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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 126, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/126\hfil Twin periodic solutions]
{Twin periodic solutions of predator-prey dynamic system on
time scales}

\author[R.-H. He, H.-X. Li, L. Zhang\hfil EJDE-2009/126\hfilneg]
{Rong-Hua He, Hong-Xu Li, Liang Zhang}  % in alphabetical order

\address{Rong-Hua He \newline
Department of Mathematics, Chengdu University of Information
Technology, Chengdu, Sichuan 610103, China. \newline Department of
Mathematics, Sichuan University, Chengdu, Sichuan 610064,  China}
\email{ywlcd@cuit.edu.cn}

\address{Hong-Xu Li \newline
Department of Mathematics, Sichuan University, Chengdu, Sichuan
610064,  China} 
\email{hoxuli@sohu.com}

\address{Liang Zhang \newline
Department of Mathematics, Sichuan University, Chengdu, Sichuan
610064,  China}
\email{lzhang1979@yahoo.com.cn}

\thanks{Submitted December 2, 2008. Published October 4, 2009.}
\subjclass[2000]{92D25, 39A12}
\keywords{Time scales; delayed predator-prey dynamic
system; \hfill\break\indent periodic solutions; continuation theorem}

\begin{abstract}
 In this article, we consider a delayed predator-prey dynamic system
 with type IV functional responses on time scales. Sufficient
 criteria for the existence of at least two periodic solutions are
 established by using the well-known continuation theorem due to
 Mawhin. An example is given to illustrate the main result.
\end{abstract}

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

\section{Introduction}

 In studying the interaction between predators and their
prey, it is crucial to determine what specific form of the
functional response that describes the mount of prey consumed per
predator per unit of time is biological plausible provides a sound
basis for theoretical development.

In this paper, we consider the following delay predator-prey dynamic
equation with type IV functional responses on time scale
$\mathbb{T}$:
\begin{equation}
\begin{gathered}
y_1^\Delta (t)  =  b_1(t)-a_1(t)\exp \{y_1(t-\tau _1(t))\}
-\frac{c(t)\exp \{y_2(t-\gamma (t))\}}{\exp \{2y_1(t)\}/ n+\exp
\{y_1(t)\}+a}, \\
y_2^\Delta (t)  =  -b_2(t)+\frac{a_2(t)\exp \{y_1(t-\tau
_2(t))\}}{\exp \{2y_1(t)(t-\tau _2(t))\}/ n+\exp \{y_1(t-\tau
_2(t))\}+a},
\end{gathered}  \label{e1.1} %a3
\end{equation}
where for $i=1, 2$; $c, \gamma , a_i, b_i, \tau _i\in C_{rd}(\mathbb{T}
) $ are $\omega $-priodic functions with $c(t)\geq 0, \gamma (t)\geq 0,
a_i(t)\geq 0, \tau _i(t)\geq 0, \overline{c}>0$, and ${\bar b_i}>0$,
$n $ and $a$ are positive constants, $C_{rd}(\mathbb{T})$ will be
defined later.

Calculus on time scales was initiated by Stefan Hilger in 1990
with the motivation of providing a unified approach to continuous
and discrete analysis. Since then the theory of dynamic equations on
time scales has become a new important mathematical branch, and it
has been applied in various directions (see, eg.,
\cite{a1,a2,b1,b2,b3,h1,k1,l1,p1,s1,w1,w2,z1,z2,z3}
 and the refs cited therein).

On the other hand, the Mawhin's continuation is a powerful tool when
deal with the existence of periodic solutions for population models,
and much work have been done (see, e.g.,
\cite{c1,h2,l2,w2,z4} and the
references cited therein). However, to the best of our knowledge, the
study on the existence of multiple periodic solutions for population
models on time scales are scarce.

Motivated and inspired by the above excellent work, in this paper,
we establish some sufficient criteria for the existence of at least
two periodic solutions for system \eqref{e1.1} by using Mawhin technique.

\section{Preliminaries}

We first provide without proof several definitions and
results from the calculus on time scales which are useful in the
following argument. For more details, we refer the authors to
\cite{b3}.

