, besides delays, impulsive effects likewise exist widely in many evolution processes. In the last two decades, the theory of impulsive differential equations has been extensively investigated due to its widespread applications [13–16].
Therefore, it is more realistic to investigate Nicholsontype delay systems with impulsive effects. However, to the best of our knowledge, few authors [17] have considered the conditions for existence and uniqueness of positive periodic solution for system (1.2) with impulsive effects. Thus, techniques and methods on the existence and uniqueness of a positive periodic solution for system (1.2) with impulsive effects should be developed and explored.
 (H_{1}):

\(0< t_{0}< t_{1}< t_{2}< \cdots\), \(t_{i}\), \(i=1,2,\ldots\) are fixed impulsive points with \(\lim_{k\rightarrow\infty}t_{k}=\infty\);
 (H_{2}):

\(\{b_{k}\}\) is a real sequence, and \(b_{k}>1\), \(k=1,2,\ldots\) ;
 (H_{3}):

\(\alpha_{i}(t)\), \(\beta_{i}(t)\), \(c_{ij}(t)\), \(\gamma_{ij}(t)\), \(\tau_{ij}(t)\), and \(\prod_{0< t_{k}< t}(1+b_{k})\) are periodic functions with common period \(\omega>0\), \(i=1,2\), \(j=1,2,\ldots,m\), \(k=1,2,\ldots\) .
Here and in the sequel, we assume that a product equals unit if the number of factors is equal to zero.
Let \(\tau=\max\{\tau_{ij}^{+}\}\), \(\tau_{ij}^{+}=\max_{0\leq t\leq\omega}\tau_{ij}(t)\), \(i=1,2\), \(j=1,2,\ldots,m\). If \(y_{i}(t)\) is defined on \([t_{0}\tau, \sigma]\) with \(t_{0}, \sigma\in R\), then we define \(y_{t}\in C([\tau, 0], R)\) as \(y_{t}=(y_{t}^{1}, y_{t}^{2})\) where \(y_{t}^{i}(\theta)=y_{i}(t+\theta)\) for \(\theta\in[\tau,0]\) and \(i=1,2\).
The remaining parts of this paper is organized as follows. In Section 2, we introduce some notation, definitions, and lemmas. In Section 3, we first establish the equivalence between the solution (or positive periodic solution) of a Nicholsontype delay system with impulses and that of the corresponding Nicholsontype delay system without impulses. Then, we give some criteria ensuring the existence and uniqueness of positive periodic solutions of Nicholsontype delay systems with and without impulses. In Section 4, an example and its simulation are provided to illustrate our results obtained in the previous sections. Finally, some conclusions are drawn in Section 5.
2 Preliminaries
Definition 2.1
 (i)
\(y(t)\) is absolutely continuous on the intervals \((t_{0},t_{1}]\) and \((t_{k},t_{k+1}]\), \(k=1,2,\ldots\) ;
 (ii)
for all \(t_{k}\), \(k=1,2,\ldots\) , \(y(t_{k}^{+})\) and \(y(t_{k}^{})\) exist, and \(y(t_{k}^{})=y(t_{k})\);
 (iii)
\(y(t)\) satisfies the differential equation of (1.3) in \([t_{0},\infty) \backslash\{t_{k}\}\) and the impulsive conditions for all \(t=t_{k}\), \(k=1,2,\ldots\) ;
 (iv)
\(y_{i_{t_{0}}}(s)=\varphi_{i}(s)\), \(s\in[\tau,0]\).
By a solution \(x(t)\) of Equation (2.1) with initial condition (2.2) we mean an absolutely continuous function \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) defined on \([t_{0},\infty)\) satisfying Equation (2.1) for \(t\geq t_{0}\) and initial condition (2.2) on \([\tau,0]\).
Similarly to the method of [18], we have the following:
Lemma 2.1
Assume that (H_{1})(H_{3}) hold. Then
(i) if \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2), then \(y(t)=(\prod_{0< t_{k}< t}(1+b_{k})x_{1}(t),\prod_{0< t_{k}< t}(1+b_{k})x_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4) on \([\tau,\infty)\);
(ii) if \(y(t)=(y_{1}(t),y_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (1.3) with initial condition (1.4), then \(x(t)=(\prod_{0< t_{k}< t}(1+b_{k})^{1}y_{1}(t), \prod_{0< t_{k}< t}(1+b_{k})^{1}y_{2}(t))^{T}\) is a solution (or positive ωperiodic solution) of Equation (2.1) with initial condition (2.2) on \([\tau,\infty)\).
Proof
From the above analysis we know that the conclusion of Lemma 2.1 is true. This completes the proof. □
Lemma 2.2
Proof
Clearly, by Lemma 2.1, we only need to prove that every solution \(x(t)\) of Equation (2.1) with (2.2) is positive and bounded on \([t_{0},\infty)\). In order to show that, we only need to see Lemma 2.3 in [11].
Lemma 2.3
(Cone fixed point theorem [19])
 (i)
\(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{1}\) and \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{2}\), or
