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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 158, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/158\hfil Positive periodic solutions]
{Positive periodic solutions for an impulsive
 ratio-dependent predator-prey system with delays}

\author[Y. Liu, Q. Wang\hfil EJDE-2008/158\hfilneg]
{Yan Liu, Quanyi Wang}  % in alphabetical order

\address{Yan Liu \newline
School of mathematical sciences, Huaqiao University,
Quanzhou, Fujian 362021, China}
\email{liuyanlxly@yahoo.com.cn}

\address{Quanyi Wang \newline
School of mathematical sciences, Huaqiao University, 
Quanzhou, Fujian 362021, China}
\email{qywang@hqu.edu.cn}

\thanks{Submitted May 28, 2008. Published December 3, 2008.}
\thanks{Supported by grant Z0511026 from the Natural Science
 Foundation of \hfill\break\indent Fujian Province of China}
\subjclass[2000]{34K45, 34K13, 92D25}
\keywords{Positive periodic solution; ratio-dependent;
 time delay; \hfill\break\indent
 predator-prey system; impulse; coincidence degree theory}

\begin{abstract}
 In this paper, we study a periodic ratio-dependent predator-prey
 system of two species with impulse and multiple time delays.
 By means of analysis techniques and the continuation theorem
 of coincidence degree theory, we obtain sufficient conditions
 for the existence of positive periodic solutions of the system.
 Our results extend  previous results obtained in \cite{l2}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

 The existence of positive periodic solutions of
predator-prey models has been extensively studied by many
mathematicians and biologists in recent years. Some authors have
already obtained many good conclusions, see \cite{c1,f2,w1,w2}.

However in many cases, especially when predators have to search, share
or compete for food, a more suitable general predator-prey model
should be based on the ratio-dependent theory. This roughly states
that the per capita predator growth rate should be a function of
the ratio of prey to predator abundance, see \cite{c2,f1}.

In addition, there are numerous examples of evolutionary systems
which at certain instants in time are subjected to rapid changes
(for example, those due to seasonal effects of weather, food supply,
hunting or harvesting seasons, etc). Those short-time perturbations
are often assumed to be in the form of impulses in the modelling
process. Consequently, impulsive differential equations provide a
natural description of such systems. Because equations of this kind
are found in many fields such as chemotherapy, population dynamics,
optimal control, ecology, biotechnology and physics, they have
attracted the interest of many researchers, see \cite{b1,l1,l2}
and the references cited therein.

For the above reasons, Liu and Li \cite{l2} considered the following
ratio-dependent predator-prey system with only one time delay and
impulsive effects
\begin{equation}
\begin{gathered}
x'=x(t)(b_1(t)-a_1(t)x(t)-\frac{c(t)y(t)}{m_1(t)y(t)+x(t)}),\quad
t\neq{t_k},\\
y'=y(t)(-b_2(t)+\frac{a_2(t)x(t-\tau)}{m_2(t)y(t-\tau)+x(t-\tau)}),\quad
t\neq{t_k},\\
x(t^+_k)-x(t^-_k)=c_kx(t_k),\\
y(t^+_k)-y(t^-_k)=d_ky(t_k),\quad t={t_k},\\
(x(0+),y(0+))=(x_0,y_0),\\
(x(t),y(t))=(\varphi_1(t),\varphi_2(t))>0,\quad {-\tau}\leq{t}<0,
\end{gathered}\tag{e1.1}\label{e1.1}
\end{equation}
where $x(t), y(t)$ represent the densities of prey and predator at
time $t$, respectively; $\tau$ is a positive constant time delay,
${b_1(t)}$, ${a_1(t)}$, ${m_1(t)}$, ${b_2(t)}$, ${c(t)}$,
${a_2(t)}$, ${m_2(t)}>0$
 are continuous T-periodic functions, $Z_+=\{1,2,\dots\}$.
The initial functions are
$\varphi(t)=(\varphi_1(t),\varphi_2(t))$,  where
$0<t_1<t_2<\dots<t_k<\dots$ and $\lim_{k\to\infty}t_k=+\infty$.
 Assume that ${c_k}$, ${d_k}$ $(k\in Z_+)$ are constants
and there exists an integer $q>0$ such that $c_{k+q}=c_k$,
$d_{k+q}=d_k$, $t_{k+q}=t_k+T$, $0<t_{k+1}-t_k<T$.  Liu and Li
\cite{l2} obtained the following result.

\noindent {\bf thmA}(\cite{l2}) \label{thmA} {\it Assume that the
following conditions hold:
\begin{gather*}
\overline{b_1}T+\ln(\prod_{k=1}^{q}(1+c_k))>\overline{({\frac{c}{m_1}})}T,\\
\overline{b_2}T-\ln(\prod_{k=1}^{q}(1+d_k))>0, \quad
\overline{a_2}>\overline{b_2},\quad T>\tau, \\
{a_2}^{l}(T-\tau)-\overline{b_2}T+\ln(\prod_{k=1}^{q}(1+d_k))>0,\\
{c}^{l}-{m_1}^{u}[\overline{b_1}+{\frac{1}{T}}\ln(\prod_{k=1}^{q}(1+c_k))]>0\,.
\end{gather*}
Then system ({e1.1}) has at least one positive T-periodic solution.}


However, this theorem is not valid because the condition
\[
{c}^{l}-{m_1}^{u}[\overline{b_1}+{\frac{1}{T}}\ln(\prod_{k=1}^{q}(1+c_k))]>0
\]
contradicts the condition
\[
\overline{b_1}T+\ln(\prod_{k=1}^{q}(1+c_k))>\overline{({\frac{c}{m_1}})}T
\]
because
\[
\frac{{c}^{l}}{{m_1}^{u}}\leq\overline{({\frac{c}{m_1}})}.
\]
Thus, the existence of a solution to ({e1.1}) has not been proved.
Moreover, there are also some mistakes in the course of the proof of
{thmA}, such as the computations of
$K_P(I-Q)N\begin{pmatrix}u(t)\\v(t)\end{pmatrix}$ (see \cite[p.
719]{l2}) and $QNX$ (see \cite[p. 722]{l2}).