A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of
the real numbers, and it inherits the topology from the real numbers
with the standard topology. Let $\omega >0$ is a constant.
Throughout this paper, the time scale we considered is always
assumed to be $\omega $-priodic (i.e., $t\in \mathbb{T}$ implies
$t\pm \omega \in \mathbb{T}$) and unbounded above and below. Set
\[
\kappa =\min \{ \mathbb{R^+\cap\mathbb{T}}\} , \quad
\mathbb{I}_\omega =[\kappa , \kappa +\omega ]\cap \mathbb{T}.
\]

\begin{definition} \label{def2.1} \rm
The forward jump operator $\sigma :\mathbb{T}
\to \mathbb{T}$ and the backward jump operator
$\rho :\mathbb{T}\to \mathbb{T}$ are defined by
\[
\sigma (t):=\inf\{s\in \mathbb{T}:s\geq t\}, \quad
\rho (t):=\sup\{s\in \mathbb{T}:s\leq t\},
\]
respectively, for any $t\in \mathbb{T}$.
If $\sigma (t)=t$, then $t$ is
called right-dense (otherwise: right-scattered), and if $\rho (t)=t$, then
$t$ is called left--dense (otherwise left-scattered).
\end{definition}

\begin{definition} \label{def2.2} \rm
 Assume that $f:\mathbb{T}\to \mathbb{R}$
and fix $t\in \mathbb{T}$. Then $f$ is called differential at $t\in
\mathbb{T}$ if there exists  $c\in\mathbb{R}$ such that given any
$\varepsilon >0$, there is an open neighborhood $U$ of $t$
satisfying
\[
|[f(\sigma (t))-f(s)]-c[\sigma (t)-s]|\leq\varepsilon |\sigma (t)-s|,
\quad  s\in U.
\]
In this case, $c$ is called the delta (or Hilger) derivative of $f$
at $t\in\mathbb{T}$, and is denoted by $c=f^\Delta (t)$.
\end{definition}

\begin{remark} \label{rmk2.1} \rm
We say that $f$ is delta (Hilger) differential
on $\mathbb{T}$ if $f^\Delta (t)$ exists for all $t\in \mathbb{T}$. A
function $F:\mathbb{T}\to \mathbb{R}$ is called an antiderivative of
$f: \mathbb{T}\to \mathbb{R}$ provided that $F^\Delta (t)=f(t)$
for all $t\in \mathbb{T}$. Then we define
\[
\int_r^sf(t)\Delta t=F(s)-F(r), \quad r, s\in \mathbb{T}.
\]
\end{remark}

\begin{definition} \label{def2.3} \rm
A function $f:\mathbb{T}\to\mathbb{R}$ is
called rd-continuous if it is continuous at right-dense points in
$\mathbb{T}$ and its left-sided limits exist(finite) at left-dense
points in $\mathbb{T}$. The set of rd-continuous functions
$f:\mathbb{T}\to\mathbb{R}$ will be
denoted by $C_{rd}=C_{rd}(\mathbb{T})=C_{rd}[\mathbb{T}, \mathbb{R})$.
\end{definition}

\begin{remark} \label{rmk2.2} \rm
Every rd-continuous function has an
antiderivative. Every continuous function is rd-continuous.
\end{remark}

\begin{lemma} \label{lem2.1}
If $a, b\in \mathbb{T}$, $\alpha, \beta\in
\mathbb{R}$ and $f, g\in C_{rd}(\mathbb{T})$, then

\begin{itemize}
\item[(C1)]  $\int_a^b[\alpha f(t)+\beta g(t)]\Delta t=\alpha
\int_a^bf(t)\Delta t+\beta \int_a^bg(t)\Delta t$;

\item[(C2)]  if $f(t)\geq 0$ for all $a\leq t\leq b$, then
$\int_a^bf(t)\Delta t\geq 0$;

\item[(C3)]  if $|f(t)|\leq g(t)$ on $[a, b):=\{t\in \mathbb{T}
:a\leq t<b\}$, then $|\int_a^bf(t)\Delta t|\leq
\int_a^bg(t)\Delta t$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{b1}] \label{lem2.2}
 Let $t_1, t_2\in I_\omega $ and $t\in \mathbb{T}$.
If $g:\mathbb{T}\to \mathbb{R}$ is $\omega -periodic$, then
\[
g(t)\leq g(t_1)+\int_{\mathbb{I}_\omega}|g^\Delta (s)|
\Delta s\quad\text{and}\quad
g(t)\geq g(t_2)-\int_{\mathbb{I}_\omega}|g^\Delta (s)|\Delta s.
\]
\end{lemma}

