\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 123, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/123\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of positive periodic solutions
for first-order singular systems with impulse effects}

\author[B.-X. Yang \hfil EJDE-2013/123\hfilneg]
{Bian-Xia Yang}  % in alphabetical order

\address{Bian-Xia Yang \newline
School of Mathematics and Statistics, Lanzhou University,
Lanzhou, Gansu 730000, China}
\email{yanglina7765309@163.com}

\thanks{Submitted January 24, 2013. Published May 17, 2013.}
\subjclass[2000]{34A34, 34A37}
\keywords{Positive periodic solution; singular systems; impulsive; 
\hfill\break\indent fixed point theorem}

\begin{abstract}
 In this article, we consider the existence and multiplicity of
 positive periodic solutions for a first-order singular system
 with impulse effects.  The proof of our main result
 is based on Krasnoselskii's fixed point theorem in a cone.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 Impulsive differential equations have  wide applicability in physics,
population dynamics, ecology, biological systems, biotechnology,
industrial robotic, pharmacokinetics, optimal control, etc. 
The reason for this applicability arises from the fact that impulsive 
differential problems are an appropriate model for describing process 
which at certain moments change their state rapidly and which cannot 
be described using the classical differential equation. Therefore, 
the study of impulsive differential equation has gained prominence 
and it is a rapidly growing field, see \cite{b1,b2,c1,e1,f1}
and the references therein.

In 2008, Chu and Nieto \cite{c1} studied first-order impulsive
periodic boundary-value problem (BVP)
\begin{equation}
\begin{gathered}
u'(t)+a(t)u(t)=f(t,u(t))+e(t),\quad  t\in\mathbb{J'},\\
u(t_k^+)=u(t_k^-)+I_k(u(t_k)),\quad k=1,\dots, p,\quad u(0)=u(1),
\end{gathered}\label{e1.1}
\end{equation}
where $0 = t_0 < t_1 < \dots < t_p < t_{p+1} =1$,
$\mathbb{J'}=[0,1]\backslash\{t_1,\dots, t_p\}$,
$a, e \in C(\mathbb{R},\mathbb{R})$ are 1-periodic functions,
$I_k\in C(\mathbb{R},\mathbb{R})$, $k=1,\dots, p$. The nonlinearity function
$f(t, u)\in C( \mathbb{J'}\times \mathbb{R})$ is 1-periodic in $t$,
 and $f(t, u) $ is left continuous at $t=t_k$,
the right limit $f(t_k^+,u)$ exists. Using the
Leray-Schauder nonlinear alternative  and a truncation
technique, under some conditions, they obtained the existence of at
least one non-trivial 1-periodic solution of  \eqref{e1.1}.

In 2011, Wang \cite{w1} studied the first-order nonautonomous singular
$n$-dimensional system
\begin{equation}
u'_i(t)+a_i(t)u_i(t)=\lambda b_i(t)f_i( u_1(t),\dots, u_n(t)),\quad
i=1, \dots n.\label{e1.2}
\end{equation}
By using the fixed point theorem in cones, the author established
the following result, under the assumptions:
\begin{itemize}
\item[(A1)] $a_i, b_i\in C(\mathbb{R}, \mathbb{R}_+)$ are $\omega$-periodic
functions  such that $\int_0^\omega a_i(t)dt>0$ and 
$\int_0^\omega b_i(t)dt>0$, for  $i=1, \dots, n$;


\item[(A2)] $f_i\in C(\mathbb{R}_+^n\backslash\{0\},\mathbb{R}_+\backslash\{0\})$,
$ i=1, \dots, n$, and $\lim_{|\mathbf{u}|\to 0}f_j( \mathbf{u})=\infty$
for some $j=1,\dots, n$.
\end{itemize}

\begin{theorem} \label{thmA}
 Let {\rm (A1), (A2)} hold.
Then
\begin{itemize}
\item[(i)] there exists a $\lambda_0>0$, such that \eqref{e1.2} has a positive
$\omega$-periodic solution for $0<\lambda<\lambda_0$;

\item[(ii)]  if $\lim_{|\mathbf{u}|\to\infty}\frac{f_i(\mathbf{u})}{|\mathbf{u}|}=0$,
$ i=1,\dots, n$, then, for all $\lambda>0$, \eqref{e1.2} has a positive
$\omega$-periodic solution;

\item[(iii)] if $\lim_{|\mathbf{u}|\to\infty}\frac{f_i(
\mathbf{u})}{|\mathbf{u}|}=\infty, i=1, \dots, n$, then, for
sufficiently small $\lambda>0$, \eqref{e1.2} has two positive
$\omega$-periodic solutions.
\end{itemize}
\end{theorem}

Here $ \mathbb{R}_+=[0,\infty)$, $\mathbb{R}_+^n=\Pi_{i=1}^n\mathbb{R}_+$, 
$\mathbf{u}= (u_1, u_2, \dots, u_n)\in \mathbb{R}_+^n$, 
$|\mathbf{u}|=\sum_{i=1}^n|u_i|$.