In the actual environment, scientific researches suggest that time
delays  often occur in the course of the interaction of species in
many ecological systems. So, in the present paper, we study the
following two-species ratio-dependent predator-prey system with
multiple time delays and impulsive effects
\[
\begin{gathered}
{x'(t)}=x(t) \big[
b_1(t)-a_1(t)x(t-\tau_1(t))-\frac{c(t)y(t-\sigma_1(t))}
{m_1(t)y(t-\sigma_1(t))+x(t-\tau_2(t))}
\big],\quad t\neq{t_k},\\
{y'(t)}=y(t)\big[-b_2(t)+\frac{a_2(t) x(t-\tau_3(t))}
{m_2(t)y(t-\sigma_2(t))+x(t-\tau_3(t))}\big],
\quad t\neq{t_k},\\
x(t^+_k)-x(t^-_k)=c_{1k}x(t_k),\\
y(t^+_k)-y(t^-_k)=c_{2k}y(t_k),\quad t={t_k},\quad k=1,2,\dots,\\
(x(0+),y(0+))=(x_0,y_0),\\
(x(t),y(t))=(\varphi_1(t),\varphi_2(t))>0,\quad {-\tau}\leq{t}\leq0,
\end{gathered}\tag{e1.2}
\]
where $x(t),y(t)$ represent the densities of prey and predator at
time $t$, respectively; $a_1(t), a_2(t), b_1(t),b_2(t),c(t),m_1(t)$
and $ m_2(t)$ are all positive continuous $\omega$-periodic
functions; $\sigma_1(t),\sigma_2(t),\tau_1(t),\tau_2(t)$ and
$\tau_3(t)$ are all nonnegative continuous $\omega$-periodic
functions, $\tau=\max_{0\leq
t\leq\omega}\{\sigma_1(t),\sigma_2(t),\tau_1(t),
\tau_2(t),\tau_3(t)\}$. For the study of ({e1.2}), we always assume
that
\begin{itemize}
\item[(H1)] $\{{c_{ik}}\}$ is a real sequence and $1+c_{ik}>0$, $i=1,2$,
 $k=1,2,\dots$;
\item[(H2)] There exists an integer $q>0$ such that $c_{i(k+q)}=c_{ik}$,
$i=1,2$, $k=1,2,\dots$;
\item[(H3)] $0<t_1<t_2<\dots<t_q<\omega$ are fixed impulsive points in a
period and $t_{k+q}= t_k+\omega$, $k=1,2,\dots$.
\end{itemize}
In what follows, we shall use the following notation
\[
f^M=\max_{t\in[0,\omega]}f(t),\quad
f^l=\min_{t\in[0,\omega]}f(t),\quad
\overline{f}=\frac{1}{\omega}\int^\omega_0 f(t)dt,
\]
where $f(t)$ is a continuous $\omega$-periodic function.

\section{Preliminaries}

 In this section, we first introduce the continuation theorem
of coincidence degree theory \cite{g1}, which will be used in this paper.

Let $X, Z$ be real Banach spaces; let
$L:\mathop{\rm Dom}L\subset{X}\to{Z}$ be a linear mapping and
$N:{X}\to{Z}$ a continuous mapping. The mapping $L$ is
called a Fredholm mapping of index zero if
$\mathop{\rm dim}\ker L =\mathop{\rm codim\;Im}L<{+\infty}$ and
 $\mathop{\rm Im}L$ is closed in $Z$. If $L$ is
a Fredholm mapping of index zero and there exist continuous
projectors $P:{X}\to{X}$ and $Q:{Z}\to{Z}$ such that
$\mathop{\rm Im}P=\ker L$, $\ker Q=\mathop{\rm Im}L$,
$X=\ker L \oplus \ker P$ and
$Z=\mathop{\rm Im}L \oplus \mathop{\rm Im}Q$, then the
restriction $L_P$ of $L$ to $\mathop{\rm Dom}L\cap \ker P$
is one-to-one and onto $\mathop{\rm Im}L$, so that its
(algebraic) inverse
$K_P:\mathop{\rm Im}L \to \mathop{\rm Dom}L \cap \ker P$
is defined. Let $\Omega$ be an open bounded subset of $X$,
the mapping $N$ is called $L$-compact on $\overline{\Omega}$ if
 $QN:\overline{\Omega}\to{Z}$ and
$K_P(I-Q)N:\overline{\Omega}\to{X}$  are compact. Since
$\mathop{\rm Im}Q$ is isomorphic to $\ker L$, there exists an
isomorphism $J:\mathop{\rm Im}Q \to \ker L$.
The following results appears in \cite{g1}.


\noindent{\bf lem2.1}(\cite{g1})
 {\it Let $\Omega\subset X$ be an open bounded set. Let $L$ be a Fredholm
mapping of index zero and
$N:X\to Z $ a continuous operator which is $L$-compact on
$\overline{\Omega}$. Assume
\begin{itemize}
\item[(a)] for each $\lambda \in (0,1)$,
$x\in \partial \Omega\cap \mathop{\rm Dom} L$, $Lx\neq\lambda N x$;
\item[(b)] for each $x\in\partial\Omega\cap \ker  L,QNx\neq 0$;
\item[(c)] $\deg\{JQN,\Omega\cap \ker  L,0\}\neq 0$.
\end{itemize}
Then the operator equation $Lx=Nx$ has at least one solution in
$\overline\Omega\cap \mathop{\rm Dom}L$.}


To prove the main conclusion by means of the continuation
theorem, we need to introduce some function spaces.

Let $J_1\subset \mathbb{R}$, and
 $ PC(J_1,\mathbb{R})$ be the set of functions $\psi:{J_1}\to\mathbb{R}$ such that
$\psi(t)$ is continuous for $t\in J_1$,  $t\neq t_k$,  and is continuous
from the left for $t\in J_1$, and $\psi(t^+_k)$ exists for $k=1,2,\dots$.
Also define
\[
PC^1(J_1,\mathbb{R})=\{\psi:{J_1}\to\mathbb{R},\psi'\in PC(J_1,\mathbb{R})\}.
\]
Now we define
\begin{align*}
X&=\big\{u(t)=(u_1(t),u_2(t))^T:u_i(t)\in{PC([0, \omega],\mathbb{R})}\text{ for }
t\in[0, \omega], \\
&\quad u_i(t+\omega)=u_i(t) \text{ for } t\in \mathbb{R}, i=1,2\big\}
\end{align*}
 and
$Z=X\times{\mathbb{R}^{2q}}$,
where
\[
\mathbb{R}^{2q}=\underbrace{{\mathbb{R}^2}\times{\mathbb{R}^2}
\times{\dots}\times{\mathbb{R}^2}}_q.
\]
Denote
\begin{gather*}
\|u\|=\max\big\{\sup_{t\in{[0,\omega]}}|{u_1(t)}|,
\sup_{t\in{[0,\omega]}}|{u_2(t)}|\big\} \quad  \text{for } u\in{X}
\\
\|z\|=\|{u}\|+\sum_{k=1}^q{\|{r_k}\|}\quad \text{for }
z={(u,r_1,r_2,\dots,r_q)}\in{Z},
\end{gather*}
where
\[
r_k=\begin{pmatrix} r_{1k}\\r_{2k}\end{pmatrix}
 \in{\mathbb{R}^2}, \quad
\|{r_k}\|=\max\{|{r_{1k}}|,|{r_{2k}}|\}, \quad
k=1,2,\dots,q\,.
\]
Then $($X$,\|{\cdot}\|)$ and $($Z$,\|{\cdot}\|)$ are both Banach spaces.


