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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 02, pp. 1--11.\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 2004 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2004/02\hfil Multiple periodic solutions]
{Multiple periodic solutions of a discrete time predator-prey
systems with type IV functional responses}

\author[Z. Liu, A. Chen, J. Cao, F. Chen, \hfil EJDE--2003/02\hfilneg]
{Zhigang Liu, Anping Chen, Jinde Cao, \& Fengde Chen} 


\address{Zhigang Liu \hfill\break
Department of Mathematics, Xiangnan University, Chenzhou, Hunan
423000, China} \email{lzglzglzg4359@sina.com}

\address{Anping Chen \hfill\break
Department of Mathematics, Xiangnan University, Chenzhou, Hunan
423000, China} \email{anping\_chen@hotmail.com}

\address{Jinde Cao\hfill\break Department of Mathematics, Southeast
University, Nanjing 210096, China}
\email{jdcao@seu.edu.cn}

\address{Fengde Chen \hfill\break
Department of Mathematics, Fuzhou University, Fuzhou,
Fujian  350002, China}
\email{fdchen@fzu.edu.cn}


\date{}
\thanks{Submitted September 2, 2003. Published January 2, 2004.}
\subjclass[2000]{39A11, 92B05}
\keywords{Periodic solution, delayed predator-prey, coincidence degree, \hfill\break\indent
type IV functional response,  non-autonomous difference equation}

\begin{abstract}
  By using the continuation theorem of Mawhin's coincidence degree theory,
  some sufficient conditions are obtained ensuring the existence of
  multiple positive periodic solutions of a discrete time
  predator-prey systems with type IV functional responses.
\end{abstract}

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

\date{}

\section{Introduction}

Recently, a Lotka-Volterra model with Holloing Type functional
response has been extensively studied by number of papers (see
papers [1]-[7], [9]-[12], [15], [18], [21]-[23] and the references
cited therein). The model is described by the following system
\[
\begin{gathered}
x'_{1}(t)=x_{1}(t)\big[b_{1}(t)-a_{1}(t)x_{1}(t)-
\frac{c(t)x_{2}(t)}{m(t)x_{2}(t)+x_{1}(t)}\big],\\
x'_{2}(t)=x_{2}(t)\big[-b_{2}(t)+\frac{a_{2}(t)x_{1}(t)}{m(t)x_{2}(t)+x_{1}(t)}
\big],
\end{gathered}\eqno(1.1)
\]
where $x_{1}(t)$ and $x_{2}(t)$ represent the densities of the
prey and the predator, respectively, $b_{1}(t)$, $c(t)$,
$b_{2}(t)$ and $a_{2}(t)$ are the prey intrinsic growth rate,
capture rate, death rate of the predator, and the conversion rate,
respectively, $b_{1}(t)/a_{1}(t)$ gives the carrying capacity of
the prey, and $m$ is the half saturation constant, the functional
response $x/(m(t)y+x)$ is ratio-dependent.

When the prey group has defence or toxicity, the functional response
in a predator-prey model should be type IV. Kot [19] proposed the
following predator-prey model with a type IV functional response
\[
\begin{gathered}
x'_{1}(t)=x_{1}(t)\big[b_{1}(t)-a_{1}(t)x_{1}(t-\tau_{1}(t))-
\frac{c(t)x_{2}(t-\sigma(t))}{\frac{x^{2}_{1}(t-\tau_{2}(t))}{i}
+x_{1}(t-\tau_{2}(t))+a}\big],\\
x'_{2}(t)=x_{2}(t)\big[-b_{2}(t)+\frac{a_{2}(t)x_{1}(t-\tau_{3}(t))}
{\frac{x^{2}_{1}(t-\tau_{4}(t))}{i}
+x_{1}(t-\tau_{4}(t))+a}\big],
\end{gathered}  \eqno(1.2)
\]
where $c,\sigma,a_{i},b_{i}$ ($i=1,2$) and $\tau_{j}$
($j=1,2,3,4$) are continuous $\omega$-periodic functions with
$c(t)\geq0$, $\sigma(t)\geq0$, $a_{i}(t)\geq0$ and
$\tau_{j}(t)\geq0$, $\int_{0}^{\omega}c(t)dt>0$ and
$\int_{0}^{\omega}b_{i}(t)dt>0$, $i$ and $a$ are positive
constants.

Recently, many authors studied the existence of positive periodic
solutions in population models by using the powerful and effective
method of coincidence degree. Chen [8] has established the results
of the existence of multiple positive periodic solutions by
applying the continuation theorem for system (1.2) in the case
$\tau_{2}(t)=0$.

When the populations have non-overlapping generations, discrete
time model described by difference equations is more appropriate
than the continuous one. In [9] and [28], authors studied the
periodic solutions of some difference equations by using
coincidence degree theory. However, no work has been done for the
multiple positive periodic solutions of discrete time
predator-prey model with type IV functional responses.

