In the sequel, we will use the following notations. For an ω-periodic integrable function \(f(t):{\mathbf{R}}\rightarrow{\mathbf{R}}_{+}\), let
$$\bar{f}=\frac{1}{\omega}\int_{0}^{\omega}f(t)\,dt, \qquad \hat{f}=\frac {1}{\omega}\sum_{0\leq t_{k}< \omega}f(t_{k}). $$
For a positive constant r and \(y\in P\),
$$\begin{aligned}& f_{r}^{u}(t)=\sup_{\|y\|=r, y\in P} \frac{f(t,y)}{y}, \qquad f_{r}^{l}(t)=\inf_{\| y\|=r, y\in P} \frac{f(t,y)}{y}, \\& I_{r}^{u}(t)=\sup_{\substack{{\|y\|=r, y\in P}\\{k=1,2,\ldots,\rho}}} \frac {I_{k}(t,y)}{y}, \qquad I_{r}^{l}(t)=\inf_{\substack{{\|y\|=r, y\in P}\\{k=1,2,\ldots,\rho}}} \frac{I_{k}(t,y)}{y}. \end{aligned}$$
Clearly from (H2) and (H5), \(f_{r}^{u}\), \(f_{r}^{l}\), \(I_{r}^{u}\), and \(I_{r}^{l}:{\mathbf{R}}\rightarrow{\mathbf{R}}_{+}\) are positive bounded Lebesgue measurable ω-periodic functions.
We are now in a position to state and prove our results of the existence of positive ω-periodic solution for (\({E}_{i}\)).
Theorem 3.1
Assume that (H1)-(H5) hold and there exist positive constants
\(r_{1}\)
and
\(r_{2}\)
with
\(r_{1}< r_{2}\)
such that
$$ B_{i}\omega\bigl(\lambda\bar{f}_{r_{1}}^{u}+\mu \hat{I}_{r_{1}}^{u}\bigr)\leq 1 $$
(3.1)
and
$$ A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{2}}^{l}+\mu \hat{I}_{r_{2}}^{l}\bigr)\geq 1, $$
(3.2)
then (\({E}_{i}\)) has a positive
ω-periodic 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
$$\begin{aligned} (T_{i}y) (t) \leq& B_{i}\biggl[\lambda\int _{t}^{t+\omega}\frac{f(s,y(s-\tau(s)))}{y(s-\tau (s))}y\bigl(s-\tau(s)\bigr)\, ds \\ &{} +\mu\sum_{t\leq t_{k}< t+\omega}\frac{I_{k}(t_{k},y(t_{k}-\tau(t_{k})))}{y(t_{k}-\tau (t_{k}))}y \bigl(t_{k}-\sigma(t_{k})\bigr)\biggr] \\ \leq& B_{i}\omega \bigl[\lambda\bar{f}_{r_{1}}^{u} \|y\|+\mu\hat{I}_{r_{1}}^{u}\|y\| \bigr]. \end{aligned}$$
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
$$\begin{aligned} (T_{i}y) (t) \geq& A_{i}\biggl[\lambda\int _{t}^{t+\omega}\frac{f(s,y(s-\tau(s)))}{y(s-\tau (s))}y\bigl(s-\tau(s)\bigr)\, ds \\ &{}+\mu\sum_{t\leq t_{k}< t+\omega}\frac{I_{k}(t_{k},y(t_{k}-\tau(t_{k})))}{y(t_{k}-\tau (t_{k}))}y \bigl(t_{k}-\sigma(t_{k})\bigr)\biggr] \\ \geq& A_{i}\omega \bigl[\lambda\bar{f}_{r_{2}}^{l} \beta\|y\|+\mu\hat{I}_{r_{2}}^{l}\beta \|y\| \bigr]. \end{aligned}$$
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 ω-periodic 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 (H1)-(H5) 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
ω-periodic 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
ω-periodic solution
\(y(t)\)
with
\(r_{1}\leq \|y\|\leq r_{2}\).
Theorem 3.2
Assume that (H1)-(H5) hold and there exist positive constants
\(r_{1}\)
and
\(r_{2}\)
with
\(r_{1}< r_{2}\)
such that
$$B_{i}\omega\bigl(\lambda\bar{f}_{r_{2}}^{u}+\mu \hat{I}_{r_{2}}^{u}\bigr)\leq1 $$
and
$$A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{1}}^{l}+\mu \hat{I}_{r_{1}}^{l}\bigr)\geq1, $$
then (\({E}_{i}\)) has a positive
ω-periodic solution
\(y(t)\)
with
\(r_{1}\leq \|y\|\leq r_{2}\).