\noindent{\bf def2.1}(\cite{b1}) \label{def2.1} \rm  {\it The set
$F\subset PC([0, \omega],\mathbb{R})$ is said to be
quasi-equicontinuous in $[0, \omega]$ if for any $\varepsilon>0$
there exists $\delta>0$ such that if $x\in F$, $k\in Z_+$, $t_1,
t_2\in(t_{k-1}, t_k)\cap[0, \omega]$, $|t_1-t_2|<\delta$, then
$|x(t_1)-x(t_2)|<\varepsilon$.}


\noindent{\bf lem2.2}(\cite{b1}) \label{lem2.2}
 {\it The set $F \subset PC([0, \omega],\mathbb{R})$ is relatively compact if and
only if
\begin{itemize}
\item[(1)] $F$ is bounded; that is,
$\|\psi\|=\sup\{|\psi|: t\in [0, \omega]\}\leq M$ for each $\psi\in F$
and some $M>0$;

\item[(2)] $F$ is quasi-equicontinuous in $[0, \omega]$.
\end{itemize}}


\section{Existence of positive $\omega$-periodic solutions}

 In this section, we demonstrate the existence of a positive
$\omega$-periodic solution of ({e1.2}).

\noindent {\bf thm3.1} \label{thm3.1} {\it Assume that {\rm
(H1)--(H3)} hold, and further assume the following conditions:
\begin{itemize}
\item[(1)] $\overline{b_1}\omega+\sum_{k=1}^{q}{\ln(1+c_{1k})}>
\overline{({\frac{c}{m_1}})}\omega$
\item[(2)] $a_2^l\omega+\sum_{k=1}^{q}{\ln(1+c_{2k})}>\overline{b_2}\omega$
\item[(3)] $\overline{b_2}\omega>\sum_{k=1}^{q}{\ln(1+c_{2k})} $
\end{itemize}
Then ({e1.2}) has at least one positive $\omega$-periodic solution.}


\begin{proof} Let ${x(t)}=e^{u_1(t)}$,
${y(t)}=e^{u_2(t)}$, then system ({e1.2}) can be rewritten as
\begin{equation}
\begin{gathered}
u_1'(t)=b_1(t)-a_1(t)e^{u_1(t-\tau_1(t))}
-\frac{c(t)e^{u_2(t-\sigma_1(t))}}
{m_1(t)e^{u_2(t-\sigma_1(t))}}+e^{u_1(t-\tau_2(t))},\quad
t\neq{t_k},\\
{u_2'(t)}=-b_2(t)+\frac{a_2(t)e^{u_1(t-\tau_3(t))}}
{m_2(t)e^{u_2(t-\sigma_2(t))}+e^{u_1(t-\tau_3(t))}},\quad
t\neq{t_k},\\
\Delta{u_1(t_k)}={u_1(t^+_k)}-{u_1(t_k)}=\ln(1+c_{1k}),\\
\Delta{u_2(t_k)}={u_2(t^+_k)}-{u_2(t_k)}=\ln(1+c_{2k}), \quad k=1, 2,\dots.
\end{gathered}\tag{e3.1}
\end{equation}
For the sake of simplicity, we denote
\begin{gather*}
f_1(t,u(t))=b_1(t)-a_1(t)e^{u_1(t-\tau_1(t))}-
\frac{c(t)e^{u_2(t-\sigma_1(t))}}{m_1(t)e^{u_2(t-\sigma_1(t))}
+e^{u_1(t-\tau_2(t))}},
\\
f_2(t,u(t))=-b_2(t)+ \frac{a_2(t)e^{u_1(t-\tau_3(t))}}
{m_2(t)e^{u_2(t-\sigma_2(t))}+e^{u_1(t-\tau_3(t))}},
\\
 \Delta{u(t_k)}={u(t^+_k)}-{u(t_k)}
=\begin{pmatrix}\Delta{u_1(t_k)}\\ \Delta{u_2(t_k)}\end{pmatrix},
 \quad k=1,2,\dots,q,
\\
 u(t)=(u_1(t), u_2(t))^T,\quad
 C_{1k}=\ln(1+c_{1k}),\quad
 C_{2k}=\ln(1+c_{2k}).
\end{gather*}
It is obvious that if system ({e3.1}) has an $\omega$-periodic
solution $u^*(t)=(u^*_1(t), u^*_2(t))^T$, then
$(x^*(t),y^*(t))^T=(e^{u^*_1(t)}, e^{u^*_2(t)})^T$ is a positive
$\omega$-periodic solution of system ({e1.2}). So, to complete the
proof, it suffices to show that the system ({e3.1}) has one
$\omega$-periodic solution.

To apply {lem2.1} for establishing the existence of
$\omega$-periodic solutions of system ({e3.1}), now let
\[
\mathop{\rm Dom}L=\{u(t)=(u_1(t), u_2(t))^T \in X: (u_1(t), u_2(t))^T\in
PC^1([0, \omega], \mathbb{R})\},
\]
 and take $L:\mathop{\rm Dom}L$$\subset{X}\to{Z}$ as follows:
\[
{u}\to{(u',\triangle{u(t_1)},\dots,
\triangle{u(t_q)})},
\]
 and define $N: X\to{Z}$ by
\[
Nu= \Bigg(\begin{pmatrix}  f_1(t,u(t))\\ f_2(t,u(t))  \end{pmatrix},
  \begin{pmatrix}   C_{11}\\ C_{21} \end{pmatrix},
  \dots,\begin{pmatrix}  C_{1q}\\ C_{2q} \end{pmatrix}\Bigg)
\] for $u=(u_1,u_2)^T\in X$.
 Evidently, we have
\begin{gather*}
\ker L=\{u:{u}\in{X},u=c\in{\mathbb{R}^2}\}, \\
\mathop{\rm Im}L=\big\{z=(u,r_1,r_2,\dots,r_q)\in{Z}:
\frac{1}{\omega}\big(\int_{0}^{\omega}u(t)dt+\sum_{k=1}^{q}{r_k}\big)=0
\big\}.
\end{gather*}
So, $\mathop{\rm Im}L$ is closed in $Z$, and
$\dim\ker L=2=\mathop{\rm codim\;Im}L$.  Hence, $L$ is a
Fredholm mapping of index zero.

