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Existence and nmultiplicity of positive periodic solutions for impulsive functional differential equations with two parameters
Boundary Value Problems volume 2015, Article number: 212 (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.
1 Introduction
In this paper, we consider the following impulsive functional differential equations with two parameters:
Throughout this paper, we use \(i=1,2\), \(k\in{\mathbf{Z}}\), where Z denotes the set of all integers, \({\mathbf{R}}=(\infty, \infty)\) and \({\mathbf{R}}_{+}=[0, \infty)\).
For system (\({E}_{i}\)) we introduce the hypotheses:
 (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
Throughout this paper, we will use the following notation:
and
Let E be the Banach space defined by
with norm \(\y\=\sup_{0\leq t\leq\omega}y(t)\). Define P to be a cone in E by
For a positive constant r, we also define \(\Omega_{r}\) by
Finally, we define two operators \(T_{i}:P\rightarrow P\), \(i=1,2\), as
where
Note that, from (H_{3}) and (2.2), we have
A function \(y:{\mathbf{R}}\rightarrow{\mathbf{R}}\) is said to be solution of (\({E}_{i}\)) if the following conditions are satisfied:

(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
Let E be a Banach space and P be a cone in E. Assume that \(\Omega_{1}\) and \(\Omega_{2}\) are bounded open subsets of E with \(0\in\Omega_{1}\), \(\bar{\Omega }_{1}\subset\Omega_{2}\), and let T be a completely continuous operator such that either

(i)
\(\Ty\\geq\y\\), \(y\in P\cap\partial\Omega_{1}\) and \(\Ty\\leq\y\\), \(y\in P\cap\partial\Omega_{2}\)
or

(ii)
\(\Ty\\leq\y\\), \(y\in P\cap\partial\Omega_{1}\) and \(\Ty\\geq\y\\), \(y\in P\cap\partial\Omega_{2}\),
then T has a fixed point in \(P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\).
3 Existence of positive periodic solutions
In the sequel, we will use the following notations. For an ωperiodic integrable function \(f(t):{\mathbf{R}}\rightarrow{\mathbf{R}}_{+}\), let
For a positive constant r and \(y\in P\),
Clearly from (H_{2}) and (H_{5}), \(f_{r}^{u}\), \(f_{r}^{l}\), \(I_{r}^{u}\), and \(I_{r}^{l}:{\mathbf{R}}\rightarrow{\mathbf{R}}_{+}\) are positive bounded Lebesgue measurable ωperiodic functions.
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
Assume that (H_{1})(H_{5}) hold and there exist positive constants \(r_{1}\) and \(r_{2}\) with \(r_{1}< r_{2}\) such that
and
then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).
Proof
Consider the Banach space E defined in Section 2 and P in E. Define two open sets \(\Omega_{r_{1}}\) and \(\Omega_{r_{2}}\) with \(r_{1}< r_{2}\). If \(y\in P\cap\partial\Omega_{r_{1}}\), then \(\beta\y\\leq y(t)\leq\y\=r_{1}\). From (2.1)_{ i } and (2.2) we have
Hence from (3.1), we obtain \(\T_{i}y\\leq\y\\).
On the other hand, if \(y\in P\cap\partial\Omega_{r_{2}}\), then \(\beta\y\\leq y(t)\leq\y\=r_{2}\). From (2.1)_{ i } and (2.2) we have
In view of (3.2), we obtain \(\T_{i}y\\geq(T_{i}y)(t)\geq\y\\).
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
Assume that (H_{1})(H_{5}) hold and there exist positive constants \(r_{1}\) and \(r_{2}\) with \(r_{1}< r_{2}\);

(i)
if \(\lambda=0\) and
$$\frac{1}{A_{i}\omega\beta\hat{I}_{r_{2}}^{l}}\leq\mu\leq\frac{1}{B_{i}\omega\hat {I}_{r_{1}}^{u}}, $$then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\);

(ii)
if \(\mu=0\) and
$$\frac{1}{A_{i}\omega\beta\bar{f}_{r_{2}}^{l}}\leq\lambda\leq\frac{1}{B_{i}\omega \bar{f}_{r_{1}}^{u}}, $$then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).
Theorem 3.2
Assume that (H_{1})(H_{5}) hold and there exist positive constants \(r_{1}\) and \(r_{2}\) with \(r_{1}< r_{2}\) such that
and
then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).
Corollary 3.2
Assume that (H_{1})(H_{5}) hold and there exist positive constants \(r_{1}\) and \(r_{2}\) with \(r_{1}< r_{2}\);

(i)
if \(\lambda=0\) and
$$\frac{1}{A_{i}\omega\beta\hat{I}_{r_{1}}^{l}}\leq\mu\leq\frac{1}{B_{i}\omega\hat {I}_{r_{2}}^{u}}, $$then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\);