In the remainder of this section we list  well known
elements and result which can be found in \cite{g1}.
Let $X$ and $Z$ are two Banach spaces. Consider a operator equation:
\begin{equation}
Lx=\lambda Nx, \quad \lambda \in (0,1),  \label{b1}
\end{equation}
where $L:\mathop{\rm Dom}L\cap X$ $\to Z$ is a linear operator,
$N:X\to Z$ is a continuous operator and $\lambda $ is a parameter.
Let $P$ and $Q$ denote
two projectors $P:X\to X$ and $Q:Z\to Z$ such that
$\mathop{\rm Im}P=\mathop{\rm Ker}L$ and
$\mathop{\rm Im}L=\mathop{\rm Ker}Q=\mathop{\rm Im}(I-Q)$.
It follows that $L|_{\mathop{\rm Dom}L\cap \mathop{\rm Ker}P}:(I-P)
X\to \mathop{\rm Im}L$ is invertible. We denote the inverse
of this map by $Kp$. 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 $Kp(I-Q)N:\overline{\Omega }\to X$ is compact. Because
$\mathop{\rm Im}Q$ is isomorphic to $\mathop{\rm Ker}L$,
there exists an isomorphism $J:\mathop{\rm Im}Q\to \mathop{\rm Ker}L$.

Recall that operator $L$ will be called a Fredholm operator of index
zero if $\dim \mathop{\rm Ker}L=\mathop{\rm codim}\mathop{\rm Im}L<\infty$,
 and $\mathop{\rm Im}L$ is closed in $Z$.

\begin{lemma}[Continuation Theorem \cite{g1}] \label{lem2.3}
 Let $L$  be a Fredholm mapping of index zero and let $N$
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)]  $QNx\neq 0$ for each
$x\in \partial \Omega \cap \mathop{\rm Ker}L$;

\item[(c)] $\deg \{ JQN, \Omega , \theta \} \neq 0$.

\end{itemize}
Then the equation $Lx=Nx$ has at least
one solution lying in $\mathop{\rm Dom}L\cap \bar{\Omega }$.
\end{lemma}

To facilitate the discussion below, throughout this paper we adopt the
following notation
\[
\bar{g}=\frac 1\omega \int_{I_\omega }g(s)\Delta s, \quad
|\bar b_i|=\frac 1\omega \int_{I_\omega}|b_i(s)|\Delta s,
\]
where i=1, 2; $g\in C_{rd}(\mathbb{T})$ is an $\omega$-periodic real
function, i.e., $g(t+\omega )=g(t)$ for all $t\in \mathbb{T}$. And
the other symbols appearing in the sequel are denoted accordingly.
Set
\begin{gather*}
u_{1\pm }:=\frac{n({\bar a_2}-{\bar b_2})
\pm \sqrt{n^2({\bar a_2}-{\bar b_2}) ^2-4{\bar b_2}^2an}}{2
\overline{b_2}},
\\
\begin{aligned}
l_{\pm }&:=\frac{1}{2{\bar b_2}}
\Big(n[{\bar a_2}\exp\{ ({ \bar
b_1}+{|\bar b_1| }) \omega\} -{\bar b_2}] \\
&\quad \pm \sqrt{n^2[{ \bar
a_2}\exp\{( {{\bar b_1}+{|\bar b_1| }}) {\omega }\} -{\bar b_2}]
^2-4{\bar b_2}^2an}\Big),
\end{aligned} \\
\begin{aligned}
v_{\pm }&:=\frac{1}{2{\bar b_2}\exp\{ \omega({\bar b_1}+{|\bar b_1|
})\}}
\Big(n[{\bar a_2}-{\bar b_2}
\exp\{ {({\bar b_1}+{|\bar b_1| })\omega }\} ] \\
&\quad \pm \sqrt{n^2[{\bar a_2}-{\bar b_2}
\exp\{( {{\bar b_1}+{|\bar b_1| }}) {\omega }\}] ^2
-4{\bar b_2}^2an\exp\{ 2\omega({|\bar b_1| }+{\bar b_1})\}}\Big).
\end{aligned}
\end{gather*}
 From the above six positive numbers, one  obtains
\begin{equation}
l_{-}<u_{1-}<v_{-}<v_{+}<u_{1+}<l_{+}.  \label{b2}
\end{equation}