Set two projectors $P:{X}\to{X}$ and $Q:{Z}\to{Z}$
as follows:
\begin{gather*}
Pu=\frac{1}{\omega}\int_{0}^{\omega}u(t)dt, \quad
(\forall{u}={(u_1,u_2)^T}\in{X}),
\\
Qz=Q(u,r_1,\dots,r_q)=\Big( \frac{1}{\omega}
\big(\int_{0}^{\omega}u(t)dt+\sum_{k=1}^{q}{r_k}\big),0,0,
\dots,0 \Big),\\
 (\forall{z}={(u,r_1,r_2,\dots,r_q)}\in{Z})\,.
\end{gather*}
It is easy to see that $P$ and $Q$ are continuous
projectors, such that
\begin{gather*}
\mathop{\rm Im}P=\ker L,\quad  \ker Q=\mathop{\rm Im}L, \\
X=\ker L \oplus \ker P,\quad Z=\mathop{\rm Im} L\oplus
\mathop{\rm Im} Q.
\end{gather*}
Furthermore, through an easy computation, we find that the
inverse $K_P$ of $L_P$ (the restriction of $L$ to
 ${\mathop{\rm Dom}L}\cap{\ker P}$) has the form
$K_P:\mathop{\rm Im}L \to{{\mathop{\rm Dom}L}\cap{\ker P}}$,
\[
K_P(z(t))=\int_{0}^{t}u(s)ds+\sum_{0<t_k<t}r_k
-\frac{1} {\omega}
\big[\int_{0}^{\omega}
\int_{0}^{t}u(s)dsdt+\sum_{k=1}^{q}r_k(\omega-t_k)\big]
\]
for  $ z={(u,r_1,r_2,\dots,r_q)}\in{Z}$.
Accordingly, $QN:{X}\to{Z}$ and $K_P(I-Q)N:{X}\to{X}$ read
\[
QNu=\left(\begin{pmatrix}
     \frac{1}{\omega}\big(\int_{0}^{\omega}f_1(s,u(s)\big)ds
 +\sum_{k=1}^{q}C_{1k})\\
 \frac{1}{\omega}
   \big(\int_{0}^{\omega}f_2(s,u(s))ds+\sum_{k=1}^{q}C_{2k}\big)
   \end{pmatrix}
,0,0,\dots,0 \right),
\]
and
\begin{align*}
K_P(I-Q)Nu &=\begin{pmatrix}
\int_{0}^{t}f_1(s,u(s))ds+\sum_{0<t_k<t}C_{1k}\\
\int_{0}^{t}f_2(s,u(s))ds+\sum_{0<t_k<t}C_{2k}
\end{pmatrix} \\
 &\quad -\frac{1}{\omega}\begin{pmatrix}
\int_{0}^{\omega}\int_{0}^{t}f_1(s,u(s))dsdt+
\sum_{k=1}^{q}C_{1k}(\omega-t_k)\\
\int_{0}^{\omega}\int_{0}^{t}f_2(s,u(s))dsdt
+\sum_{k=1}^{q}C_{2k}(\omega-t_k)\end{pmatrix}\\
&\quad-(\frac{t}{\omega}-\frac{1}{2})
\begin{pmatrix}
\int_{0}^{\omega}f_1(s,u(s))ds+\sum_{k=1}^{q}C_{1k}\\
\int_{0}^{\omega}f_2(s,u(s))ds+\sum_{k=1}^{q}C_{2k}
\end{pmatrix}.
\end{align*}
Using the Lebesgue convergence theorem, it is
easy to see that $QN$ and $K_P(I-Q)N$  are continuous. Moreover,
from the expressions of  $QNu$ and $K_P(I-Q)Nu$, it is easy to see
that $QN(\overline \Omega)$ and $K_P(I-Q)N(\overline \Omega)$ are
bounded for any open bounded set $\Omega\subset{X}$. Furthermore, we
have that
\[
\frac{d}{dt}(QNu)=(0, 0, \dots, 0), \quad t\neq t_k,\; k=1,2,\dots
\]
and
\[
\frac{d}{dt}(K_P(I-Q)Nu) =\begin{pmatrix}
f_1(t,u(t))-\frac{1}{\omega}(\int_{0}^{\omega}f_1(s,u(s))ds
 +\sum_{k=1}^{q}C_{1k})\\
f_2(t,u(t))-\frac{1}{\omega}(\int_{0}^{\omega}f_2(s,u(s))ds
 +\sum_{k=1}^{q}C_{2k})\end{pmatrix}
\]
for $t\neq t_k$, $k=1,2,\dots$, and ${u}\in{X}$. It follows from
these expressions that the sets $\{\frac{d}{dt}(QNu):u\in{\overline
\Omega}\}$  and $\{\frac{d}{dt}(K_P(I-Q)Nu): u\in{\overline
\Omega}\}$ are bounded. So we have that $QN(\overline \Omega)$ and
$K_P(I-Q)N(\overline \Omega)$ are equi-continuous in $[0,\omega]$.
It follows from {lem2.2} that $QN(\overline \Omega)$ and
$K_P(I-Q)N(\overline \Omega)$ are compact.  Therefore $N$ is
$L$-compact on $\overline \Omega$.

 Corresponding to the operator equation
 $Lu=\lambda{Nu}$ with  $\lambda\in(0,1)$, we have
\begin{equation}
\begin{gathered}
{u_1'(t)}=\lambda\big[
b_1(t)-a_1(t)e^{u_1(t-\tau_1(t))}-\frac{c(t)e^{u_2(t-\sigma_1(t))}}
{m_1(t)e^{u_2(t-\sigma_1(t))}+e^{u_1(t-\tau_2(t))}}\big],\quad
t\neq{t_k},\\
{u_2'(t)}=\lambda\big[-b_2(t)+\frac{a_2(t)e^{u_1(t-\tau_3(t))}}
{m_2(t)e^{u_2(t-\sigma_2(t))}+e^{u_1(t-\tau_3(t))}}\big],\quad
t\neq{t_k},\\
\Delta{u_1(t_k)}=\lambda{\ln(1+c_{1k})},\\
\Delta{u_2(t_k)}=\lambda{\ln(1+c_{2k})},\quad  k=1, 2,\dots.\\
\end{gathered}\tag{e3.2}\label{e3.2}
\end{equation}