(ii)
if \(\mu=0\) and
$$\frac{1}{A_{i}\omega\beta\bar{f}_{r_{1}}^{l}}\leq\lambda\leq\frac{1}{B_{i}\omega \bar{f}_{r_{2}}^{u}}, $$then (\({E}_{i}\)) has a positive ωperiodic solution \(y(t)\) with \(r_{1}\leq \y\\leq r_{2}\).
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
Assume that (H_{1})(H_{5}) hold and there exist \(n+1\) positive constants \(r_{m}\), \(m=1,2,\ldots, n+1\), with \(r_{1}< r_{2}<\cdots<r_{n+1}\) such that one of the following conditions is satisfied:
and
then (\({E}_{i}\)) has n positive ωperiodic solutions \(y_{1}, y_{2}, \ldots,y_{n}\) with \(\y_{1}\\leq\y_{2}\\leq \cdots\leq\y_{n}\\).
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
Consider the following impulse differential equation satisfying (H_{1})(H_{5}):
where
where \(m=0,2,4,\ldots,n\), n is an even, and ξ is a constant. \(p, q\in L ({\mathbf{R}}, {\mathbf{R}}_{+})\) are ωperiodic functions.
Now, we show that \((3.5)'_{i}\) has n positive ωperiodic functions.
Let
Thus, for \(r>0\) and \(y\in P\),
where \(\delta=\inf_{\y\=\frac{\pi}{2}, y\in P}\{\sin y\}\). Clearly \(\delta>0\),
First, we choose \(\lambda+\mu>0\) enough small to satisfy
then we choose ξ sufficiently large such that
Similarly, we can obtain
and
Therefore condition (3.4) of Theorem 3.3 is satisfied. By Theorem 3.3, \((3.5)'_{i}\) has n positive ωperiodic solutions \(y_{1}, y_{2}, \ldots,y_{n}\) with \(r_{1}\leq\y_{1}\\leq r_{2}\leq\y_{2}\\leq r_{3}\leq\cdots\leq r_{n}\leq\y_{n}\\leq r_{n+1}\). The proof is completed.
Remark 3.2
Theorems 3.1 and 3.2 generalize and improve, respectively, Theorems 3.1 and 3.2 in [17].
4 Applications of main results
Let \(r\rightarrow0\) or \(r\rightarrow\infty\), \(\bar {f}_{0}^{u}\), \(\bar{f}_{0}^{l}\), \(\hat{I}_{0}^{u}\), \(\hat{I}_{0}^{l}\), \(\bar{f}_{\infty}^{u}\), \(\bar{f}_{\infty}^{l}\), \(\hat{I}_{\infty}^{u}\), and \(\hat{I}_{\infty}^{l}\) denote respectively the corresponding upper and lower limits of \(\bar{f}_{r}^{u}\), \(\bar{f}_{r}^{l}\), \(\hat{I}_{r}^{u}\), and \(\hat{f}_{r}^{l}\). In particular, let
Theorem 4.1
Assume that (H_{1})(H_{5}) hold and one of the following conditions is satisfied:
and
then (\({E}_{i}\)) has a positive ωperiodic solution.
Proof
From (4.1) we can choose positive constants \(r_{1}\) and \(r_{2}\) with \(r_{1}< r_{2}\) such that
and
By Theorem 3.1, it follows that (\({E}_{i}\)) has a positive ωperiodic solution.
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
Assume that (H_{1})(H_{5}) hold and there exists \(r>0\) such that

(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
From (2.1), for any \(y\in P\cap\partial\Omega_{r}\) and λ, μ satisfying (4.3) we have
This yields
(i) If \(\bar{f}_{0}=\hat{I}_{0}=0\), then we choose \(r_{1}\) with \(0< r_{1}< r\) so that for λ and μ satisfying (4.3)
Thus from (2.1) we obtain
which implies that, for \(y\in P\cap\partial\Omega_{r_{1}}\),
It follows from Theorem 3.2 that (\({E}_{i}\)) has a positive ωperiodic solution.
If \(\bar{f}_{\infty}=\hat{I}_{\infty}=0\), then we choose \(r_{2}>r\) so that for λ and μ satisfying (4.3)
Hence from (2.1) we have
which implies that, for \(y\in P\cap\partial\Omega_{r_{2}}\),
It follows from Theorem 3.2 that (\({E}_{i}\)) has a positive ωperiodic solution.
(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
Assume that (H_{1})(H_{5}) hold and there exists \(r>0\) such that

(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.
Remark 4.1
Theorems 4.1 and 4.2 generalize and improve, respectively, Corollaries 4.2 and 4.3 in [17], Theorems 3.13.3 in [6], and Theorem 1.3 in [17].
5 Discussion
In Sections 3 and 4 of this paper, by using the behaviors of the quotient of \(\frac{f(t,y)}{y}\) at any point \(y\in[0, \infty)+\{\infty\}\) we have proved the existence and nmultiplicity of positive ωperiodic solutions for impulsive functional differential equation (\(E_{i}\)) which are general enough to incorporate some periodic mathematical and ecological models. The method and technique are based on the application of the famous Krasnoselskii fixed point theorem on the cone of Banach space. In particular, when \(\lambda=0\) in (\(E_{i}\)), Corollaries 3.1 and 3.2 in Section 3, the existence of positive ωperiodic solutions is caused by impulsive effects. This is different from the corresponding continuous system. In fact, when \(\lambda=0\), (\(E_{i}\)) reduce to
If the condition of Theorem 3.3 holds, then (∗) has n positive ωperiodic solutions by Theorem 3.3. But the equation \(y'(t)=(1)^{i}h(t,y)\) has no periodic solution.
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].
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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).
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Meng, Q., Yan, J. Existence and nmultiplicity of positive periodic solutions for impulsive functional differential equations with two parameters. Bound Value Probl 2015, 212 (2015). https://doi.org/10.1186/s1366101504782
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DOI: https://doi.org/10.1186/s1366101504782