The main purpose of this paper is to propose a discrete analogue
of system (1.2) and to obtain sufficient conditions for the
existence of its multiple positive periodic solutions by employing
coincidence degree theory and some analysis technique. This is the
first time that a discrete time predator-prey model with a type IV
functional response has been studied by using this way.

The rest of this paper is organized as follows. In Section 2, we
propose a discrete predator-prey model with type IV functional
responses described by difference equations with the help of
differential equations with piecewise constant arguments. In
section 3, we shall establish easily verifiable sufficient
criteria for the existence of multiple positive periodic solutions
of the difference equations derived in Section 2.

\section{Discrete analogue of system (1.2)}

 Let us consider the following equation with piecewise arguments, it is considered as
 a semi-discretization of (1.2)
\[ \begin{gathered}
\frac{1}{x_{1}(t)}\frac{dx_{1}(t)}{dt}
=b_{1}([t])-a_{1}([t])x_{1}([t]-\tau_{1}([t]))-\!
\frac{c([t])x_{2}([t]-\sigma([t]))}{\frac{x^{2}_{1}([t]-\tau_{2}([t]))}{i}
+x_{1}([t]-\tau_{2}([t]))+a},\\
\frac{1}{x_{2}(t)}\frac{dx_{2}(t)}{dt}
=-b_{2}([t])+\frac{a_{2}([t])x_{1}([t]-\tau_{3}([t]))}
{\frac{x^{2}_{1}([t]-\tau_{4}([t]))}{i}
+x_{1}([t]-\tau_{4}([t]))+a},\quad t\neq 0,1,2,\cdots,
\end{gathered} \eqno(2.1)
\]
where $[t]$ denotes the integer part $t$, $t\in(0,+\infty)$.

By a solution of (2.1), we mean a function $x=(x_{1},x_{2})^{T}$,
which is defined for $t\in [0,+\infty)$, and possesses the
following properties:
\begin{enumerate}
\item $x$ is continuous on $[0,+\infty)$.

\item The derivative $\frac{dx_{1}(t)}{dt}$, $\frac{dx_{2}(t)}{dt}$
exist at each point $t\in[0,+\infty)$ with the possible exception
of the points $t\in\{0,1,2,\cdots\}$, where left-sided derivatives
exist.

\item The equations in (2.1) are satisfied on each interval $[k,k+1)$
with $k=0,1,2,\cdots$.
\end{enumerate}
For $k\leq t<k+1,k=0,1,2,\cdots$, integrating (2.1) from $k$ to
$t$, we obtain,
\[\begin{gathered}
\begin{aligned}
x_{1}(t)&=x_{1}(k)\exp\big\{\big[b_{1}(k)-a_{1}(k)x_{1}(k-\tau_{1}(k))\\
&\quad -\frac{c(k)x_{2}(k-\sigma(k))}{\frac{x^{2}_{1}(k-\tau_{2}(k))}{i}
+x_{1}(k-\tau_{2}(k))+a}\big](t-k)\big\},
\end{aligned}\\
x_{2}(t)=x_{2}(k)\exp\big\{\big[-b_{2}(k)+\frac{a_{2}(k)x_{1}(k-\tau_{3}(k))}
{\frac{x^{2}_{1}(k-\tau_{4}(k))}{i}
+x_{1}(k-\tau_{4}(k))+a}\big](t-k)\big\}.
\end{gathered} \eqno(2.2)
\]
Letting $t\to k+1$, we have
\[ \begin{gathered}
\begin{aligned}
x_{1}(k+1)&=x_{1}(k)\exp\big\{\big[b_{1}(k)-a_{1}(k)x_{1}(k-\tau_{1}(k))\\
&\quad-\frac{c(k)x_{2}(k-\sigma(k))}{\frac{x^{2}_{1}(k-\tau_{2}(k))}{i}
+x_{1}(k-\tau_{2}(k))+a}\big]\big\},
\end{aligned}\\
x_{2}(k+1)=x_{2}(k)\exp\big\{\big[-b_{2}(k)+\frac{a_{2}(k)x_{1}(k-\tau_{3}(k))}
{\frac{x^{2}_{1}(k-\tau_{4}(k))}{i}
+x_{1}(k-\tau_{4}(k))+a}\big]\big\},
\end{gathered} \eqno(2.3)
\]
for $k=0,1,2,\cdots$. (2.3) is a discrete analogue of system
(1.2).

