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Existence and nmultiplicity of positive periodic solutions for impulsive functional differential equations with two parameters
 Qiong Meng^{1}Email author and
 Jurang Yan^{1}
 Received: 29 July 2015
 Accepted: 10 November 2015
 Published: 18 November 2015
Abstract
In this paper, we employ the wellknown Krasnoselskii fixed point theorem to study the existence and nmultiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. The form including an impulsive term of the equations in this paper is rather general and incorporates as special cases various problems which have been studied extensively in the literature. Easily verifiable sufficient criteria are obtained for the existence and nmultiplicity of positive periodic solutions of the impulsive functional differential equations.
Keywords
 functional differential equation
 impulse effect
 Krasnoselskii fixed point theorem
 positive periodic solution
MSC
 34K20
1 Introduction
 (H_{1}):

\(\lambda\geq0\) and \(\mu\geq0\) with \(\lambda+\mu>0\) are parameters.
 (H_{2}):

\(f:{\mathbf{R}}\times{\mathbf{R}}_{+}\rightarrow{\mathbf{R}}_{+}\) satisfies the Caratheodory condition, that is, \(f(t,y)\) is locally Lebesgue measurable ωperiodic (\(\omega>0\)) function in t for each fixed y and continuous in y for each fixed t, τ, and \(\sigma:{\mathbf{R}}\rightarrow{\mathbf{R}}\) are locally bounded Lebesgue measurable ωperiodic functions.
 (H_{3}):

There exist ωperiodic functions \(a_{1}\) and \(a_{2}: {\mathbf{R}}\rightarrow{\mathbf{R}}\) which are locally bounded Lebesgue measurable so that \(a_{1}(t)y\leq h(t,y)\leq a_{2}(t)y\) for all \(y>0\) and \(\lim_{y\rightarrow0^{+}}\frac{h(t,y)}{y}\) exists, \(\int_{0}^{\omega }a_{1}(t)\, dt>0\).
 (H_{4}):

There exists a positive integer ρ such that \(t_{k+\rho }=t_{k}+\omega\).
 (H_{5}):