Inspired by \cite{c1,w1}, in this paper, we are concerned with the existence 
and multiplicity of the positive $1$-periodic solution of the following
first-order singular $n$-dimensional system with impulse effect
\begin{equation}
\begin{gathered}
u'_i(t)+a_i(t)u_i(t)=\lambda b_i(t)f_i(t,u_1(t),\dots,
u_n(t))+\lambda e_i(t),\quad t\in\mathbb{J'},\\
u_i(t_k^+)=u_i(t_k^-)+\lambda I_i^k(u_1 (t_k), \dots, u_n(t_k)),\quad
 k=1,\dots, p,\\
 u_i(0)=u_i(1),\quad  i=1, \dots, n,
\end{gathered} \label{e1.3}
\end{equation}
where $\lambda>0$ is a parameter, $\mathbb{J'}$ is defined as above.
By a positive $1$-periodic solution, we mean a positive $1$-periodic
function in $C^1(\mathbb{R},\mathbb{R}^n)$ solving corresponding
systems \eqref{e1.3} and each component is positive for all $t$.

We will use the following assumptions:
\begin{itemize}
\item[(H1)] $a_i, e_i\in C(\mathbb{R},\mathbb{R})$, 
$b_i\in C(\mathbb{R},\mathbb{R}_+\backslash\{0\})$ are
1-periodic functions and \\ $\int_0^1a_i(t)dt>0$ for $i=1, \dots, n$;

\item[(H2)] $f_i\in C(\mathbb{J}'\times(\mathbb{R}_+^n\backslash\{0\}),
\mathbb{R}_+\backslash\{0\})$
is 1-periodic in $t$. Moreover, $f_i(t, u) $ is left continuous at
$t= t_k$ and the right limit $f_i(t_k^+,u)$ exists, $i=1, \dots,
n$;

\item[(H3)] $I_i^k\in C( \mathbb{R}_+^n,\mathbb{R}_+ )$, 
$k=1,\dots, p$, $i=1, \dots, n$.
\end{itemize}
Using Krasnoseskii's fixed point theorem in cone, we obtain 
the following result.

\begin{theorem} \label{thm1.1}
 Let {\rm (H1)--(H3)} hold. Assume that
$\lim_{|\mathbf{u}|\to 0}f_i(t, \mathbf{u})=\infty$, 
$i=1, \dots, n$ uniformly with respect to $t\in[0,1]$. Then
\begin{itemize}
\item[(i)] there exists a $\lambda_1>0$, such that \eqref{e1.3} has a positive
$1$-periodic solution for $0<\lambda<\lambda_1$;

\item[(ii)] if $\lim_{|\mathbf{u}|\to\infty}\frac{f_i(t,
\mathbf{u})}{|\mathbf{u}|}=0$ and 
$\lim_{|\mathbf{u}|\to \infty}f_i(t, \mathbf{u})=\infty$  
 uniformly with respect to
$t\in[0,1], \lim_{\\mathbf{u}\to\infty}
\frac{I_i^k(\mathbf{u})}{{\bf |u|}}=0$ for
$i=1, \dots, n$, $k=1, \dots, p$, then, there exists $\lambda_2>0$,
such that \eqref{e1.3} has a positive $1$-periodic solution for
$\lambda>\lambda_2$;

\item[(iii)] if $\lim_{|\mathbf{u}|\to \infty}
\frac{f_i(t,\mathbf{u})}{|\mathbf{u}|}=\infty, i=1, \dots, n$ 
uniformly with respect to $t\in[0,1]$, then, for  sufficiently small
$\lambda>0$, \eqref{e1.3} has two positive $1$-periodic solutions.
\end{itemize}
\end{theorem}

We  remark that $e_i$ may take negative values in this paper;
nevertheless, we still obtain the existence and multiplicity of 
positive $1$-periodic solution of \eqref{e1.3}.

\begin{remark} \label{rmk1,1} \rm
 If $I_k= 0$ for $ k=1,\dots, p$, $f_i(t,\mathbf{u})=f_i(\mathbf{u})$, 
$e_i=0$ for $i=1, \dots, n$, then system \eqref{e1.3}
reduces to \eqref{e1.2}. In this case, we  need only
 $\lim_{|\mathbf{u}|\to 0}f_j( \mathbf{u})=\infty$ for some 
$j=1,\dots, n$; so  (i), (iii) of Theorem \ref{thm1.1} reduce to the (i),
 (iii) of Theorem \ref{thmA}, respectively. 
Hence,  Theorem \ref{thm1.1} extends  Theorem \ref{thmA}.