Throughout this paper, we are interested only in solutions
$(x_{1}(k),x_{2}(k))^{T}$ of (2.3) with the initial conditions of
the form
$$
x_{i}(s)\geq0,\quad x_{i}(0)>0,\quad s=-m,-m+1,\cdots,0,\quad
i=1,2, \eqno(2.4)
$$
where $m=\max_{k\in
I_{\omega}}\{\tau_{1}(k),\tau_{2}(k),\tau_{3}(k),
\tau_{4}(k),\sigma(k)\}$, $\tau_{i}(k)$ and $\sigma(k)$ are
integers.  For given initial conditions (2.4), we may prove that
(2.3) has a unique solution $(x_{1}(k),x_{2}(k))^{T}$ defined on
$\{-m,\cdots,-1,0,1,2,\cdots\}$ and satisfying
$$
x_{i}(k)>0,\quad i=1,2;k=0,1,2,\cdots.
$$


\section{Existence of multiple positive periodic solutions}

In this section, we shall apply the continuation theorem of
Mawhin's coincidence degree theory to establish our main results.

Let $\mathbb{Z}$, $\mathbb{Z}^{+}$, $\mathbb{R}$, $\mathbb{R}^{+}$, and
$\mathbb{R}^{2}$ denote the sets of all
integers, nonnegative integers, real numbers, nonnegative real
numbers, and two-dimensional Euclidean vector space, respectively.

Throughout this paper, we will use the following notation:
$$
I_{\omega}=\{0, 1, \cdots,\omega-1\},\quad
\overline{g}=\frac{1}{\omega}\sum_{k=0}^{\omega-1}g(k),\quad
\overline{G}=\frac{1}{\omega}\sum_{k=0}^{\omega-1}|g(k)|,
$$
where $\{g(k)\}$ is an $\omega$-periodic sequence of real numbers
defined for $k\in\mathbb{Z}$.
\par
In system (2.3), we always assume that $b_{i}:\mathbb{Z}\to \mathbb{R}$ and
$c,\sigma,a_{i},\tau_{j}:\mathbb{Z}\to \mathbb{R}^{+}$ are
$\omega$-periodic, i.e.,
\begin{gather*}
a_{i}(k+\omega)=a_{i}(k),\quad b_{i}(k+\omega)=b_{i}(k),
\quad c(k+\omega)=c(k),\\
\sigma(k+\omega)=\sigma(k),\quad \tau_{j}(k+\omega)=\tau_{j}(k),
\end{gather*}
for any $k\in\mathbb{Z}$, $i=1,2$; $j=1,2,3,4$ and $\overline{c}>0$,
$\overline{b}_{i}>0$, $i$ and $a$ are positive constants, where
$\omega$, a fixed positive integer, denotes the prescribed common
period of the parameters in (2.3).

For the reader's convenience, we first summarize a few concepts
from the book by Gaines and Mawhin [14].

Let $X$ and $Y$ be normed vector spaces. Let
$L: \mathop{\rm Dom}L\subset X\to Y$ be a linear mapping and $N: X\to Y$ be a
continuous mapping. The mapping $L$ will be 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, then
there exist continuous projectors $P: X\to X$ and $Q:
Y\to Y$ such that $\mathop{\rm Im}P=\ker L$ and
$\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$.
It follows that $L|\mathop{\rm Dom}L\cap\ker P:
(I-P)X\to\mathop{\rm Im}L$ is invertible and its inverse is
denoted by $K_{p}$. If $\Omega$ is a bounded open subset of $X$,
the mapping $N$ is called $L$-compact on $\overline{\Omega}$ if
$(QN)(\overline{\Omega})$ is bounded and $K_{p}(I-Q)N:
\overline{\Omega}\to X$ is compact. Because $\mathop{\rm Im}Q$ is
isomorphic to $\ker L$, there exists an isomorphism $J:
\mathop{\rm Im}Q\to\ker L$.

In the proof our existence result, we need the following lemmas.

\begin{lemma}[{Continuation theorem [14]}] \label{lm3.1}
Let $L$ be a Fredholm mapping of index zero and $N$
 be $L$-compact on $\overline{\Omega}$. Suppose
\begin{itemize}
\item[(a)] For each $\lambda\in (0,1)$,
 every solution $x$ of $Lx=\lambda Nx$ is such that $x\notin\partial\Omega$;
\item[(b)] $QNx\neq0$ for each $x\in \partial\Omega\cap \ker L$ and
  $\deg \{JQN,\Omega \cap \ker L,0\}\neq0$.
\end{itemize}
Then the operator equation $Lx=Nx$ has at least one solution lying
in $\mathop{\rm Dom}L\cap\overline{\Omega}$.
\end{lemma}

\begin{lemma}[{\cite[Lemma 3.2]{s9}}] \label{lm3.2}
Let $g:\mathbb{Z}\to \mathbb{R}$ be an $\omega$-periodic, i.e.,  $g(k+\omega)=g(k)$.
Then for any fixed $k_{1}, k_{2}\in I_{\omega}$, and any $k\in\mathbb{Z}$,
 one has
 \begin{gather*}
 g(k)\leq g(k_{1})+\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|,\\
 g(k)\geq g(k_{2})-\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|.
 \end{gather*}
\end{lemma}