 Suppose that $u(t)={(u_1(t),u_2(t))^T}\in{X}$ is an
$\omega$-periodic solution of ({e3.2}) for a certain
$\lambda\in(0,1)$. Integrating ({e3.2}) over the interval [0,
$\omega$], we obtain
\[
\int_{0}^{\omega}\big[
b_1(t)-a_1(t)e^{u_1(t-\tau_1(t))}-\frac{c(t)e^{u_2(t-\sigma_1(t))}}
{m_1(t)e^{u_2(t-\sigma_1(t))}+e^{u_1(t-\tau_2(t))}}\big]dt
=-\sum_{k=1}^{q}\ln(1+c_{1k}),
\]
\[
\int_{0}^{\omega}\big[-b_2(t)+\frac{a_2(t)e^{u_1(t-\tau_3(t))}}
{m_2(t)e^{u_2(t-\sigma_2(t))}+e^{u_1(t-\tau_3(t))}}\big]dt
=-\sum_{k=1}^{q}\ln(1+c_{2k}),
\]
which yield
\begin{equation}
\begin{aligned}
&\int_{0}^{\omega}\big[
a_1(t)e^{u_1(t-\tau_1(t))}+\frac{c(t)e^{u_2(t-\sigma_1(t))}}
{m_1(t)e^{u_2(t-\sigma_1(t))}+e^{u_1(t-\tau_2(t))}}\big]dt\\
&=\int_{0}^{\omega}b_1(t)dt+\sum_{k=1}^{q}\ln(1+c_{1k}),
\end{aligned}\tag{e3.3}\label{e3.3}
\end{equation}
\begin{equation}
 \int_{0}^{\omega}\frac{a_2(t)e^{u_1(t-\tau_3(t))}}
{m_2(t)e^{u_2(t-\sigma_2(t))}+e^{u_1(t-\tau_3(t))}}dt
=\int_{0}^{\omega}b_2(t)dt -\sum_{k=1}^{q}\ln(1+c_{2k}).
\tag{e3.4}\label{e3.4}
\end{equation}
In view of ({e3.2}), ({e3.3}) and ({e3.4}), we have
\begin{gather}
{\int_{0}^{\omega}|{u_1}'(t)| dt}
\leq {2\int_{0}^{\omega}{b_1(t)}dt +\sum_{k=1}^{q}\ln(1+c_{1k})}
= 2\overline{b_1}\omega+\sum_{k=1}^{q}\ln(1+c_{1k}),
\tag{e3.5}\label{e3.5} \\
{\int_{0}^{\omega}|{u_2}'(t)|dt} \leq {2\int_{0}^{\omega}b_2(t)dt
-\sum_{k=1}^{q}\ln(1+c_{2k})} = 2\overline{b_2}\omega-
\sum_{k=1}^{q}\ln(1+c_{2k}). \tag{e3.6}\label{e3.6}
\end {gather}
Since $u(t)={(u_1(t),u_2(t))^T}\in{X}$, there exist
${\eta_i, \xi_i}\in{[0,\omega]}$, $(i=1,2)$ such that
\begin{gather*}
 u_i(\eta^+_i)=\sup_{t\in[0,\omega]}u_i(t)\quad
 \text{or}\quad u_i(\eta^-_i)=\sup_{t\in[0,\omega]}u_i(t),\\
u_i(\xi^+_i)=\inf_{t\in[0,\omega]}u_i(t)\quad
\text{or}\quad
u_i(\xi^-_i)=\inf_{t\in[0,\omega]}u_i(t),\quad  (i=1,2).
\end{gather*}
Whichever they are, for the sake of simplicity, we can
denote them as follows:
\[
u_i(\eta_i)=\sup_{t\in[0,\omega]}u_i(t), \quad
u_i(\xi_i)=\inf_{t\in[0,\omega]}u_i(t),\quad (i=1,2).
\]
Furthermore, it follows from ({e3.3}) that
\[
\overline{b_1}\omega+\sum_{k=1}^{q}\ln(1+c_{1k}) \geq
\int_{0}^{\omega}a_1(t)e^{u_1(t-\tau_1(t))}dt
 \geq\overline{a_1}\omega e^{u_1(\xi_1)},
\]
\begin{align*}
&\overline{b_1}\omega+\sum_{k=1}^{q}\ln(1+c_{1k})\\
&= \int_{0}^{\omega}a_1(t)e^{u_1(t-\tau_1(t))}dt+\int_{0}^{\omega}
\frac{c(t)e^{u_2(t-\sigma_1(t))}}
{m_1(t)e^{u_2(t-\sigma_1(t))}+e^{u_1(t-\tau_2(t))}}dt\\
&\leq\overline{a_1}\omega
e^{u_1(\eta_1)}+\overline{(\frac{c}{m_1})}\omega.
\end{align*}
So, from the condition (1), we have
\begin{gather}
u_1(\xi_1)\leq\ln{\frac{\overline{b_1}\omega+\sum_{k=1}^{q}\ln(1+c_{1k})}
{\overline{a_1}\omega}}=:H_1, \tag{e3.7}\label{e3.7}
\\
u_1(\eta_1)\geq\ln{\frac{\overline{b_1}\omega-\overline{(\frac{c}{m_1})}
\omega +\sum_{k=1}^{q}\ln(1+c_{1k})}{\overline{a_1}\omega}}=:H_2.
\tag{e3.8}\label{e3.8}
\end{gather}
Hence, ({e3.5})--({e3.8}) yield
\begin{equation}
\begin{aligned}
u_1(t)
&\leq u_1(\xi_1)+{\int_{0}^{\omega}|{u_1}'(t)|dt}
 +\sum_{k=1}^{q}|{\ln(1+c_{1k})}|\\
&\leq H_1+2{\overline{b_1}}\omega+\sum_{k=1}^{q}
\big[\ln(1+c_{1k}) +|{\ln(1+c_{1k})}|\big]
=:H_3,
\end{aligned}\tag{e3.9}\label{e3.9}
\end{equation}
\begin{equation}
\begin{aligned}
u_1(t)
&\geq  u_1(\eta_1)-{\int_{0}^{\omega}|{u_1}'(t)|dt}
  -\sum_{k=1}^{q}|{\ln(1+c_{1k})}| \\
&\geq H_2-2{\overline{b_1}}\omega-\sum_{k=1}^{q}\big[
\ln(1+c_{1k}) +|{\ln(1+c_{1k})}|\big]
 =:H_4.\end{aligned}\tag{e3.10}\label{e3.10}
\end{equation}
It follows from the two equations above  that
\begin{equation}
\sup_{t\in[0,\omega]}|{u_1(t)}| \leq\max\{| H_3 |, | H_4 |\}=:K_1.
\tag{e3.11}\label{e3.11}
\end{equation}
On the other hand, from ({e3.4}), we can easily get
\begin{equation}
\overline{b_2}\omega-\sum_{k=1}^{q}\ln(1+c_{2k}) \leq
\int_{0}^{\omega}\frac{a_2(t)e^{u_1(\eta_1)}}
{m_2(t)e^{u_2(\xi_2)}+e^{u_1(\eta_1)}}dt
\leq\frac{a^M_2e^{u_1(\eta_1)}\omega}{m^l_2e^{u_2(\xi_2)}
+e^{u_1(\eta_1)}}, \tag{e3.12} \label{e3.12}
\end{equation}
and
\begin{equation}
\overline{b_2}\omega-\sum_{k=1}^{q}\ln(1+c_{2k})
\geq \int_{0}^{\omega}\frac{a_2(t)e^{u_1(\xi_1)}}
{m_2(t)e^{u_2(\eta_2)}+e^{u_1(\xi_1)}}dt\\
\geq \frac{a^l_2e^{u_1(\xi_1)}\omega}{m^M_2e^{u_2(\eta_2)}