\section{Main result}

 In this section, our emphasis is focused on the existence
of at least two periodic solutions for  \eqref{e1.1}. Before
formulate the main result, we first embed our problem into the frame
of Lemma \ref{lem2.3}.
Set
\[
X=Z=\{y=(y_1(t), y_2(t))^T\in C(\mathbb{T},
\mathbb{R}^2)|y_i(t+\omega )=y_i(t),\; i=1, 2,\; t\in \mathbb{T}\}.
\]
Then $X, Z$ are Banach spaces  endowed with the norm
$\|y\|=\sum_{i=1}^2\max_{t\in I_\omega }|y_i(t)|$. Define
\begin{gather*}
Ny(t)=(\Phi _1(t), \Phi _2(t))^T, \quad
Ly(t)=(y_1^\Delta (t), y_2^\Delta (t))^T,
\\
Py=Qy=\Big(\frac 1\omega \int_{\mathbb{I}_\omega}y_1(t)\Delta t,
\frac 1\omega \int_{\mathbb{I}_\omega}y_2(t)\Delta t\Big)^T,
\end{gather*}
where $y\in X$,
\begin{gather*}
\Phi _1(t) =  b_1(t)-a_1(t)\exp \{y_1(t-\tau
_1(t))\}-\frac{c(t)\exp \{y_2(t-\gamma (t))\}}{\exp
\{2y_1(t)\}/ n+\exp \{y_1(t)\}+a}, \\
\Phi _2(t) =  -b_2(t)+\frac{a_2(t)\exp
\{y_1(t-\tau _2(t))\}}{\exp \{2y_1(t-\tau _2(t))\}/
n+\exp \{y_1(t-\tau _2(t))\}+a}.
\end{gather*}
Obviously,
\begin{gather*}
\mathop{\rm Ker}L = \{ y\in X:y=h=(h_1, h_2)^T\in \mathbb{R}^2{, }\ t\in
\mathbb{T}\} , \\
\mathop{\rm Im}L = \big\{ y\in Z:\int_{\mathbb{I}_\omega}y_i(t)\Delta t=0, t\in
\mathbb{T}, \ i=1,2\big\} ,
\end{gather*}
$P, Q$ are continuous projectors such that
\[
\mathop{\rm Im}P=\mathop{\rm Ker}L, \quad
\mathop{\rm Im}L=\mathop{\rm Ker}Q=\mathop{\rm Im}(I-Q),
\]
the set $\mathop{\rm Im}L$ is closed in $Z$, and
\[
dim\mathop{\rm Ker}L=2=\mathop{\rm codim}\mathop{\rm Im}L.
\]
Hence, $L$ is a Fredholm mapping of index zero. Furthermore, the
generalized inverse (to $L$)
$K_p:\mathop{\rm Im}L\to \mathop{\rm Dom}L\cap \mathop{\rm Ker}P$
exists and is given by
\[
K_py=\Big(\int_\kappa ^ty_1(s)\Delta s-\frac 1\omega
\int_{\mathbb{I}_\omega}\int_\kappa ^ty_1(s)\Delta s\Delta t,
\int_\kappa ^ty_2(s)\Delta s-\frac 1\omega
\int_{\mathbb{I}_\omega}\int_\kappa ^ty_2(s)\Delta s\Delta t\Big)^T.
\]
Thus
\begin{gather*}
QNy=\Big(\frac 1\omega \int_{\mathbb{I}_\omega}\Phi _1(
s)\Delta s, \frac 1\omega \int_{\mathbb{I}_\omega}\Phi
_2(s)\Delta s\Big)^T, \\
K_p(I-Q)Ny=\Big(\Theta _1(t), \Theta _2(t)\Big)^T,
\end{gather*}
where,
\begin{gather*}
\Theta _1(t)=\int_\kappa ^t\Phi _1(s)\Delta s-\frac
1\omega \int_{\mathbb{I}_\omega}\int_\kappa ^t\Phi _1(s)\Delta
s\Delta t-\Big(t-\kappa -\frac 1\omega
\int_{\mathbb{I}_\omega}(t-\kappa
)\Delta t\Big)\bar{\Phi}_1,
\\
\Theta _2(t)=\int_\kappa ^t\Phi _2(s)\Delta s-\frac
1\omega \int_{\mathbb{I}_\omega}\int_\kappa ^t\Phi _2(s)\Delta
s\Delta t-\Big(t-\kappa -\frac 1\omega
\int_{\mathbb{I}_\omega}(t-\kappa )\Delta t\Big)\bar{\Phi}_2.
\end{gather*}