\begin{proof}
 It is only necessary to prove that the inequalities hold for any
 $k\in I_{\omega}$. For the first inequality, it is easy to see the first
 inequality holds if $k=k_{1}$.
If $k>k_{1}$, then
 $$
 g(k)-g(k_{1})=\sum_{s=k_{1}}^{k-1}(g(s+1)-g(s))
 \leq\sum_{s=k_{1}}^{k-1}|g(s+1)-g(s)|\leq\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|,
 $$
and hence,
$ g(k)\leq g(k_{1})+\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|$.
If $k<k_{1}$, then
 $$
 g(k_{1})-g(k)=\sum_{s=k}^{k_{1}-1}(g(s+1)-g(s))
 \geq-\sum_{s=k}^{k_{1}-1}|g(s+1)-g(s)|\geq-\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|,
 $$
equivalently,
$ g(k)\leq g(k_{1})+\sum_{s=0}^{\omega-1}|g(s+1)-g(s)|$.
Now we can claim that the first inequality is valid.

The proof of the second inequality is exactly the same as that
carried out above and the details are omitted here. The proof is
complete.
\end{proof}

Define
$$
l_{2}=\{y=\{y(k)\}: y(k)\in \mathbb{R}^{2}, k\in\mathbb{Z}\}.
$$
For $a=(a_{1}, a_{2})^{T}\in \mathbb{R}^{2}$, define $|a|=\max\{|a_{1}|,
|a_{2}|\}$. Let $l^{\omega}\subset l_{2}$ denote the subspace of
all $\omega$-periodic sequences equipped with the usual supremum
norm $\|\cdot\|$, i.e., for $y=\{y(k): k\in\mathbb{Z}\}\in l^{\omega}$,
$\|y\|=\max_{k\in I_{\omega}}|y(k)|$.
It is difficult to show that $l^{\omega}$ is a finite-dimensional
Banach space.

Let the linear operator $S:l^{\omega}\to \mathbb{R}^{2}$ be defined
by
$$
S(y)=\frac{1}{\omega}\sum_{k=0}^{\omega-1}y(k),\quad
y=\{y(k):k\in\mathbb{Z}\}\in l^{\omega}.
$$
Then we obtain two subspaces $l^{\omega}_{0}$ and $l^{\omega}_{c}$
of $l^{\omega}$ defined by
\begin{gather*}
l^{\omega}_{0}=\{y=\{y(k)\}\in l^{\omega}:S(y)=0\}\,\\
l_{c}^{\omega}=\{y=\{y(k)\}\in l^{\omega}: y(k)\equiv\beta,
\mbox{ for some $\beta\in \mathbb{R}^{2}$ and for all $k\in\mathbb{Z}$}\},
\end{gather*}
respectively. Denote by $L:l^{\omega}\to l^{\omega}$ the
difference operator given by $Ly=\{(Ly)(k)\}$ with
$$
(Ly)(k)=y(k+1)-y(k),\quad \mbox{for $y\in l^{\omega}$ and $k\in\mathbb{Z}$}.
$$
Let a linear operator $K:l^{\omega}\to l_{c}^{\omega}$ be
defined by $Ky=\{(Ky)(k)\}$ with
$$
(Ky)(k)\equiv S(y),\quad \mbox{for $y\in l^{\omega}$ and $k\in\mathbb{Z}$}.
$$
Then we have the following lemma. [28].

\begin{lemma}[{[28]}] \label{lm3.3}
\begin{itemize}
\item[(i)] Both $l_{0}^{\omega}$ and $l_{c}^{\omega}$ are closed linear
subspaces of $l^{\omega}$ and
$l^{\omega}=l_{0}^{\omega}\oplus l_{c}^{\omega}$, $\mathop{\rm dim}l_{c}^{\omega}=2$.

\item[(ii)] $L$ is a bounded linear operator with
$\ker L=l_{c}^{\omega}$ and $\mathop{\rm Im}L=l_{0}^{\omega}$.

\item[(iii)] $K$ is a bounded linear operator with
$\ker (L+K)=\{0\}$ and $\mathop{\rm Im}(L+K)=l^{\omega}$.
\end{itemize}
\end{lemma}