 If $n=1$, $\lambda=1$, $b_i(t)=1$, then system \eqref{e1.3}
reduces to  \eqref{e1.1}. So, Theorem \ref{thm1.1} partially improves the result 
of \cite{c1}.
\end{remark}

The rest of this paper is organized as follows. In Section 2, some
notation and preliminaries are given. In Section 3, we give the proof
of  main result. At last, an example is presented to
illustrate the main result.


\section{Preliminaries}

 Denote
\begin{align*}
PC[0,1]=\Big\{&u: \text{$u$ is continuous on $\mathbb{J'}$, left continuous at
$t= t_k$,}\\
&\text{and the right limit $u(t_k^+)$ exists for $k=1, \dots, p$}.
\end{align*}
Let  $E=\Pi_{i=1}^nPC[0,1]$ which is a Banach space under the norm
\[
\|\mathbf{u}\|=\sum_{i=1}^n\sup_{t\in[0,1]}|u_i(t)|.
\]
Denote the cone 
\begin{align*}
K=\big\{&\mathbf{u}=(u_1,\dots, u_n)\in E: u_i(t)\geq 0,\,
 t\in[0,1],i=1, \dots, n, \text{ and}\\
&\min_{t\in[0,1]}|\mathbf{u}(t)|\geq\sigma\|\mathbf{u}\|\big\},
\end{align*} 
where
\begin{equation}
\sigma=\min_{i=1,\dots, n}\{\sigma_i\},\quad
\sigma_i=\frac{m_i}{M_i}, i=1, \dots, n.\label{e2.1}
\end{equation}
The constants $m_i, M_i$ will be defined by \eqref{e2.3} below.

Let $T_\lambda:K\backslash\{0\}\to E$  be
a map with components $(T_\lambda^1, \dots, T_\lambda^n)$:
\begin{equation}
T_\lambda^i\mathbf{u}(t)
=\lambda\int_0^1G_i(t,s)\Big[b_i(s)f_i(s,\mathbf{u}(s))
+e_i(s)\Big]ds+\lambda\sum _{k=1}^pG_i(t,t_k) I_i^k(\mathbf{u}(t_k)),\label{e2.2}
\end{equation}
where
$$
G_i(t,s) =\begin{cases}
\frac{e^{-A_i(t)+A_i(s)}}{1-e^{-A_i(1)}},
& 0\leq s\leq t \leq 1,\\
\frac{e^{-A_i(1)-A_i(t)+A_i(s)}}{1-e^{-A_i(1)}},
&0\leq t< s\leq 1,
\end{cases}
$$
with $A_i(t)=\int_0^ta_i(s)ds$, (see \cite{c1} for  details).
 It is easy to see that  (H1) implies that $G_i(t,s)>0$.

Clearly, $\mathbf{u}\in E\backslash \{0\}$ is a solution of \eqref{e1.3} 
if and only if it is a fixed point of $T_{\lambda}$. Also note that
each component $u_i(t)$ of any nonnegative periodic solution $\mathbf{u}(t)$ 
is strictly positive for all $t$ because of
the positiveness of $G_i(t,s)$ and assumptions (H1)--(H3).

For convenience, throughout this paper, we denote
\begin{equation}
M_i=\sup_{t,s\in [0,1]} G_i(t,s),\quad
m_i=\inf_{t,s\in [0,1]} G_i(t,s)\label{e2.3}
\end{equation}
and
$$
|\mathbf{u}|=\sum_{i=1}^n|u_i|, \quad \text{where }
 \mathbf{u}=(u_1, u_2, \dots, u_n)\in \mathbb{R}^n.
$$
For $r>0$, define $\Omega_r=\{\mathbf{u}\in K: \|\mathbf{u}\|<r\}$.
Then $\partial\Omega_r=\{\mathbf{u}\in K: \|\mathbf{u}\|=r\}$.
We now look at several properties of the operator $T_\lambda$.

 \begin{lemma} \label{lem2.1}
 Assume that {\rm (H1)--(H3)} hold.
\begin{itemize}
\item[(i)] If $\lim_{|\mathbf{u}|\to 0}f_i(t, \mathbf{u})=\infty$ 
uniformly with respect to $t\in[0,1]$ for $i=1, \dots, n$, 
then there is a $\delta>0$,  such that for $r\in(0, \delta)$, 
$T_\lambda: \bar{\Omega}_r\backslash\{0\}\to K$ is completely
 continuous.

\item[(ii)] If $\lim_{|\mathbf{u}|\to \infty}f_i(t, \mathbf{u})=\infty$ 
uniformly with respect to $t\in[0,1]$ for $i=1, \dots, n$, then there 
is a $\Delta>0$,  such that for $R>\Delta$,  
$T_\lambda: K\backslash\Omega_R\to K$ is completely  continuous.