It is easy to show that $QN$ and $K_p(I-Q)$ are continuous. By using
the Arzela-Ascoli theorem, one can show that
$K_p(I-Q)(\bar{\Omega})$ is
relatively compact for any open bounded set $\Omega \in X$. Moreover,
$QN(\bar{\Omega })$ is bounded. Thus, $N$ is $L$-compact on
$\bar{\Omega }$ for any open bounded set $\Omega \subset X$.

After the above preparations, we now state and prove our main
result.

\begin{theorem} \label{thm3.1}
System \eqref{e1.1} has at least two $\omega$-periodic
 solutions if the following conditions hold.
\begin{itemize}
\item[(i)]  ${\bar a_2}>{\bar b_2}(1+2\sqrt{\frac an})
\exp \{({|\bar b_1|}+{\bar b_1})\omega \}$;

\item[(ii)]  ${\bar b_1}>{\bar a_1}l_{+}\exp \{({|
\bar b_1|}+{\bar b_1})\omega \}$.
\end{itemize}
\end{theorem}

\begin{proof}
 Corresponding to the operator equation (\ref{b1}) we have
\begin{equation}
(y_1^\Delta (t),  y_2^\Delta (t))^T=\lambda (\Phi
_1(t),  \Phi _2(t))^T.  \label{c1}
\end{equation}
Suppose that $y\in X$ is a solution of system (\ref{c1}) for a certain
$\lambda \in (0, 1)$. Integrating (\ref{c1}) over set
$\mathbb{I}_\omega$, we obtain
\begin{gather}
{\bar b_1}\omega = \int_{\mathbb{I}_\omega}\big[a_1(t)\exp
\{y_1(t-\tau _1(t))\}+\frac{c(t)\exp \{y_2(t-\gamma (t))\}}{
\exp \{2y_1(t)\}/ n+\exp \{y_1(t)\}+a}\big]\Delta t,  \label{c2}
\\
{\bar b_2}\omega = \int_{\mathbb{I}_\omega}\big[\frac{
a_2(t)\exp {y_1(t-\tau _2(t))}}{\exp \{{2y_1(t-\tau _2(t))\}}/n
+\exp {y_1(t-\tau _2(t))}+a}\big]\Delta t.  \label{c3}
\end{gather}
 From (\ref{c1})--(\ref{c3}), we have
\begin{gather}
\int_{\mathbb{I}_\omega}|y_1^\Delta (t)|\Delta t
\leq \lambda \big\{ \int_{\mathbb{I}_\omega}|b_1(t)|\Delta
t+\int_{\mathbb{I}_\omega}[-\Phi _1(t)+b_1(
t)]\Delta t\big\} <({|\bar b_1|}+{\bar b_1})\omega ,\label{c4}
\\
\int_{\mathbb{I}_\omega}|y_2^\Delta (t)|\Delta t
\leq \lambda \big\{ \int_{\mathbb{I}_\omega}|b_2(t)|\Delta
t+\int_{\mathbb{I}_\omega}[\Phi _2(t)+b_2(
t)]\Delta t\big\} <({|\bar b_2|}+{\bar b_2})\omega . \label{c5}
\end{gather}
Because $y=(y_1(t), y_2(t))^T\in X$, there exist
$\xi _i, \eta _i\in \mathbb{I}_\omega$, $i=1, 2$ such that
\begin{equation}
