In the sequel, we will use the following notations. For an ωperiodic 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 (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
$$ B_{i}\omega\bigl(\lambda\bar{f}_{r_{1}}^{u}+\mu \hat{I}_{r_{1}}^{u}\bigr)\leq 1 $$
(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
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
$$\begin{aligned} (T_{i}y) (t) \leq& B_{i}\biggl[\lambda\int _{t}^{t+\omega}\frac{f(s,y(s\tau(s)))}{y(s\tau (s))}y\bigl(s\tau(s)\bigr)\, ds \\ &{} +\mu\sum_{t\leq t_{k}< t+\omega}\frac{I_{k}(t_{k},y(t_{k}\tau(t_{k})))}{y(t_{k}\tau (t_{k}))}y \bigl(t_{k}\sigma(t_{k})\bigr)\biggr] \\ \leq& B_{i}\omega \bigl[\lambda\bar{f}_{r_{1}}^{u} \y\+\mu\hat{I}_{r_{1}}^{u}\y\ \bigr]. \end{aligned}$$
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 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
$$B_{i}\omega\bigl(\lambda\bar{f}_{r_{2}}^{u}+\mu \hat{I}_{r_{2}}^{u}\bigr)\leq1 $$
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
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:
$$ \begin{aligned} &B_{i}\omega\bigl(\lambda \bar{f}_{r_{1}}^{u}+\mu\hat{I}_{r_{1}}^{u}\bigr) \leq1, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{2}}^{l}+\mu \hat{I}_{r_{2}}^{l}\bigr)\geq1, \\ &B_{i}\omega\bigl(\lambda\bar{f}_{r_{3}}^{u}+\mu \hat{I}_{r_{3}}^{u}\bigr)\leq1, \\ &A_{i}\omega\beta\bigl(\lambda\bar{f}_{r_{4}}^{l}+\mu \hat{I}_{r_{4}}^{l}\bigr)\geq1, \\ &\ldots, \\ &B_{i}\omega\bigl(\lambda\bar{f}_{r_{n+1}}^{u}+\mu \hat{I}_{r_{n+1}}^{u}\bigr)\leq1 \end{aligned} $$
(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
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}):
$$y'(t)=(1)^{i}\bigl[h\bigl(t, y(t)\bigr)f\bigl(t, y \bigl(t\tau(t)\bigr)\bigr)\bigr],\hspace{125pt} (3.5)'_{i}, \ i=1,2 $$
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 ωperiodic functions.
Now, we show that \((3.5)'_{i}\) has n positive ωperiodic 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 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].