\item[(iii)] If  $T_\lambda: K\backslash\{0\}\to K$, then for 
$\mathbf{u}\in K$ with $\|\mathbf{u}\|=r$, we have
\begin{gather}
\|T_\lambda\mathbf{u}\|\geq\frac{\lambda\hat{m}_r}{2}
\sum_{i=1}^n m_i\int_0^1b_i(s)ds, \label{e2.4} \\
\|T_\lambda\mathbf{u}\|\leq\lambda\sum_{i=1}^nM_i\Big(\hat{M}_r\int_0^1b_i
(s)ds+\int_0^1|e_i(s)|ds+p\tilde{M}_r\Big),  \label{e2.5}
\end{gather}
where $\hat{m}_r=\min\{f_i(t,\mathbf{u}): t\in[0,1], 
\mathbf{u}\in \mathbb{R}_+^n $ with
$\sigma r\leq |\mathbf{u}|\leq r, i=1, \dots, n\}$,
\begin{gather*}
\hat{M}_r=\max\{f_i(t,\mathbf{u}): t\in[0,1], \mathbf{u}\in
\mathbb{R}_+^n \text{ with }  \sigma r\leq |\mathbf{u}|\leq r, i=1, \dots, n\},
\\
\tilde{M}_r=\max\{I_i^k(u): \mathbf{u}\in \mathbb{R}_+^n \ \text{ with }
\sigma r\leq|\mathbf{u}|\leq r, \, k=1,\dots,p, i=1, \dots, n\}.
\end{gather*}
\end{itemize}
\end{lemma}

\begin{proof}
  (i) We split $b_i(t)f_i(t, \mathbf{u})+e_i(t)$ into two
 terms $\frac{1}{2}b_i(t)f_i(t, \mathbf{u})$ and  
$\frac{1}{2}b_i(t)f_i(t, \mathbf{u})\\+e_i(t)$.
Then the first term is always positive  and used to carry out the
estimates of the operator. We will make the second term
$\frac{1}{2}b_i(t)f_i(t, \mathbf{u})+e_i(t)$ positive by choosing
appropriate domains of $f_i$.

Noting that $b_i(t)$ is continuous and positive 
on $[0,1]$, and $\lim_{|\mathbf{u}|\to 0} f_i(t, \mathbf{u})=\infty$, 
for $i=1, \dots, n$, 
there exists  $\delta>0$, such that
$$
f_i(t, \mathbf{u})\geq 2\frac{\max_{t\in[0,1]}
\{|e_i(t)|\}+1}{\min_{t\in[0,1]}b_i(t)},  \quad t\in[0,1],\;
 \mathbf{u}\in \mathbb{R}^n,\;  0<|\mathbf{u}|\leq\delta.
$$
 Now for $r\in (0, \delta)$ and $\mathbf{u}\in \bar{\Omega}_r\backslash\{0\}, 
t\in[0,1]$, we have
\begin{align*}
b_i(t)f_i(t, \mathbf{u}(t))+e_i(t)
&\geq \frac{1}{2}b_i(t)f_i(t,\mathbf{u}(t))+e_i(t)\\
&\geq b_i(t)\frac{\max_{t\in[0,1]}\{|e_i(t)|\}+1}{\min_{t\in[0,1]}\{b_i(t)\}}
+e_i(t)>0,
\end{align*}
and
\begin{align*}
\min_{t\in[0,1]}(T_\lambda^i \mathbf{u})(t)
&\geq \lambda\int_0^1 m_i\Big[b_i(s)f_i(s,\mathbf{u}(s))+e_i(s)\Big]ds+\lambda
m_i\sum_{k=1}^p I_i^k (\mathbf{u}(t_k))\\
&=\lambda \sigma_i\int_0^1M_i\Big[ b_i(s)f_i(s,\mathbf{u}(s))+e_i(s)\Big]ds
+\lambda \sigma_i \sum_{k=1}^p M_iI_i^k(\mathbf{u}(t_k))\\
&\geq \sigma_i\sup_{t\in[0,1]}|T_\lambda^i \mathbf{u}|.
\end{align*}
Thus, $T_\lambda(\bar{\Omega}_r\backslash\{0\})\subset K$. According to Arzela-Ascoli
theorem and the hypothesis (H1)-(H3), we know that
$T_\lambda: \bar{\Omega}_r\backslash\{0\}\to K$ is completely continuous.