y_i(\xi _i)=\min_{t\in \mathbb{I}_\omega}\{y_i(t)\}, \quad
y_i(\eta _i)=\max_{t\in \mathbb{I}_\omega}\{y_i(t)\}. \label{c6}
\end{equation}
 From (\ref{c3}) and (\ref{c6}) we get
\begin{align*}
{\bar b_2}\omega
&\leq \int_{\mathbb{I}_\omega}\big[\frac{
a_2(t)\exp \{y_1(\eta _1)\}}{\exp \{2y_1(\xi _1)\}/n
 +\exp \{y_1(\xi _1)\}+a}\big]\Delta t \\
&=\frac{{\bar a_2}\omega \exp \{y_1(\eta _1)\}}{\exp
\{2y_1(\xi _1)\}/n+\exp \{y_1(\xi _1)\}+a},
\end{align*}
which implies
\begin{equation}
y_1(\eta _1)\geq \ln \big[\frac{{\bar b_2}}{{\bar a_2}}(\exp
\{2y_1(\xi _1)\}/n+\exp {y_1(\xi _1)}+a)\big]. \label{c7}
\end{equation}
By virtue of (\ref{c4}), (\ref{c7}) and Lemma \ref{lem2.2}, we get
\begin{align*}
y_1(t) &\geq y_1(\eta _1)-\int_{\mathbb{I}_\omega}|y_1^\Delta
(t)|\Delta t \\
&>\ln [\frac{{\bar b_2}}{{\bar a_2}}(\exp \{2y_1(\xi
_1)\}/n+\exp {y_1(\xi _1)}+a)]-({|\bar b_1|}+
{\bar b_1})\omega .
\end{align*}
In particular,
\[
y_1(\xi _1)>\ln \big[\frac{{\bar b_2}}{{\bar a_2}}(\exp
\{2y_1(\xi _1)\}/n+\exp {y_1(\xi _1)}+a)\big]-({|\bar
b_1|}+{\bar b_1})\omega .
\]
or
\begin{equation}
\frac{{\bar b_2}}n\exp \{2y_1(\xi _1)\}-\big({\bar a_2}\exp \{(
{|\bar b_1|}+{\bar b_1})\omega \}-{\bar b_2}\big)\exp
\{y_1(\xi _1)\}+{\bar b_2}a<0. \label{c8}
\end{equation}
According to (i), we have
\begin{equation}
\ln l_{-}<y_1(\xi _1)<\ln l_{+}.  \label{c9}
\end{equation}
 From (\ref{c3}), we have
\begin{align*}
{\bar b_2}\omega
&\geq \int_{\mathbb{I}_\omega}\big[\frac{
a_2(t)\exp \{y_1(\xi _1)\}}{\exp \{2y_1(\eta _1)\}/n
+\exp \{y_1(\eta _1)\}+a} \big]\Delta t\\
&=\frac{{\bar a_2}\omega \exp \{y_1(\xi _1)\}}{\exp
\{2y_1(\eta _1)\}/n+\exp \{y_1(\eta _1)\}+a};
\end{align*}
that is,
\[
y_1(\xi _1)<\ln \big[\frac{{\bar b_2}}{{\bar a_2}}(\exp
\{2y_1(\eta _1)\}/n+\exp {y_1(\eta _1)}+a)\big].
\]
Which together with (\ref{c4}) and Lemma \ref{lem2.2} lead to
\begin{align*}
y_1(t) &\leq y_1(\xi _1)+\int_{\mathbb{I}_\omega}|y_1^\Delta
(t)|\Delta t \\
&< \ln \big[\frac{\overline{b_2}}{\overline{a_2}}(\exp \{2y_1(\eta
_1)\}/n+\exp {y_1(\eta _1)}+a)\big]+({|\bar b_1|}+
{\bar b_1})\omega .
\end{align*}
Hence, we have
\[
y_1(\eta _1)<\ln \big[\frac{{\bar b_2}}{{\bar a_2}}(\exp