+e^{u_1(\xi_1)}}. \tag{e3.13}\label{e3.13}
\end{equation}
Thus, from ({e3.11})--({e3.13}) and the conditions (2) and (3), one
has
\begin{gather}
u_2(\xi_2) \leq \ln\frac{ \big[ a^M_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k}) \big]e^{K_1}}{m^l_2 \big[
\overline b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]} =H_5+K_1,
\tag{e3.14}\label{e3.14}
\\
u_2(\eta_2) \geq \ln\frac{\big[a^l_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]e^{-K_1}}
{m^M_2\big[\overline b_2\omega- \sum_{k=1}^{q}\ln(1+c_{2k})\big]}
=H_6-K_1, \tag{e3.15}\label{e3.15}
\end{gather}
where
\begin{gather*}
H_5=\ln\frac{\big[a^M_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]}
{m^l_2 \big[ \overline b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]},
\\
H_6=\ln\frac{\big[a^l_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]}
{m^M_2\big[\overline
b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]}.
\end{gather*}
Furthermore, from  ({e3.6}), ({e3.14}) and ({e3.15}), it follows
that
\begin{equation}
\begin{aligned}
u_2(t)
&\leq  {u_2(\xi_2)}+{\int_{0}^{\omega}|{u_2}'(t)|
dt}+\sum_{k=1}^{q}|{\ln(1+c_{2k})}| \\
&\leq H_5+K_1+2\overline {b_2}\omega-\sum_{k=1}^{q}{\ln(1+c_{2k})}
+\sum_{k=1}^{q}|{\ln(1+c_{2k})}|
=:H_7,\end{aligned}\tag{e3.16}\label{e3.16}
\end{equation}
\begin{equation}
\begin{aligned}
u_2(t)
&\geq{u_2(\eta_2)}-{\int_{0}^{\omega}|{u_2}'(t)|
dt}-\sum_{k=1}^{q}|{\ln(1+c_{2k})}| \\
&\geq H_6-K_1-2\overline b_2\omega+\sum_{k=1}^{q}{\ln(1+c_{2k})}
-\sum_{k=1}^{q}|{\ln(1+c_{2k})}| =:H_8.
\end{aligned}\tag{e3.17}\label{e3.17}
\end{equation}
Therefore,
\begin{equation}
\sup_{t\in[0,\omega]}|{u_2(t)}| \leq\max\{| H_7 |, | H_8 |\}=:K_2.
\tag{e3.18}\label{e3.18}
\end{equation}
Set $ K=1+K_1+K_2+|{H_1}|+|{H_2}|+|{H_5}|+|{H_6}|$. Clearly, $K$ is
independent of $\lambda$ $(\lambda\in(0,1))$. Then it follows from
({e3.11}) and ({e3.18}) that
\begin{equation}
\|{u}\|\leq{K}. \tag{e3.19}\label{e3.19}
\end{equation}
 Suppose $u=(u_1,u_2)^T\in{\mathbb{R}^2}$. Then from the expression of
$QNu$ , we  obtain
\begin{equation}
\begin{aligned}
&QN \begin{pmatrix} u_1\\ u_2\end{pmatrix}\\
&=\Bigg( \begin{pmatrix}
 \overline b_1-\overline a_1e^{u_1}
-\frac{1}{\omega}\int_{0}^{\omega}\frac{c(t)e^{u_2}}
{m_1(t)e^{u_2}+e^{u_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{1k})\\
-\overline b_2+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u_1}}
{m_2(t)e^{u_2}+e^{u_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})
\end{pmatrix}
,0,\dots,0
\Bigg).
\end{aligned}\tag{e3.20}\label{e3.20}
\end{equation}
Consider the  equation
\begin{equation}
\begin{gathered}
 {\overline{ b_1}-\overline {a_1}e^{u_1}-\frac{1}{\omega}
\int_{0}^{\omega}\frac{c(t)e^{u_2}}{m_1(t)e^{u_2}+e^{u_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{1k})=0,} \\
{-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u_1}}
{m_2(t)e^{u_2}+e^{u_1}}dt+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0.}
\end{gathered}\tag{e3.21}\label{e3.21}
\end{equation}
 By  analysis  similar to the one for ({e3.7}), ({e3.8}),
({e3.14}) and ({e3.15}), it is not difficult to see that any
solution $u^*=(u^*_1,u^*_2)^T\in{\mathbb{R}^2}$ of  ({e3.21})
exists, it must satisfy: $H_2\leq{u^*_1}\leq H_1$,
\begin{align*}
u^*_2 &\leq  \ln\frac{\big[ a^M_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]e^{u^*_1}}
{m^l_2\big[\overline b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]}\\
&=\ln\frac{\big[a^M_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]}
{m^l_2\big[\overline b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]}+u^*_1\\
&=H_5+H_1,
\end{align*}
\begin{align*}
u^*_2 &\geq \ln\frac{\big[ a^l_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]e^{u^*_1}}
{m^M_2\big[\overline b_2\omega
-\sum_{k=1}^{q}\ln(1+c_{2k})\big]}\\
&=\ln\frac{\big[a^l_2\omega-\overline
b_2\omega+\sum_{k=1}^{q}\ln(1+c_{2k})\big]}
{m^M_2\big[\overline
b_2\omega-\sum_{k=1}^{q}\ln(1+c_{2k})\big]}+u^*_1\\&=H_6+H_2,
\end{align*}
which yield
\begin{equation}
\|{u^*}\|\leq K. \tag{e3.22}\label{e3.22}
\end{equation}
Put $\Omega=\{u=(u_1,u_2)^T\in X:\|{u}\|<{K_0}\}$,  where $K_0=K+1$.
Then it follows from ({e3.19}) that condition (a) of {lem2.1} is
satisfied.