(ii)  If $\lim_{|\mathbf{u}|\to\infty}f_i(t,\mathbf{u})=\infty$, there is
an $\hat{R}>0$, such that
$$
f_i(t, \mathbf{u})\geq 2\frac{\max_{t\in[0,1]}
\{|e_i(t)|\}+1}{\min_{t\in[0,1]}\{b_i(t)\}},\quad t\in[0,1],\;
 \mathbf{u}\in \mathbb{R}^n,\ |\mathbf{u}|\geq\hat{R}.
$$
Let $\Delta=\frac{\hat{R}}{\sigma}$. Then for $R>\Delta,\mathbf{u}\in
K\backslash\Omega_R$, we have that
$\min_{t\in[0,1]}|\mathbf{u}(t)|\geq\sigma\|\mathbf{u}\|\geq\hat{R}$, 
and therefore 
$$
b_i(t)f_i(t, \mathbf{u})+e_i(t)
\geq\frac{1}{2}b_i(t)f_i(t,\mathbf{u})+e_i(t)>0,\quad t\in[0,1].
$$
Similar to (i), we have that  $T_\lambda: K\backslash\Omega_R\to K$ 
is completely continuous.

(iii) If  $\mathbf{u}\in K$ with $\|\mathbf{u}\|=r$, then for 
$t\in[0,1]$, $\sigma r\leq |\mathbf{u}(t)|\leq r$, so 
$\hat{m}_r\leq f_i(t,\mathbf{u}(t))\leq \hat{M}_r$, $t\in [0,1]$, 
 and $I_i^k(u)\leq\tilde{M}_r$, $k=1,\dots,p$, $i=1, \dots, n$.
 By the definition of
$T_\lambda \mathbf{u}$, we have
\begin{align*}
\|T_\lambda\mathbf{u}\|
&=\sum_{i=1}^n \sup _{t\in [0,1]}T_\lambda^i\mathbf{u}(t)\\
&\geq\frac{1}{2}\lambda\sum_{i=1}^n m_i\int^1_0
 b_i(s)f_i(s,\mathbf{u}(s))ds\\
&\geq \frac{\lambda\hat{m}_r}{2}\sum_{i=1}^nm_i\int^{1}_0  b_i(s)ds,
\end{align*}
and
\begin{align*}
 \|T_\lambda \mathbf{u}\|
&= \sum_{i=1}^n\sup_{t\in[0,1]}T_\lambda^i\mathbf{u}(t)\\
&\leq \lambda\sum_{i=1}^n M_i \Big(\int_0^1 b_i(s)f_i(s, \mathbf{u}(s))ds+\int_0^1|e_i(s)|ds+\sum_{k=1}^p I_i^k(\mathbf{u}(t_k))\Big)\\
&\leq \lambda\sum_{i=1}^n M_i \Big(\hat{M}_r\int_0^1
b_i(s)ds+\int_0^1|e_i(s)|ds+p\tilde{M}_r\Big).
\end{align*}
\end{proof}

The following well-known  fixed point theorem is
crucial in our arguments.

\begin{lemma}[\cite{g1,k1}] \label{lem2.2}
 Let $E$ be a Banach space and $K$ a
cone in $E$.  Assume that $\Omega_1, \Omega_2$ are bounded open
subsets of $E$ with $0\in\Omega_1, \bar{\Omega}_1\subset \Omega_2$,
and let
$$ 
T: K\cap(\bar{\Omega}_2\backslash\Omega_1)\to K
$$ 
be completely continuous operator such that either
\begin{itemize}
\item[(i)] $\|Tu\|\geq \|u\|, u\in K\cap\partial\Omega_1$ and
$\|Tu\|\leq\|u\|, u\in K\cap\partial\Omega_2$; or

\item[(ii)] $\|Tu\|\leq \|u\|, u\in K\cap\partial\Omega_1$ and
$\|Tu\|\geq\|u\|, u\in K\cap\partial\Omega_2$.
\end{itemize}
Then $T$ has a fixed point in
$K\cap(\bar{\Omega}_2\backslash\Omega_1)$.
\end{lemma}