\{2y_1(\eta _1)\}/n+\exp {y_1(\eta _1)}+a)\big]
+({|\bar b_1|}+{\bar b_1})\omega .
\]
or
\[
\frac{{\bar b_2}}n\exp \{2y_1(\eta _1)\}-\big({\bar a_2}\exp \{-(
{|\bar b_1|}+{\bar b_1})\omega \}-{\bar b_2}\big)\exp
\{y_1(\eta _1)\}+{\bar b_2}a>0.
\]
Similarly, we can show that
\begin{equation}
y_1(\eta _1)<\ln v_{-}\quad\text{or}\quad
y_1(\eta _1)>\ln v_{+}.  \label{c10}
\end{equation}
 From (\ref{c4}), (\ref{c9}) and Lemma \ref{lem2.2},
\begin{equation}
y_1(t)\leq y_1(\xi _1)+\int_{\mathbb{I}_\omega}|y_1^\Delta
(t)|\Delta t<\ln l_{+}+({|\bar b_1|}+{\bar b_1})\omega
:=P_1. \label{c11}
\end{equation}
On the other hand, (\ref{c2}) and (\ref{c6}) yield
\begin{gather}
{\bar b_1}\omega \geq \frac{{\bar c}\omega \exp \{y_2(\xi _2)\}}{
\exp \{2P_1\}/n+\exp \{P_1\}+a},  \label{c12}
\\
{\bar b_1}\omega \leq {\bar a_1}\omega \exp \{P_1\}+\frac{{\bar c
}\omega \exp \{y_2(\eta _2)\}}a.  \label{c13}
\end{gather}
It follows from (\ref{c12}) that
\[
y_2(\xi _2)\leq \ln \big[\frac{{\bar b_1}}{{\bar c}}(\exp
\{2P_1\}/n+\exp \{P_1\}+a)\big].
\]
This, together with (\ref{c5}) and Lemma \ref{lem2.2}, yields
\begin{equation}
\begin{aligned}
y_2(t) &\leq y_2(\xi _2)+\int_{\mathbb{I}_\omega}|y_2^\Delta
(t)|\Delta t   \\
&<\ln \big[\frac{{\bar b_1}}{{\bar c}}(\exp \{2P_1\}/n+\exp
\{P_1\}+a)\big]+({|\bar b_2|}+{\bar b_2}
)\omega :=P_2.  \label{c14}
\end{aligned}
\end{equation}
Moreover, because of (ii), it follows from (\ref{c13}) that
\[
y_2(\eta _2)\geq \ln \big[\frac a{{\bar c}}({\bar b_1}-\exp
\{({\bar B_1}+{\bar b_1})\omega {\bar a_1}l_{+}\})\big],
\]
which, combined with Lemma \ref{lem2.2}, gives
\begin{equation}
\begin{aligned}
y_2(t) &\geq y_2(\eta _2)-\int_{\mathbb{I}_\omega}|y_2^\Delta
(t)|\Delta t   \\
&>\ln [\frac a{{\bar c}}({\bar b_1}-\exp \{({
\bar B_1}+{\bar b_1})\omega {\bar a_1}l_{+}\})]-({
|\bar b_2|}+{\bar b_2})\omega :=P_3.  \label{c15}
\end{aligned}
\end{equation}
It follows from (\ref{c14}) and (\ref{c15}) that
\begin{equation}
\max_{t\in \mathbb{I}_\omega}|y_2(t)|<\max \{|P_2|, |P_3|\}:=P.
\label{c16}
\end{equation}
Obviously, $\ln l_{\pm }$, $\ln v_{\pm }$, $P_1$ and $P$ are
independent of the choice of $\lambda \in (0, 1)$.