 Furthermore, for each
$u=(u_1,u_2)^T\in{\partial\Omega}\cap{\ker L} ={\partial\Omega}\cap{\mathbb{R}^2}$,
we know that $u=(u_1,u_2)^T$ is a constant
 vector in $\mathbb{R}^2$ with $\|{u}\|=K_0$, and then can
 directly get $QNu\neq0$ by ({e3.20})-({e3.22}).
This shows that condition (b) of {lem2.1} is satisfied.

 Finally, let us prove that the condition (c) of {lem2.1} is satisfied.
Define
 $\phi:\overline \Omega\cap{\ker L}\times[0,1]\to{\mathbb{R}^2}$ by
\begin{align*}
\phi(u_1,u_2,\eta)
&=\begin{pmatrix}
 \overline b_1-\overline{a_1}e^{u_1}
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{1k})\\
-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u_1}}
{m_2(t)e^{u_2}+e^{u_1}}dt+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})\\
\end{pmatrix} \\
&\quad +\eta \begin{pmatrix}
   -\frac{1}{\omega}
\int_{0}^{\omega}\frac{c(t)e^{u_2}}{m_1(t)e^{u_2}+e^{u_1}}dt\\
0 \end{pmatrix},
\end{align*}
where $u=(u_1,u_2)^T\in{\overline\Omega}\cap{\ker L}$,
$\eta\in[0,1]$. First, we will prove that $\phi(u_1,u_2,\eta)\neq0$
when $u=(u_1,u_2)^T\in{\partial\Omega}\cap{\ker L},
\eta\in[0,1]$. Assume the conclusion is not true, then there exists
a constant vector $u=(u_1,u_2)^T$ with $\|{u}\|=K_0$, such that
$\phi(u_1,u_2,\eta)=0$ for a certain $\eta\in[0,1]$; i.e.,
\begin{equation}
\begin{gathered}
 {\overline{ b_1}-\overline {a_1}e^{u_1}+\frac{1}{\omega}
 \sum_{k=1}^{q}\ln(1+c_{1k})
 -\frac{\eta}{\omega}
\int_{0}^{\omega}\frac{c(t)e^{u_2}}{m_1(t)e^{u_2}+e^{u_1}}dt
=0,} \\
{-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u_1}}
{m_2(t)e^{u_2}+e^{u_1}}dt+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0.}
\end{gathered}\tag{e3.23}\label{e3.23}
\end{equation}
 By a similar discussion on the solutions of ({e3.21}), it
is easy to see that any solution $u=(u_1,u_2)^T\in{\mathbb{R}^2}$ of
({e3.23}) must satisfy
\begin{equation}
\|{u}\|\leq{K}<{K_0}, \tag{e3.24}\label{e3.24}
\end{equation}
 which contradicts $\|{u}\|=K_0$ ($\forall
u\in{\partial\Omega}\cap{\ker L}$). This shows that
$\phi(u_1,u_2,\eta)\neq0$ when $u=(u_1,u_2)^T\in
 {\partial\Omega}\cap{\mathbb{R}^2}$ and $\eta\in[0,1]$.
We next prove that the equation
$\phi(u_1,u_2,0)=0$ in ${\Omega}\cap{\ker L}$ has a unique
solution $u^{**}=(u^{**}_1,u^{**}_2)^T$. In fact,
$\phi(u^{**}_1,u^{**}_2,0)=0$ means
\begin{equation}
\begin{gathered}
 {\overline{ b_1}-\overline {a_1}e^{u^{**}_1}
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{1k})=0,} \\
-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u^{**}_1}}
{m_2(t)e^{u^{**}_2}+e^{u^{**}_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0;\end{gathered}\tag{e3.25}\label{e3.25}
\end{equation}
 that is,
\begin{equation}
\begin{gathered}
{u^{**}_1 =\ln
\frac{\overline{b_1}+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{1k})}
{\overline {a_1}}},\\
{-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u^{**}_1}}
{m_2(t)e^{u^{**}_2}+e^{u^{**}_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0.}
\end{gathered}\tag{e3.26}\label{e3.26}
\end{equation}
For $u_2\in \mathbb{R}$, let
\begin{equation}
g(u_2)={-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}
\frac{a_2(t)e^{u^{**}_1}}
{m_2(t)e^{u_2}+e^{u^{**}_1}}dt+\frac{1}{\omega}
\sum_{k=1}^{q}\ln(1+c_{2k})}, \tag{e3.27}\label{e3.27}
\end{equation}
 then ${g(u_2)}\in C^1(\mathbb{R},\mathbb{R})$ and
\[
g'(u_2)=-\frac{1}{\omega}\int_{0}^{\omega}
\frac{a_2(t)m_2(t)e^{u^{**}_1}e^{u_2}}
{\big[m_2(t)e^{u_2}+e^{u^{**}_1}\big]^2}dt<0;
\]
 that is,  $g(u_2)$ is strictly monotonous decreasing in
$\mathbb{R}$. From the conditions (2) and (3), it is easy to obtain
\begin{gather*}
g(+\infty)=-\overline{b_2}+\frac{1}{\omega}
\sum_{k=1}^{q}\ln(1+c_{2k})<0,\\
g(-\infty)=\overline{a_2}-\overline{b_2}+\frac{1}{\omega}
\sum_{k=1}^{q}\ln(1+c_{2k})>0.
\end{gather*}
 Therefore, there exists a unique ${u^{**}_2}\in \mathbb{R}$ such
 that $g(u^{**}_2)=0$; that is,
 \begin{equation}
-\overline{b_2}+\frac{1}{\omega}\int_{0}^{\omega}\frac{a_2(t)e^{u^{**}_1}}
{m_2(t)e^{u^{**}_2}+e^{u^{**}_1}}dt
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0.\tag{e3.28}
\label{e3.28}
\end{equation}
 Then  by integral mean value theorem, there
exists a ${t_0}\in[0,\omega]$ such that
\begin{equation}
-\overline{b_2}+\frac{a_2(t_0)e^{u^{**}_1}}
{m_2(t_0)e^{u^{**}_2}+e^{u^{**}_1}}
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})=0.\tag{e3.29}\label{e3.29}
\end{equation}
 It follows from ({e3.29}) and  conditions (2), (3) that
\begin{equation}
{u^{**}_2}=\ln\frac{\big[ a_2(t_0)-\overline{b_2}
+\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})\big]}
{m_2(t_0)\big[\overline{b_2}
-\frac{1}{\omega}\sum_{k=1}^{q}\ln(1+c_{2k})\big]} +u^{**}_1.
\tag{e3.30}\label{e3.30}
\end{equation}
 Thus, from ({e3.24}) and ({e3.30}), we obtain
\[
H_2\leq{u^{**}_1}\leq H_1,\quad {H_2+H_6}\leq{u^{**}_2}\leq
{H_1+H_5},
\]
 which imply
\[
\|{u^{**}}\|\leq K<{K_0};
\]
i.e., the equation $\phi(u_1,u_2,0)=0$ in
${\Omega}\cap{\ker L}$ has a unique solution
$u^{**}=(u^{**}_1,u^{**}_2)^T$.