\section{Proof of  main results}

\begin{proof}[Proof of  Theorem \ref{thm1.1}]
 (i) By  Lemma \ref{lem2.1} (i), there is a $\delta>0$, such that if $0<r<\delta$, then
$T_\lambda: \bar{\Omega}_r\backslash\{0\}\to K$ is completely continuous. 
Now, for a fixed number $r_1\in(0,\delta)$, if we choose 
$$
\lambda_1=\frac{r_1}{\sum_{i=1}^nM_i\Big(\hat{M}_{r_1}\int_0^1b_i
(s)ds+\int_0^1|e_i(s)|ds+p\tilde{M}_{r_1}\Big)},
$$ 
for $\lambda\in(0,\lambda_1)$, \eqref{e2.5} implies
\begin{equation}
\|T_\lambda\mathbf{u}\|<\|\mathbf{u}\|,\quad
 \mathbf{u}\in \partial\Omega_{r_1}.\label{e3.1}
\end{equation}
On the other hand, for
$\lambda\in(0,\lambda_1)$, in view of the assumption
$\lim_{|\mathbf{u}|\to 0}f_i(t, \mathbf{u})=\infty$, there is a positive number
$r_2<r_1$, such that
$$
f_i(t,\mathbf{u})\geq\eta|\mathbf{u}|, \quad t\in[0,1],\;
 \mathbf{u}\in \mathbb{R}^n \text{ with } 0<|\mathbf{u}|\leq r_2,
$$
 where $\eta>0$ is chosen so that
$$
\frac{\lambda\eta\sigma}{2}\min_{i=1, \dots, n}\{m_i\int_0^1b_i(s)ds\}>1.
$$
Thus, for $\mathbf{u}\in\partial\Omega_{r_2}, $ we have
$$
f_i(t, \mathbf{u}(t))\geq \eta | \mathbf{u}(t)|,\quad t\in[0,1].
$$
and
\begin{equation} \label{e3.2}
\begin{aligned}
\|T_\lambda\mathbf{u}\|
&\geq \sup_{t\in[0,1]}T_\lambda^i\mathbf{u}(t)\\
&= \sup_{t\in[0,1]}\lambda\Big(\int_0^1G_i(t,s)
\big[b_i(s)f_i(s,\mathbf{u}(s))+e_i(s)\big]ds+\sum_{k=1}^pG_i(t,t_k)I_i^k(\mathbf{u}(t_k))\Big)\\
&\geq \frac{1}{2}\lambda
\sup_{t\in[0,1]}\int_0^1G_i(t,s)b_i(s)f_i(s,\mathbf{u}(s))ds\\
&\geq \frac{1}{2}\lambda m_i\int^1_0
 b_i(s)f_i(s,\mathbf{u}(s))ds\\
 &\geq \frac{1}{2}\eta \lambda m_i\int^{1}_0
 b_i(s)| \mathbf{u}(s)|ds\\
&\geq \frac{1}{2}\eta\lambda m_i\sigma\int^{1}_0
 b_i(s)ds\|\mathbf{u}\|>\|\mathbf{u}\|.
\end{aligned}
\end{equation}
So from Lemma \ref{lem2.2}, \eqref{e3.1}, \eqref{e3.2}, we obtain that
$T_\lambda$ has a fixed point
$\mathbf{u}\in \bar{\Omega}_{r_1}\backslash\Omega_{r_2}$.
The fixed point $\mathbf{u}$ is the desired
positive $1$-periodic solution of \eqref{e1.3}.

(ii)  According to  Lemma \ref{lem2.1} (ii),
there is a $\Delta>0$, such that for $R>\Delta$,
 $T_\lambda: K\backslash\Omega_R\to K$ is completely continuous.
 Now for a fixed number
$R_1>\Delta$, if we choose
 $$
\lambda_2=\frac{2R_1}{\hat{m}_{R_1}\sum_{i=1}^n m_i\int_0^1b_i(s)ds},
$$
for $\lambda>\lambda_2$, \eqref{e2.4} means that
\begin{equation}
\|T_\lambda\mathbf{u}\|\geq\frac{\lambda\hat{m}_{R_1}}{2}
\sum_{i=1}^n m_i\int_0^1b_i(s)ds>R_1=\|\mathbf{u}\|,\quad
\mathbf{u}\in\partial\Omega_{R_1}.\label{e3.3}
\end{equation}
On the other hand. Since
$\lim_{|{\bf u}|\to\infty}\frac{f_i(t, \mathbf{u})}{|\mathbf{u}|}=0$,
$\lim_{|{\bf u }|\to\infty}\frac{I_i^k(\mathbf{u})}{|\mathbf{u}|}=0$,
for a fixed $ \lambda>\lambda_2$, we can choose
$$
R_2>\max\big\{2R_1, \ \ \ 2\lambda\sum_{i=1}^n M_i\int_0^1 |e_i(s)|ds\big\},
$$
so that
$$
f_i(t,\mathbf{u})\leq\epsilon |\mathbf{u}| \text{ and }
I_i^k(\mathbf{u})\leq\epsilon |\mathbf{u}|  \text{ for }
 t\in[0,1], \; \mathbf{u}\in \mathbb{R}^n \text{ with }
 |\mathbf{u}|\geq \sigma R_2,
$$
where the constant $\epsilon>0$ satisfies
$$
\lambda \epsilon\sum_{i=1}^n M_i \Big(\int_0^1 b_i(s)ds+ p \Big)<\frac{1}{2}.
$$
From the definition of $T_\lambda$, for $\mathbf{u}\in\partial\Omega_{R_2}$,
we have
\begin{equation}
\begin{aligned}
& \|T_\lambda \mathbf{u}\|\\
&= \sum_{i=1}^n\sup_{t\in[0,1]}T_\lambda^i\mathbf{u}(t)\\
&\leq \lambda\sum_{i=1}^n M_i\Big( \int_0^1 b_i(s)
 f_i(s, \mathbf{u}(s))ds+\int_0^1|e_i(s)|ds+
\sum_{k=1}^p  I_i^k(\mathbf{u}(t_k))\Big)\\
&\leq \lambda \sum_{i=1}^nM_i\Big(r_2\epsilon\int_0^1 b_i(s)ds
+\int_0^1|e_i(s)|ds+ pr_2\epsilon\Big)
< R_2=\|\mathbf{u}\|.
\end{aligned}  \label{e3.4}
\end{equation}
By Lemma \ref{lem2.2}, \eqref{e3.3}, \eqref{e3.4}, we have that
  $T_\lambda$ has a fixed point
$\mathbf{u}\in\bar{\Omega}_{R_2}\backslash\Omega_{R_1}$.
 The fixed point $\mathbf{u}$ is the desired
positive $1$-periodic solution of \eqref{e1.3}.