\(I_{k}:{\mathbf{R}}\times{\mathbf{R}}_{+}\rightarrow{\mathbf{R}}_{+}\) satisfies the Caratheodory condition and are ωperiodic functions in t. \(\{t_{k}\}\), \(k \in{\mathbf{Z}}\), is an increasing sequence of real numbers with \(\lim_{k\rightarrow\pm\infty} t_{k} =\pm\infty\). Moreover, \(I_{k+\rho }(t_{k+\rho}, y)=I_{k}(t_{k}, y)\) for all k.
The periodic system (\({E}_{i}\)) include many periodic mathematical ecological models with or without impulse effects. This type of equations has been proposed as models for a variety of physiological precesses and conditions including production of blood cells, respiration, and cardiac arrhythmias; see [1–5]. The study of positive periodic solutions for impulsive functional differential equations has attracted considerable attention, and research results emerge continuously; see [6–16].
The purpose of this paper is to obtain some weaker conditions for the global existence of positive periodic solutions of (\({E}_{i}\)) and the number \(n\geq1\) of periodic solutions. Following the technique in [6, 17] and by using the wellknown Krasnoselskii fixed point theorem, we show that, for \(\lambda+\mu>0\), the number \(n\geq1\) of positive ωperiodic solutions of (\({E}_{i}\)) can be determined by the behaviors of the quotient of \(\frac{f(t, y)}{y}\) at any point \(y\in(0, \infty)\) and \(y\rightarrow0^{+}\), \(y\rightarrow\infty\), \(t\in{\mathbf{R}}\). In particular, for \(\lambda=0\) and \(\mu>0\), the global existence of positive ωperiodic solutions of (\({E}_{i}\)) is caused completely by impulse effects. These results are new and they generalize and improve those in [6–8, 17]. For \(n=1\), the results of this paper also improve those in [7–10].
The paper is organized as follows. In Section 2, we give some lemmas to prove the main results of this paper and several preliminaries are given. In Section 3, the existence theorems for the numbers 1, 2 and \(n>2\) of positive periodic solutions of (\({E}_{i}\)) are proved by using the wellknown fixed point theorem due to Krasnoselskii addressing the quotient of \(\frac{f(t, y)}{y}\) and \(\frac{I_{k}(t, y)}{y}\) at \(y>0\), \(t\in{\mathbf{R}}\). An example is also given. In Section 4, we employ the results obtained in Section 3 to prove that the number 1 or 2 of positive periodic solutions of (\({E}_{i}\)) can be determined by \(\frac{f(t, y)}{y}\) and \(\frac{I_{k}(t, y)}{y}\) when \(y\rightarrow0_{+}\) and \(y\rightarrow\infty\), \(t\in{\mathbf{R}}\).
2 Preliminaries
 (i)
\(y(t)\) is absolutely continuous on each \((t_{k}, t_{k+1})\);
 (ii)
for each \(k\in{\mathbf{Z}}\), \(y(t_{k}^{+})\) and \(y(t_{k}^{})\) exist and \(y(t_{k}^{})=y(t_{k})\);
 (iii)
\(y(t)\) satisfies the differential equation in (\({E}_{i}\)) for almost everywhere on R;
 (iv)
\(y(t_{k})\) satisfies the impulse condition in (\({E}_{i}\)).
In the proofs of the main theorems of this paper we will use the following lemmas and theorem. The proofs of the lemmas are similar to those of the lemmas in [6]. We omit them.
Lemma 2.1
Assume that (H_{1})(H_{5}) hold. Then \(T:P\rightarrow P\) is well defined and is completely continuous.
Lemma 2.2
Assume that (H_{1})(H_{5}) hold. The existence of positive ωperiodic solutions of (\({E}_{i}\)) is equivalent to that of nonzero fixed points of T in P.
Theorem K
 (i)
\(\Ty\\geq\y\\), \(y\in P\cap\partial\Omega_{1}\) and \(\Ty\\leq\y\\), \(y\in P\cap\partial\Omega_{2}\)
 (ii)
\(\Ty\\leq\y\\), \(y\in P\cap\partial\Omega_{1}\) and \(\Ty\\geq\y\\), \(y\in P\cap\partial\Omega_{2}\),
3 Existence of positive periodic solutions
We are now in a position to state and prove our results of the existence of positive ωperiodic solution for (\({E}_{i}\)).
Theorem 3.1
Proof
By Theorem K, \(T_{i}\) has a positive fixed point \(P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\). It follows from Lemma 2.2 that (\({E}_{i}\)) has a positive ωperiodic solution with \(r_{1}\leq\y\\leq r_{2}\). The proof of Theorem 3.1 is complete. □
When \(\lambda=0\) or \(\mu=0\), from Theorem 3.1, we obtain immediately the following result.
Corollary 3.1
 (i)if \(\lambda=0\) andthen (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\);$$\frac{1}{A_{i}\omega\beta\hat{I}_{r_{2}}^{l}}\leq\mu\leq\frac{1}{B_{i}\omega\hat {I}_{r_{1}}^{u}}, $$
 (ii)if \(\mu=0\) andthen (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).$$\frac{1}{A_{i}\omega\beta\bar{f}_{r_{2}}^{l}}\leq\lambda\leq\frac{1}{B_{i}\omega \bar{f}_{r_{1}}^{u}}, $$
Theorem 3.2
Corollary 3.2
 (i)if \(\lambda=0\) andthen (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\);$$\frac{1}{A_{i}\omega\beta\hat{I}_{r_{1}}^{l}}\leq\mu\leq\frac{1}{B_{i}\omega\hat {I}_{r_{2}}^{u}}, $$
 (ii)if \(\mu=0\) andthen (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).$$\frac{1}{A_{i}\omega\beta\bar{f}_{r_{1}}^{l}}\leq\lambda\leq\frac{1}{B_{i}\omega \bar{f}_{r_{2}}^{u}}, $$
The proof of Theorem 3.2 will be omitted since it is similar to that of Theorem 3.1.
From Theorems 3.1 and 3.2, by using the same method, we can prove the following result.
Theorem 3.3
Remark 3.1
A simple example that satisfies conditions (3.3) or (3.4) is that functions f and \(I_{k}\) are ωperiodic functions in t and Ωperiodic (\(\Omega>0\)) functions in y. Moreover, \(r_{m+1}=r_{m}+\Omega\), \(m=1,2,\ldots,n\).
Example
Now, we show that \((3.5)'_{i}\) has n positive ωperiodic functions.
4 Applications of main results
Theorem 4.1
Proof
Similarly, by Theorem 3.2, we can prove that if (4.2) holds, then (\({E}_{i}\)) has a positive ωperiodic solution. The proof of Theorem 4.1 is complete. □
Theorem 4.2
 (i)
If \(\bar{f}_{0}=\hat{I}_{0}=0\) or \(\bar{f}_{\infty}=\hat{I}_{\infty}=0\), then (\({E}_{i}\)) has a positive ωperiodic solution.
 (ii)
If \(\bar{f}_{0}=\hat{I}_{0}=\bar{f}_{\infty}=\hat {I}_{\infty}=0\), then (\({E}_{i}\)) has two positive ωperiodic solutions.
Proof
(ii) If \(\bar{f}_{0}=\hat{I}_{0}=\bar{f}_{\infty}=\hat{I}_{\infty}=0\), then we can choose \(r_{1}\) and \(r_{2}\) with \(r_{1}< r< r_{2}\) so that (4.4) and (4.5) hold. Thus by Theorem 3.3 that (\({E}_{i}\)) has respectively two positive ωperiodic solutions. The proof of Theorem 4.2 is complete. □
Similarly by using Theorems 3.2 and 3.3 we can obtain the following results.
Theorem 4.3
 (i)
If \(\bar{f}_{0}=\hat{I}_{0}=0\) or \(\bar{f}_{\infty}=\hat{I}_{\infty}=0\), then (\({E}_{i}\)) has a positive ωperiodic solution.
 (ii)
If \(\bar{f}_{0}=\hat{I}_{0}=\bar{f}_{\infty}=\hat {I}_{\infty}=0\), then (\({E}_{i}\)) has a positive ωperiodic solution.
5 Discussion
The method of this paper on impulsive differential equations is not only restricted to scalar equations, but also it can be used for systems of impulsive functional equations and impulsive Nspecies competitive systems and impulsive neutral functional differential equations; see for example [11, 20].
Declarations
Acknowledgements
The authors would like to express their sincere thanks to the referees for their helpful comments. This work was supported by the National Natural Science Foundation of China (No. 61473180).
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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