Now define the isomorphism $J:\mathop{\rm Im}Q\to${$\ker L$}  by
\[
J\Big( \frac{1}{\omega}
\Big(\int_{0}^{\omega}u(t)dt +\sum_{k=1}^{q}{r_k}\Big),0,0,\dots,0\Big)
 =\frac{1}{\omega}
\Big(\int_{0}^{\omega}u(t)dt+
\sum_{k=1}^{q}{r_k}\Big),
\]
then  $JQNu=\phi(u_1,u_2,1)$ for each
$u=(u_1,u_2)^T\in{\overline\Omega}\cap{\ker L}$.
  Using the property of topological degree,
from ({e3.25}) and ({e3.29}), we have
\begin{align*}
&\deg\{JQN,\Omega\cap\ker L,0\}\\
&=\deg\{\phi(u_1,u_2,1),\Omega\cap\ker L,0\}\\
&=\deg\{\phi(u_1,u_2,0),\Omega\cap\ker L ,0\}\\
&=\text{sign}\left|
\begin{matrix}
-\overline{a_{1}}e^{u^{**}_1} &  0  \\
\frac{a_2(t_0)m_2(t_0)e^{u^{**}_1}e^{u^{**}_2}}
{(m_2(t_0)e^{u^{**}_2}+e^{u^{**}_1})^2}&
 -\frac{a_2(t_0)m_2(t_0)e^{u^{**}_1}e^{u^{**}_2}}
{(m_2(t_0)e^{u^{**}_2}+e^{u^{**}_1})^2}  \\
 \end{matrix}\right|
\neq 0.
\end{align*}
 Thus,  condition (c) of {lem2.1} holds and by now we
have proved that all the conditions of {lem2.1} are satisfied.
Hence, system ({e3.1}) has at least one $\omega$-periodic solution.
Accordingly, system ({e1.2}) has at least one positive
$\omega$--periodic solution. This completes the proof.
\end{proof}

If we set $\sigma_1(t)=\tau_1(t)=\tau_2(t)=0$,
$\tau_3(t)=\sigma_2(t)=\tau$, $\omega=T$, then system ({e1.2}) is
simplified to system ({e1.1}) which was studied by Liu and Li in
\cite{l2}.

Obviously, our result in this paper
extends and improves greatly the result in \cite{l2}.
Finally, let us consider the  system without impulse
\begin{equation}
\begin{gathered}
{x'(t)}=x(t)\big[b_1(t)-a_1(t)x(t-\tau_1(t))-\frac{c(t)y(t-\sigma_1(t))}
{m_1(t)y(t-\sigma_1(t))+x(t-\tau_2(t))}\big],\\
{y'(t)}=y(t)\big[-b_2(t)+\frac{a_2(t)x(t-\tau_3(t))}
{m_2(t)y(t-\sigma_2(t))+x(t-\tau_3(t))}\big],
\end{gathered}\tag{e3.31}\label{e3.31}
\end{equation}
 where $ a_1(t)$, $a_2(t)$, $b_1(t)$, $b_2(t)$, $c(t)$,
$m_1(t)$ and $ m_2(t)$ are all positive continuous $\omega$-periodic
functions; $\sigma_1(t)$, $\sigma_2(t)$, $\tau_1(t)$, $\tau_2(t)$
and $\tau_3(t)$ are all continuous $\omega$-periodic functions. From
\ref{thm3.1} and its proof, we immediately get the following result.

\noindent {\bf{thm3.2}}  \label{thm3.2} {\it If system ({e3.31})
satisfies the  conditions
 \[
\overline{b_1}>\overline{(\frac{c}{m_1})},\quad a^l_2>\overline{b_2},
\]
then ({e3.31})
 has at least one positive $\omega$-periodic solution.}


\section{An example}

 In this section, we give an example that illustrates the
feasibility of our results. Consider the system
\begin{equation}
\begin{gathered}
{x'(t)}=x(t)\big[(\frac{1}{2}+|\cos t|)-x(t-|\sin
t|) -\frac{(\frac{1}{2}|\sin t|+\frac{1}{4}) y(t-|\cos{t}|)}
{y(t-|\cos{t}|)+x(t-|\sin{2t}|)}\big],\; t\neq{t_k},\\
{y'(t)}=y(t)\big[-(\frac{3}{4}+\frac{1}{2}\sin t)
+\frac{\frac{4}{\pi}x(t-|\sin{3t}|)}{2y(t-|\cos2t|)
+x(t-|\sin{3t}|)}\big],\quad t\neq{t_k},\\
{x(t^+_k)}-{x(t^-_k)}=c_{1k}{x(t_k)},\\
{y(t^+_k)}-{y(t^-_k)}=c_{2k}{y(t_k)},\quad
k=1,2,\dots,\end{gathered} \tag{4.1} \label{e4.1}
\end{equation}
where
\begin{gather*}
t_1=\frac{\pi}{2}, \quad t_2=\frac{3\pi}{2}, \quad t_{k+2}=t_k+2\pi,
\quad c_{11}=1,\quad c_{12}=-\frac{1}{2}, \\
c_{21}=1,\quad  c_{22}=-\frac{3}{4},
\quad c_{i(k+2)}=c_{ik}, \quad i=1,2,
\quad k=1,2,\dots.
\end{gather*}
 Corresponding to ({e1.2}), we have
\begin{gather*}
\omega=2\pi,\quad  a_1(t)=1,\quad
a_2(t)=\frac{4}{\pi},\quad  b_1(t)=\frac{1}{2}+|\cos
t|,\quad  b_2(t)=\frac{3}{4}+\frac{1}{2}\sin t, \\
c(t)=\frac{1}{2}|\sin t|+\frac{1}{4},\quad
m_1(t)=1,\quad  m_2(t)=2,\quad  \sigma_1(t)=|\cos t|,\\
\sigma_2(t)=|\cos 2t|,\quad
 \tau_1(t)=|\sin t|,\quad  \tau_2(t)=|\sin 2t|,\quad
\tau_3(t)=|\sin 3t|.
\end{gather*}
It is easy to obtain that,
\begin{gather*}
\overline{b_1}=\frac{1}{2\pi}\int_{0}^{2\pi} (\frac{1}{2}+|\cos
t|)dt =\frac{2}{\pi}+\frac{1}{2}, \quad
\overline{b_2}=\frac{1}{2\pi}\int_{0}^{2\pi}
(\frac{3}{4}+\frac{1}{2}\sin t)dt =\frac{3}{4}, \\
\overline{(\frac{c}{m_1})}
=\frac{1}{2\pi}\int_{0}^{2\pi}(\frac{1}{2}|\sin t|+\frac{1}{4})
dt =\frac{1}{4}+\frac{1}{\pi},\\
\sum_{k=1}^{2}\ln(1+c_{1k})=\ln2-\ln2=0,\quad
\sum_{k=1}^{2}\ln(1+c_{2k})=\ln2-\ln4=-\ln2,
\end{gather*}
and then
\begin{gather*}
\overline{b_1}\omega+\sum_{k=1}^{2}\ln(1+c_{1k})
=\pi+4>2+\frac{\pi}{2} =\overline{(\frac{c}{m_1})}\omega, \\
a^l_2\omega+\sum_{k=1}^{2}\ln(1+c_{2k})
=8-\ln2>\frac{3\pi}{2} =\overline{b_2}\omega, \\
\overline{b_2}\omega=\frac{3\pi}{2}>-\ln2=\sum_{k=1}^{2}\ln(1+c_{2k}).
\end{gather*}
 Thus, all the conditions of {thm3.1} are satisfied.
Then system ({e4.1}) has at least one positive $2\pi$-periodic
solution.

\subsection*{Acknowledgements}
The authors are grateful to the anonymous referees for their
valuable suggestions.

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\end{document}