 (iii) Since $\lim_{|\mathbf{u}|\to 0}f_i(t, \mathbf{u})=\infty$, (i) implies
\eqref{e1.3} has a positive periodic solutions 
$\mathbf{u}_1\in\bar{\Omega}_{r_1}\backslash\Omega_{r_2}$ 
for $\lambda\in (0, \lambda_1)$.

On the other hand, since 
$\lim_{|\mathbf{u}|\to\infty}\frac{f_i(t,\mathbf{u})}{|\mathbf{u}|}=\infty$,  
by Lemma \ref{lem2.1} (ii),
there is $\Delta>0$, such that if $R>\Delta$, 
$T_\lambda : K\backslash\Omega_R\to K$ is completely continuous. 
For a fixed number $R_3>\max\{\Delta, r_1\}$, if we choose 
$$
\lambda_0=\frac{R_3}{\sum_{i=1}^nM_i\Big(\hat{M}_{R_3}\int_0^1b_i
(s)ds+\int_0^1|e_i(s)|ds+p\tilde{M}_{R_3}\Big)},
$$  
for $\lambda< \lambda_0$, \eqref{e2.5} implies
\begin{equation}
\|T_\lambda\mathbf{u}\|<\|\mathbf{u}\|,\quad
\mathbf{u}\in\partial\Omega_{R_3}.\label{e3.5}
\end{equation}
Since $\lim_{|\mathbf{u}|\to\infty}\frac{f_i(t,\mathbf{u})}{|\mathbf{u}|}=\infty$
uniformly with respect to $t\in[0,1]$, there is a positive number
$\tilde{r}$ such that
$$
f_i(t, \mathbf{u})\geq\eta|\mathbf{u}|,\quad t\in[0,1], \;
\mathbf{u} \in \mathbb{R}^n  \text{ with } |\mathbf{u}|\geq\tilde{r},
$$
where $\eta>0$ is chosen so that
$$
\frac{\lambda\eta\sigma}{2}\min_{i=1, \dots, n}\{m_i\int_0^1b_i(s)ds\}>1.
$$
Let $R_4=\max\{2R_3, \frac{1}{\sigma}\tilde{r}\}>\Delta$.
If $\mathbf{u}\in \partial\Omega_{R_4}$, then
$\min_{t\in[0,1]}| \mathbf{u}(t)|\geq\sigma\|\mathbf{u}\|=\sigma R_4\geq\tilde{r}$,
 which suggests that
$$
f_i(t,\mathbf{u}(t))\geq\eta|\mathbf{u}(t)|,\quad  t\in[0,1].
$$
Similar to \eqref{e3.2}, we get
$$
\|T_\lambda\mathbf{u}\|\geq \lambda\Gamma\eta\|\mathbf{u}\|
>\|\mathbf{u}\|,\quad \mathbf{u}\in \partial\Omega_{R_4}.
$$
It follows from Lemma \ref{lem2.2} that $T_\lambda$ has a fixed point
$\mathbf{u}_2\in\bar{\Omega}_{R_4}\backslash\Omega_{R_3}$, which is a
positive $1$-periodic solution of \eqref{e1.3} for $\lambda<\lambda_0$.