For convenience, we denote $f:y\to \frac{\exp(2y)}{i}
+\exp(y)+a$. From now on, we assume that
\begin{itemize}
\item[(H1)]
$\overline{a}_{2}>\overline{b}_{2}\big(1+2\sqrt{\frac{a}{i}}\big)
\exp \left[(\overline{B}_{1}+\overline{b}_{1})\omega\right]$.
\end{itemize}
For further convenience, we introduce the following six positive numbers:
\begin{gather*}
l_{\pm}=\frac{i\{\overline{a}_{2}\exp [(\overline{B}_{1}+\overline{b}_{1})\omega]
-\overline{b}_{2}\}\pm
\sqrt{i^{2}\{\overline{a}_{2}\exp [(\overline{B}_{1}+\overline{b}_{1})\omega]
-\overline{b}_{2}\}^{2}-4ia\overline{b}_{2}^{2}}}{2\overline{b}_{2}},
\\
\begin{aligned}
u_{\pm}&=\Big(i\{\overline{a}_{2}-\overline{b}_{2}\exp [(\overline{B}_{1}
+\overline{b}_{1})\omega]\}
\\
&\quad \pm \sqrt{i^{2}\{\overline{a}_{2}-\overline{b}_{2}\exp [(\overline{B}_{1}
+\overline{b}_{1}) \omega]\}^{2}-4ia\overline{b}_{2}^{2}
\exp [2(\overline{B}_{1}+\overline{b}_{1})\omega]}\Big)\\
&\quad \div \big(2\overline{b}_{2}\exp [(\overline{B}_{1}+\overline{b}_{1})\omega]\big),
\end{aligned}\\
y_{\pm}=\frac{i(\overline{a}_{2}-\overline{b}_{2})\pm\sqrt{i^{2}(\overline{a}_{2}-\overline{b}_{2})^{2}
-4ia\overline{b}_{2}^{2}}}{2\overline{b}_{2}}.
\end{gather*}
It is not difficult to prove that
$$
l_{-}<y_{-}<u_{-}<u_{+}<y_{+}<l_{+}.\eqno(3.1)
$$
To state and prove the main result of this paper, we use the hypothesis
\begin{itemize}
\item[(H2)]
$\overline{a}_{1}l_{+}\exp [(\overline{B}_{1}+\overline{b}_{1})
\omega]<\overline{b_{1}}$.
\end{itemize}

\begin{theorem} \label{thm1}
Under the hypotheses (H1)--(H2), the system (2.3) has at least two
$\omega$-periodic positive solutions.
\end{theorem}

\begin{proof}  we make the change of variables
$$
x_{i}(k)=\exp (y_{i}(k)),\quad i=1,2. \eqno(3.2)
$$
Then (2.3) is rewritten as
\[ \begin{gathered}
y_{1}(k+1)-y_{1}(k)= b_{1}(k)-a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}-
\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))},\\
y_{2}(k+1)-y_{2}(k)=-b_{2}(k)+\frac{a_{2}(k)\exp\{y_{1}(k-\tau_{3}(k))\}}
{f(y_{1}(k-\tau_{4}(k)))},
\end{gathered} \eqno(3.3)
\]
If (3.3) has an $\omega$-periodic solution $\{y(k)\}$, then
$\{x(k)\}$: $x_{i}(k)=\exp (y_{i}(k))$ is a positive
$\omega$-periodic solution of (2.3).

Now let we define $X=Y=l^{\omega}$, $(Ly)(k)=y(k+1)-y(k)$, and
\begin{align*}
(Ny)(k)&=\begin{pmatrix}
  b_{1}(k)-a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}-
\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))} \\
-b_{2}(k)+\frac{a_{2}(k)\exp\{y_{1}(k-\tau_{3}(k))\}}
{f(y_{1}(k-\tau_{4}(k)))}
\end{pmatrix}\\
&:= \begin{pmatrix}
  \triangle_{1}(y,k) \\
  \triangle_{2}(y,k)
\end{pmatrix} ,
\end{align*}
for any $y\in X$ and $k\in\mathbb{Z}$. It follows from Lemma 3.3
that $L$ is a bounded linear operator and
$$
\ker L=l_{c}^{\omega},\quad \mathop{\rm Im}L=l_{0}^{\omega}, \quad
dim\ker L=2=\mathop{\rm codim\,Im}L,
$$
then it follows that $L$ is a Fredholm mapping of index zero.

Define
$$
Py=\frac{1}{\omega}\sum_{s=0}^{\omega-1}y(s),\quad
 y\in X,\qquad
Qz=\frac{1}{\omega}\sum_{s=0}^{\omega-1}z(s),\quad z\in Y.
$$
It is not difficult to show that $P$ and $Q$ are two continuous
projectors such that
$$
\mathop{\rm Im} P=\ker L \quad \text{and} \quad
\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q).
$$
Furthermore, the generalized inverse (of $L$) $K_{p}$:
$\mathop{\rm Im}L\to \ker P\cap\mathop{\rm Dom} L$ exists and is given by
$$
K_{p}(z)=\sum_{s=0}^{k-1}z(s)
-\frac{1}{\omega}\sum_{s=0}^{\omega-1}(\omega-s)z(s).
$$
Thus
\begin{gather*}
QNy=\Big(\frac{1}{\omega}\sum_{k=0}^{\omega-1}\triangle_{1}(y,k),
\frac{1}{\omega}\sum_{k=0}^{\omega-1}\triangle_{2}(y, k)\Big)^{T},\\
K_{p}(I-Q)Ny=(\Phi_{1}(y,k),\Phi_{2}(y,k))^{T},
\end{gather*}
where for $i=1,2$,
$$
\Phi_{i}(y,k)=\sum_{s=0}^{k-1}\triangle_{i}(y,s)
-\frac{1}{\omega}\sum_{s=0}^{\omega-1}(\omega-s)\triangle_{i}(y,s)
-\big(\frac{k}{\omega}-\frac{\omega+1}{2\omega}\big)
\sum_{s=0}^{\omega-1}\triangle_{i}(y,s).
$$
Obviously, $QN$ and $K_{p}(I-Q)N$ are continuous. Since $X$ is a
finite-dimensional Banach space, it is not difficult to show that
$\overline{K_{p}(I-Q)N(\overline{\Omega})}$ is compact for any
open bounded set $\Omega\subset X$. Moreover,
$QN(\overline{\Omega})$ is bounded. Thus, $N$ is $L$-compact on
with any open bounded set $\Omega\subset X$. The isomorphism $J$
of $\mathop{\rm Im}Q$ onto $\ker L$ can be the identity mapping, since
$\mathop{\rm Im}Q=\ker L$.