 (ii)
\(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{2}\) and \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{1}\),
3 Existence and uniqueness of positive periodic solution
Lemma 3.1
Assume that (H_{1})(H_{4}) hold. Then \(T:P\rightarrow P\) is completely continuous.
Proof
Theorem 3.1
Assume that (H_{1})(H_{4}) hold. Then Equation (1.3) with (1.4) has at least one positive ωperiodic solution.
Proof
If \(x\in P\cap\partial\Omega_{2}\), which implies that \(\x\=B\), then from (3.7) we have \(\Tx\\leq B\), and hence \(\Tx\\leq\x\\) for \(x\in P\cap\partial\Omega_{2}\).
If \(x\in P\cap\partial\Omega_{1}\), which implies that \(\x\=A\), then from (3.9) we have \(\Tx\\geq A\), and hence \(\Tx\\geq\x\\) for \(x\in P\cap\partial\Omega_{1}\).
By Lemma 2.3 the operator T has at least one fixed point in \(P\cap(\overline{\Omega_{2}} \setminus\Omega_{1})\), i.e., Equation (2.1) with (2.2) has at least one ωperiodic solution. Since \(\theta \overline{\in} P\cap(\overline{\Omega_{2}} \setminus\Omega_{1})\), Equation (2.1) with (2.2) has at least one positive ωperiodic solution. Therefore, Equation (1.3) with (1.4) has at least one positive ωperiodic solution by Lemma 2.1. This completes the proof of Theorem 3.1. □
Theorem 3.2
Proof
By Theorem 3.1 we know that Equation (2.1) with (2.2) has at least one positive ωperiodic solution. Thus, in order to prove Theorem 3.2, we only need to prove the uniqueness of a positive ωperiodic solution for Equation (2.1) with (2.2).
The following proof is similar to that of Theorem 3.2 in [11].
Remark 3.1
In Theorems 3.1 and 3.2, the conditions that ensure the existence and uniqueness of a positive ωperiodic solution for Nicholsontype delay systems with and without impulses are simple and easily to test, which is less conservative than the conditions required in some previous works [11, 12]. Moreover, the main results in this paper are totally different from that of [17].
4 An example
Example 4.1
Since \(\alpha_{1}(t)=9+\sin^{2}\pi t\), \(\alpha_{2}(t)=9+\cos^{2}\pi t\), \(\beta _{1}(t)=5+\cos^{2}\pi t\), \(\beta_{2}(t)=5+\sin^{2}\pi t\), we have \(\alpha_{1}^{}=\alpha_{2}^{}=9\), \(\beta_{1}^{+}=\beta_{2}^{+}=6\), and thus \(\frac{ \beta_{1}^{+}\beta_{2}^{+}}{\alpha_{1}^{}\alpha_{2}^{}}=\frac {4}{9}<1\).
Remark 4.1
System (4.1) is a simple form of impulsive Nicholsontype system with delays. Since \(q_{11}^{}=q_{21}^{}=\frac{7}{6}>1\), \(q_{12}^{}= q_{22}^{}=\frac{5}{4}>1\), it is clear that the condition of Theorem 3.1 in [11] and Theorem 2.1 in [12] are not satisfied. Therefore, all the results obtained in [11, 12] and the references therein cannot be applicable to system (4.1). This implies that the results of this paper are essentially new.
5 Conclusion
In this paper, a class of Nicholsontype delay systems with impulsive effects are investigated. First, an equivalence relation between the solution (or positive periodic solution) of a Nicholsontype delay system with impulses and that of the corresponding Nicholsontype delay system without impulses is established. Then, by applying the cone fixed point theorem, some criteria are established for the existence and uniqueness of a positive periodic solution of the given system. The fixed point theorem in cones is very popular in investigation of positive periodic solutions to impulsive functional differential equations [20, 21]. Our results imply that under the appropriate linear periodic impulsive perturbations, the Nicholsontype delay systems with impulses preserve the original periodic property of the Nicholsontype delay systems without impulses. Finally, an example and its simulation are provided to illustrate the main results. It is worth noting that there are only very few results [17] on Nicholsontype delay systems with impulses, and our results extend and improve greatly some earlier works reported in the literature. Furthermore, our results are important in applications of periodic oscillatory Nicholsontype delay systems with impulsive control.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11171374) and the Scientific Research Fund of Shandong Provincial of P.R. China (Grant No. ZR2011AZ001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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