Noting that
$$
r_2<\|\mathbf{u}_1\|<r_1<R_3<\|\mathbf{u}_2\|<R_4,
$$
we can conclude that $\mathbf{u}_1$ and $\mathbf{u}_2$ are the desired
distinct positive periodic solutions of \eqref{e1.3} for
$\lambda<\min\{\lambda_0, \lambda_1\}$.
\end{proof}

\subsection*{Example}
We consider the first-order singular 2-dimensional system with impulse effect
\begin{equation} \label{e4.1}
\begin{gathered}
u_1'(t)+(\sin(2\pi t)+\frac{1}{2})u_1(t)
=\lambda (\sin(2\pi t)+2)f_1(t, u_1, u_2)
+\lambda \sin(2\pi t),\\
 t\in(0,1)\backslash\{\frac{1}{2}\},
\\
u_2'(t)+(\cos(2\pi t)+\frac{1}{3})u_2(t)
=\lambda (\cos(2\pi t)+3)f_2(t, u_1, u_2)
+\lambda \cos(2\pi t),\\
   t\in(0,1)\backslash\{\frac{1}{2}\},
\\
u_1(\frac{1}{2}^+)=u_1(\frac{1}{2}^-)
 +\lambda(u_1(\frac{1}{2})+u_2(\frac{1}{2}))^{3/4}, 
\\
u_2(\frac{1}{2}^+)=u_2(\frac{1}{2}^-)
 +\lambda(u_1(\frac{1}{2})+u_2(\frac{1}{2}))^{1/2}, \\
u_1(0)=u_1(1), \quad u_2(0)=u_2(1).
\end{gathered}
\end{equation}
Let
\begin{gather*}
f_1(t, u_1, u_2)=2+\sin(2\pi t)+\frac{1}{u_1^2+u_2^3}+(u_1+u_2)^{1/2}, \\
f_2(t, u_1, u_2)=3+\sin(2\pi t)+\frac{1}{u_1+u_2^2}+(u_1+u_2)^{1/3}.
\end{gather*}
Comparing with \eqref{e1.3}, we have  $n=2$, $p=1$, $t_1=1/2$. 
Clearly  assumptions (H1)--(H3) are satisfied,  we can easily check 
that $\lim_{|\mathbf{u}|\to 0}f_i(t, \mathbf{u})=\infty$, 
$\lim_{|\mathbf{u}|\to \infty}f_i(t, \mathbf{u})=\infty$ and 
$ \lim_{|\mathbf{u}|\to \infty}\frac{f_i(t, \mathbf{u})}{\mathbf{u}}=0$, 
$i=1, 2$ uniformly with respect to $t\in[0,1]$. 
So by (i) (ii) of Theorem \ref{thm1.1}, we have: 
there exists a $\lambda_1>0$, such that \eqref{e4.1} has a positive $1$-periodic
solution for $0<\lambda<\lambda_1$ and there 
exists $\lambda_2$, such that \eqref{e4.1}
has a positive $1$-periodic solution for $\lambda>\lambda_2$.

Similarly, if we let
\begin{gather*}
f_1(t, u_1, u_2)=2+\sin(2\pi t)+\frac{1}{u_1^2+u_2^3}+(u_1+u_2)^2, \\
f_2(t, u_1, u_2)=3+\sin(2\pi t)+\frac{1}{u_1+u_2^2}+(u_1+u_2)^3.
\end{gather*}
According to (iii) of Theorem \ref{thm1.1},  for  sufficiently small
$\lambda>0$, \eqref{e4.1} has two positive $1$-periodic solutions.


\subsection*{Acknowledgements}
The author is very grateful to the anonymous referees for their valuable 
suggestions.

\begin{thebibliography}{99}
\bibitem{b1}  D. D. Bainov, P. S. Simeonov;
\emph{Impulsive differential equations: periodic
solutions and applications}, Longman, Harlow, 1993.

\bibitem{b2} D. D. Bainov, P. S. Simeonov;
\emph{Systems with Impulse effect}, Ellis Horwood, Chichester, 1989.

\bibitem{b3}  D. D. Bainov, S. G. Hristova, S. Hu, V. Lakshmikantham;
\emph{Periodic boundary-value problem for systems of first-order 
impulsive differential equations}, Differential
 Integral Equations, 2 (1989), 37-43.

\bibitem{c1}  J. Chu, J. Nieto;
\emph{Impulsive periodic solutions of first-order
singular differential equations}, Bull. London. Math. Soc. 40 (2008),
143-150.

\bibitem{e1}  L. H. Erbe, X. Liu;
\emph{Existence of periodic solutions of impulsive differential
systems}, J. Appl. Math. Stochastic Anal. 4 (1991), 137-146.

\bibitem{f1}\ M. Frigon, D. O'Regan;
\emph{Existence results for  first-order impulsive differential
equations}, J. Math. Anal. Appl. 193 (1995),  96-113.

\bibitem{g1}  D. Guo, V. Lakshmikantham;
\emph{Nonlinear Problems in Abstract Cones}, Academic Press, Orlando, FL, 1988.

\bibitem{k1}  M. Krasnoselskii;
\emph{Positive Solutions of Operator Equations},
Noordhoff, Groningen, 1964.

\bibitem{w1} H. Wang;
\emph{Positive periodic solutions of singular
systems of first order ordinary differential equations},
Appl. Math. Comput. 8(2011), 1605-1610.


\end{thebibliography}


\end{document}