From now on, we shall search for at least two appropriate open,
bounded subsets $\Omega_{1}$ and $\Omega_{2}$ in $X$.
Corresponding to the operator equation $Ly=\lambda Ny, \lambda\in
(0,1)$, we have
\[ \begin{gathered}
\begin{aligned}
y_{1}(k+1)-y_{1}(k)
&=\lambda\Big[b_{1}(k)-a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}\\
&\quad -\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))}\Big],
\end{aligned}\\
y_{2}(k+1)-y_{2}(k)=\lambda\Big[-b_{2}(k)+\frac{a_{2}(k)
\exp\{y_{1}(k-\tau_{3}(k))\}}
{f(y_{1}(k-\tau_{4}(k)))}\Big],
\end{gathered} \eqno(3.4)
\]
Assume that $y=(y_{1}(k),y_{2}(k))^{T}\in X$ is an solution of
(3.4) for a certain $\lambda\in (0,1)$. Summing on both sides of
(3.4) from 0 to $\omega-1$ about $k$, we get
\begin{align*}
0&=\sum_{k=0}^{\omega-1}(y_{1}(k+1)-y_{1}(k))\\
&=\lambda\sum_{k=0}^{\omega-1}\big[b_{1}(k)-a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}-
\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))}\big],\\
0&=\sum_{k=0}^{\omega-1}(y_{2}(k+1)-y_{2}(k))
=\lambda\sum_{k=0}^{\omega-1}\big[-b_{2}(k)+\frac{a_{2}(k)\exp\{y_{1}(k-\tau_{3}(k))\}}
{f(y_{1}(k-\tau_{4}(k)))}\big];
\end{align*}
that is,
\[ \begin{gathered}
\overline{b}_{1}\omega
=\sum_{k=0}^{\omega-1}\Big[a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}+
\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))}\Big],\\
\overline{b}_{2}\omega =
\sum_{k=0}^{\omega-1}\frac{a_{2}(k)\exp\{y_{1}(k-\tau_{3}(k))\}}
{f(y_{1}(k-\tau_{4}(k)))}.
\end{gathered}\eqno(3.5)
\]
From the first equation of (3.4), and (3.5), we have
\begin{align*}
&\sum_{k=0}^{\omega-1}|y_{1}(k+1)-y_{1}(k)|\\
&<\sum_{k=0}^{\omega-1}
\big[|b_{1}(k)|+a_{1}(k)\exp\{y_{1}(k-\tau_{1}(k))\}+
\frac{c(k)\exp\{y_{2}(k-\sigma(k))\}}{f(y_{1}(k-\tau_{2}(k)))}\big]\\
&=(\overline{B}_{1}+\overline{b}_{1})\omega;
\end{align*}
that is,
$$
\sum_{k=0}^{\omega-1}|y_{1}(k+1)-y_{1}(k)|
<(\overline{B}_{1}+\overline{b}_{1})\omega.
\eqno(3.6)
$$
Similarly, it follows from the second equation of (3.4), (3.5)
that
$$
\sum_{k=0}^{\omega-1}|y_{2}(k+1)-y_{2}(k)|<(\overline{B}_{2}+\overline{b}_{2})\omega.
\eqno(3.7)
$$
Because of $y=\{y(k)\}\in X$, there exist $\xi_{i},\eta_{i}\in
I_{\omega}$ such that
$$
y_{i}(\xi_{i})=\min_{k\in I_{\omega}}\{y_{i}(k)\}, \quad
y_{i}(\eta_{i})=\max_{k\in I_{\omega}}\{y_{i}(k)\},\quad i=1,2.
\eqno(3.8)
$$
It follows from the second equation of (3.5) and (3.8) that
$$
\overline{b}_{2}\omega\leq
\frac{\overline{a}_{2}\omega\exp\{y_{1}(\eta_{1})\}}{f(y_{1}(\xi_{1}))}.
$$
So
$$
y_{1}(\eta_{1})\geq\ln\big[\frac{\overline{b}_{2}}{\overline{a}_{2}}
f(y_{1}(\xi_{1}))\big]. \eqno(3.9)
$$