Corollary 3.2
Assume that (H1)-(H5) 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
ω-periodic 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
ω-periodic 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 (H1)-(H5) 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:
$$ \begin{aligned} &B_{i}\omega\bigl(\lambda \bar{f}_{r_{1}}^{u}+\mu\hat{I}_{r_{1}}^{u}\bigr) \leq1, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{2}}^{l}+\mu \hat{I}_{r_{2}}^{l}\bigr)\geq1, \\ &B_{i}\omega\bigl(\lambda\bar{f}_{r_{3}}^{u}+\mu \hat{I}_{r_{3}}^{u}\bigr)\leq1, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{4}}^{l}+\mu \hat{I}_{r_{4}}^{l}\bigr)\geq1, \\ &\ldots, \\ &B_{i}\omega\bigl(\lambda\bar{f}_{r_{n+1}}^{u}+\mu \hat{I}_{r_{n+1}}^{u}\bigr)\leq1 \end{aligned} $$
(3.3)
and
$$ \begin{aligned} &A_{i}\omega\beta\bigl(\lambda \bar{f}_{r_{1}}^{l}+\mu\hat{I}_{r_{1}}^{l}\bigr) \geq1, \\ &B_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{2}}^{u}+\mu \hat{I}_{r_{2}}^{u}\bigr)\leq1, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{3}}^{l}+\mu \hat{I}_{r_{3}}^{l}\bigr)\geq1, \\ &B_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{4}}^{u}+\mu \hat{I}_{r_{4}}^{u}\bigr)\leq1, \\ &\ldots, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{n+1}}^{l}+\mu \hat{I}_{r_{n+1}}^{l}\bigr)\geq1, \end{aligned} $$
(3.4)
then (\({E}_{i}\)) has
n
positive
ω-periodic 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 ω-periodic functions in t and Ω-periodic (\(\Omega>0\)) functions in y. Moreover, \(r_{m+1}=r_{m}+\Omega\), \(m=1,2,\ldots,n\).
Example
Consider the following impulse differential equation satisfying (H1)-(H5):
$$y'(t)=(-1)^{i}\bigl[h\bigl(t, y(t)\bigr)-f\bigl(t, y \bigl(t-\tau(t)\bigr)\bigr)\bigr],\hspace{125pt} (3.5)'_{i}, \ i=1,2 $$
where
$$\begin{aligned}& f(t,y)= \left \{ \textstyle\begin{array}{l@{\quad}l} \xi p(t)y \sin y, & y\in[m\pi, (m+1)\pi), \\ p(t)y|\sin y|, & y\in[(m+1)\pi, (m+2)\pi), \end{array}\displaystyle \right . \\& I_{k}(t,y)= \left \{ \textstyle\begin{array}{l@{\quad}l} \xi q(t)y \sin (y+2k\pi), & y\in[m\pi, (m+1)\pi),\\ q(t)y|\sin (y+2k\pi)|, & y\in[(m+1)\pi, (m+2)\pi), \end{array}\displaystyle \right . \end{aligned}$$
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 ω-periodic functions.
Now, we show that \((3.5)'_{i}\) has n positive ω-periodic functions.
Let
$$r_{1}=\frac{\pi}{2}, \qquad r_{2}=\pi, \qquad r_{3}=\frac{3\pi}{2}, \qquad \ldots,\qquad r_{n}= \frac{n}{2}\pi,\qquad r_{n+1}=\frac{n+1}{2}\pi. $$
Thus, for \(r>0\) and \(y\in P\),
$$f_{r_{1}}^{l}(t)=\inf_{\|y\|=\frac{\pi}{2}, y\in P}\bigl\{ \xi p(t) \sin y\bigr\} \geq\xi\delta p(t), $$
where \(\delta=\inf_{\|y\|=\frac{\pi}{2}, y\in P}\{\sin y\}\). Clearly \(\delta>0\),
$$\begin{aligned}& f_{r_{2}}^{u}(t)=\sup_{\|y\|=\pi, y\in P}\bigl\{ \bigl\vert p(t)\bigr\vert \sin y\bigr\} \leq p(t), \\& I_{r_{1}}^{l}(t)=\inf_{\substack{{\|y\|=r, y\in P}\\{k=1,2,\ldots,\rho}}}\bigl\{ \xi q(t) \sin (y+2k\pi)\bigr\} \geq\xi\delta q(t), \\& I_{r_{2}}^{u}(t)=\sup_{\substack{{\|y\|=r, y\in P}\\{k=1,2,\ldots,\rho}}}\bigl\{ q(t) \bigl\vert \sin (y+2k\pi)\bigr\vert \bigr\} \leq q(t). \end{aligned}$$
First, we choose \(\lambda+\mu>0\) enough small to satisfy
$$B_{i}\omega\bigl(\lambda\bar{f}_{r_{2}}^{u}+\mu \hat{I}_{r_{2}}^{u}\bigr)\leq B_{i}\omega (\lambda \bar{p}+\mu\hat{q})\leq1, $$
then we choose ξ sufficiently large such that
$$A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{1}}^{l}+\mu \hat{I}_{r_{1}}^{l}\bigr)\geq A_{i}\omega \beta\xi \delta(\lambda\bar{p}+\mu\hat{q})\geq1. $$
Similarly, we can obtain
$$B_{i}\omega\bigl(\lambda\bar{f}_{r_{4}}^{u}+\mu \hat{I}_{r_{4}}^{u}\bigr)\leq B_{i}\omega (\lambda \bar{p}+\mu\hat{q})\leq1 $$
and
$$\begin{aligned}& A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{3}}^{l}+\mu \hat{I}_{r_{3}}^{l}\bigr)\geq A_{i}\omega \beta\xi \delta(\lambda\bar{p}+\mu\hat{q})\geq1, \\& \ldots, \\& A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{n+1}}^{l}+\mu \hat{I}_{r_{n+1}}^{l}\bigr)\geq A_{i}\omega\beta\xi \delta(\lambda\bar{p}+\mu\hat{q})\geq1. \end{aligned}$$
Therefore condition (3.4) of Theorem 3.3 is satisfied. By Theorem 3.3, \((3.5)'_{i}\) has n positive ω-periodic 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].