Therefore, Lemma 3.2 and (3.6), (3.9) imply
\[
y_{1}(k)\geq y_{1}(\eta_{1})-\sum_{k=0}^{\omega-1}|y_{1}(k+1)-y_{1}(k)|
> \ln\big[\frac{\overline{b}_{2}}{\overline{a}_{2}}f(y_{1}(\xi_{1}))\big]-
(\overline{B}_{1}+\overline{b}_{1})\omega.
\eqno(3.10)
\]
In particular, we have
$y_{1}(\xi_{1})> \ln\big[\frac{\overline{b}_{2}}{\overline{a}_{2}}
f(y_{1}(\xi_{1}))\big]-(\overline{B}_{1}+\overline{b}_{1})\omega$,
or
$$
\frac{\overline{b}_{2}}{i}\exp(2y_{1}(\xi_{1}))-\left[\overline{a}_{2}
\exp(\overline{B}_{1}+\overline{b}_{1})\omega
-\overline{b}_{2}\right]\exp\{y_{1}(\xi_{1})\}+\overline{b}_{2}a<0.
$$
Because of (H1), we have
$$
\ln l_{-}<y_{1}(\xi_{1})<\ln l_{+}. \eqno(3.11)
$$
Similarly, we have
$$
y_{1}(\eta_{1})<\ln u_{-} \quad\mbox{or} \quad
y_{1}(\eta_{1})>\ln u_{+}. \eqno(3.12)
$$
It follows from (3.11), (3.6) and Lemma 3.2 that
\[
 y_{1}(k)\leq y_{1}(\xi_{1})+\sum_{s=0}^{\omega-1}|y_{1}(s+1)-y_{1}(s)|
< \ln l_{+}+(\overline{B}_{1}+\overline{b}_{1})\omega:=H_{12}. \eqno(3.13)
\]
On the other hand, it follows from (3.5) and (3.13) that
$$
\overline{b}_{1}\omega\geq\frac{\overline{c}\omega\exp\{y_{2}(\xi_{2})\}}
{f(\ln l_{+}+(\overline{B}_{1}+\overline{b}_{1})\omega)}
\eqno(3.14)
$$
$$
\overline{b}_{1}\omega\leq\overline{a}_{1}\omega\exp\left[\ln
l_{+}+(\overline{B}_{1}+\overline{b}_{1})\omega\right]
+\frac{\overline{c}\omega \exp\{y_{2}(\eta_{2})\}}{a}. \eqno(3.15)
$$
It follows from (3.14) that
$y_{2}(\xi_{2})\leq\ln\big\{\frac{\overline{b}_{1}}{\overline{c}}f(\ln
l_{+}+(\overline{B}_{1}+\overline{b}_{1})\omega)\big\}$.
This, combined with (3.7), gives
\[\begin{aligned}
y_{2}(k)&\leq
y_{2}(\xi_{2})+\sum_{s=0}^{\omega-1}|y_{2}(s+1)-y_{2}(s)|\\
&<\ln\Big\{\frac{\overline{b}_{1}}{\overline{c}}f(\ln
l_{+}+(\overline{B}_{1}+\overline{b}_{1})\omega)\Big\}+(\overline{B}_{2}
+\overline{b}_{2})\omega
:=H_{22}.
\end{aligned}\eqno(3.16)
\]
Moreover, because of (H2), it follows from (3.15) that
$$
y_{2}(\eta_{2})\geq\ln\frac{a\{\overline{b}_{1}-\overline{a}_{1}l_{+}\exp[(\overline{B}_{1}
+\overline{b}_{1})\omega]\}}{\overline{c}}.
$$
This, combined with (3.7) again, gives
\[\begin{aligned}
y_{2}(k)&\geq
y_{2}(\eta_{2})-\sum_{s=0}^{\omega-1}|y_{2}(s+1)-y_{2}(s)|\\
&>\ln\frac{a\{\overline{b}_{1}-\overline{a}_{1}l_{+}\exp[(\overline{B}_{1}
+\overline{b}_{1})\omega]\}}{\overline{c}}-(\overline{B}_{2}
+\overline{b}_{2})\omega
:=H_{21}.
\end{aligned}\eqno(3.17)
\]
It follows from (3.16) and (3.17) that
$$
\begin{array}{rcl}
\max_{k\in
I_{\omega}}|y_{2}(k)|<\max\{|H_{21}|,|H_{22}|\}:=H_{2}.
\end{array}\eqno(3.18)
$$
Obviously, $\ln l_{\pm}$, $\ln u_{\pm}$, $H_{12}$ and $H_{2}$ are
independent of $\lambda$.

Now, let's consider $QNy$ with $y=(y_{1},y_{2})^{T}\in \mathbb{R}^{2}$.
Note that
$$
QN(y_{1},y_{2})^{T}=\Big(\overline{b}_{1}-\overline{a}_{1}\exp(y_{1})
-\frac{\overline{c}\exp(y_{2})}{f(y_{1})},-\overline{b}_{2}
+\frac{\overline{a}_{2}\exp(y_{1})}{f(y_{1})}\Big)^{T}.
$$
Because of (H1) and (H2), we can show that $QN(y_{1},y_{2})^{T}=0$
has two distinct solutions $\widetilde{y}=(\ln
y_{-},\ln\frac{(\overline{b}_{1}-\overline{a}_{1}y_{-})f(\ln
y_{-})}{\overline{c}})$ and $\widehat{y}=(\ln
y_{+},\ln\frac{(\overline{b}_{1}-\overline{a}_{1}y_{+})f(\ln
y_{+})}{\overline{c}})$. Choose $C>0$ such that
$$
C>\max\Big\{\big|\ln\frac{(\overline{b}_{1}-\overline{a}_{1}y_{-})f(\ln
y_{-})}{\overline{c}}\big|,\big|\ln\frac{(\overline{b}_{1}
-\overline{a}_{1}y_{+})f(\ln y_{+})}{\overline{c}}\big|\Big\}.
\eqno(3.19)
$$
Let
$$
\Omega_{1}=\Big\{y=(y_{1}(k),y_{2}(k))\in X :
y_{1}(k)\in (\ln l_{-}, \ln u_{-}),\;
\max_{k\in I_{\omega}}|y_{2}(k)|<H_{2}+C\Big\},
$$
\begin{align*}
\Omega_{2}=\Big\{&y=(y_{1}(k),y_{2}(k))\in X :
\min_{k\in I_{\omega}}y_{1}(k)\in (\ln l_{-}, \ln l_{+}) ,\\
&\max_{k\in I_{\omega}}y_{1}(k)\in (\ln u_{+}, H_{12}) ,\;
\max_{k\in I_{\omega}}|y_{2}(k)|<H_{2}+C \Big\}.
\end{align*}
Then both $\Omega_{1}$ and $\Omega_{2}$ are bounded open subsets
of $X$. It follows from (3.1) and (3.19) that
$\widetilde{y}\in\Omega_{1}$ and $\widehat{y}\in\Omega_{2}$. With
the help of (3.1), (3.11)-(3.13) and (3.18)-(3.19), it is easy to
that  $\Omega_{1}\cap\Omega_{2}=\phi$ and $\Omega_{i}$ satisfies
the requirement (a) Lemma 3.1 for $i=1,2.$ Moreover, $QNy\neq0$
for $y\in \partial\Omega\cap \mathbb{R}^{2}$. A direct computation
gives
$$
\deg \{JQN,\Omega_{i}\cap \ker L,0\}=(-1)^{i+1}\neq0.
$$
Here, $J$ is taken as the identity mapping since $\mathop{\rm
Im}Q=\ker L$. So far we have proved that $\Omega_{i}$ satisfies
all the assumptions in Lemma 3.1. Hence, (3.3) has at least two
$\omega$-periodic solutions $\{y^{*}(k)\}$ and
$\{y^{\dagger}(k)\}$ with $y^{*}(k)\in \mathop{\rm
Dom}L\cap\overline{\Omega}_{1}$ and $y^{\dagger}(k)\in \mathop{\rm
Dom}L\cap\overline{\Omega}_{2}$. Obviously, $y^{*}$ and
$y^{\dagger}$ are different. Let $x^{*}_{i}(k)=\exp
(y_{i}^{*}(k))$ and $x^{\dagger}_{i}(k)=\exp
(y_{i}^{\dagger}(k)),i=1,2$. Then by (3.2),
$x^{*}(k)=(x^{*}_{1}(k),x^{*}_{2}(k))^{T}$ and
$x^{\dagger}(k)=(x^{\dagger}_{1}(k),x^{\dagger}_{2}(k))^{T}$ are
two different positive $\omega$-periodic solutions of (2.3). This
completes the proof.
\end{proof}

\subsection*{Acknowledgments}
 This work was supported by Natural Science Foundation of China
 (10371034), the Foundation for University Excellent Teacher by Chinese
Ministry of Education, the Key Project of Chinese Ministry of
Education (No.[2002]78), was also supported by the National
Natural Science Foundation of China under Grant 60373067, the
Natural Science Foundation of Jiangsu Province, China under Grant
BK2003053 and BK2003001, Qing-Lan Engineering Project of Jiangsu
Province, the Foundation of Southeast University, Nanjing, China
under grant XJ030714, was also supported by scientific research
 Foundation of Hunan provincial education department(01C009).

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\end{document}