Now, let's consider $QNh$ with $h=(h_1, h_2)^T\in \mathbb{R}^2$.
Note that
\[
QNh=\begin{pmatrix}
{\bar b_1}-{\bar a_1}\exp \{h_1\}-\frac{
{\bar c}\exp \{h_2\}}{\exp \{2h_1\}/n+\exp \{h_1\}+a} \\
-{\bar b_2}+\frac{{\bar a_2}\exp \{h_1\}}{
\exp \{2h_1\}/n+\exp \{h_1\}+a}
\end{pmatrix}
\]
In view of (i) and (ii), one can show that $QNh=0$ has two distinct
constant solutions:
\begin{gather*}
z^{\dag }=\Big(\ln u_{l-}, \ln \frac{({\bar b_1}-{\bar a_1}
u_{l-})g(u_{l-})}{{\bar c}}\Big)^T,
\\
z^{\ddag }=\Big(\ln u_{l+}, \ln \frac{({\bar b_1}-{\bar a_1}
u_{l+})g(u_{l+})}{{\bar c}}\Big)^T,
\end{gather*}
where, $g(u_{l-})=u_{l-}^2/n+u_{l-}+a$, and
$g(u_{l+})=u_{l+}^2/n+u_{l+}+a$. Chose $M>0$ such that
\begin{equation}
M>\max \big\{ |\ln \frac{(\bar{b}_1-\bar{a}_1u_{l-})g(
u_{l-}) }{\bar{c}}|, \;
|\ln \frac{(\bar{b}_1-\bar{a}_1u_{l+})g(u_{l+})}{{\bar
c}}|\big\} .  \label{c17}
\end{equation}
And set
\begin{gather*}
\Omega _1 = \big\{ y=(y_1(t), y_2(t))^T\in X:
y_1(t)\in (\ln l_{-}, \ln v_{-}), \;
\max_{t\in I_\omega }|y_2(t)|<P+M \big\} ,
 \\
\begin{aligned}
\Omega _2 = \big\{& y=(y_1(t), y_2(t))^T\in X:
\min_{t\in I_\omega }y_1(t)\in (\ln l_{-}, \ln l_{+}), \\
&\max_{t\in I_\omega }y_1(t)\in (\ln v_{+}, P_1), \;
\max_{t\in I_\omega }|y_2(t)|<P+M\big\}.
\end{aligned}
\end{gather*}
Then both $\Omega _1$ and $\Omega _2$ are open bounded subset of $X$.
It follows from (\ref{b2}) and (\ref{c17}) that
$z^{\dag }\in \Omega _1, z^{\ddag }\in \Omega _2$. With the help
of (\ref{b2}), (\ref{c9})--(\ref{c11}), and (\ref{c16})--(\ref{c17}),
it is not difficult to show that $\Omega_1\cap \Omega _2=\emptyset $
and $\Omega _i$ verifies the requirements
(a) of Lemma \ref{lem2.3} for $i=1,2$. When $y \in \partial \Omega _i\cap R^2$,
$y $ is a constant vector in $R^2$, then $QNy\neq 0$.
Moreover, after direct calculation we get the Brouwer degree
\[
\deg (JQN, \Omega _i\cap \ker L, \theta )
=(-1)^{i+1}\neq 0,
\]
for $i=1,2$, where the isomorphism $J$ can be chosen to be the
identity mapping, since $\mathop{\rm Im}Q=\mathop{\rm Ker}L$.
Up to now, we have proved that $\Omega _i$ verify all the
requirements of Lemma \ref{lem2.3}. Therefore, by Lemma \ref{lem2.3},
we derive that \eqref{e1.1} has at least two $\omega$-periodic solutions
lying in $\mathop{\rm Dom}L\cap \overline{\Omega _i}$.
The proof is complete.
\end{proof}

\begin{remark} \label{rmk 3.1}\rm
 Assume $\mathbb{T}=\mathbb{Z}$, then
\eqref{e1.1} becomes a discrete analogue of \eqref{e1.1} which
has been discussed in \cite{z4}. Therefore, our result obtained
generalized the result in the literature.
\end{remark}


\section{Example}

 Consider the following periodic predator-prey system
with a IV functional response
\begin{equation}
\begin{gathered}
\begin{aligned}
y_1^\Delta (t) &=  (\frac 1{20}+\sin t)
-(\frac 1{60}+\cos t)\exp \{y_1(t-\sin t)\} \\
&\quad -\frac{\exp \{y_2(t-\cos t\}}
 {2\exp \{y_1(t)\}+2\exp \{y_1(t)\}+1},
\end{aligned}  \\
y_2^\Delta (t)  =  -(\frac 13+\cos t)+
\frac{2(1+\cos t)\exp \{y_1(t-\cos t)\}}{2\exp \{y_1(t-\cos t)\}+2\exp
\{y_1(t-\cos t)\}+1},
\end{gathered}  \label{d1}
\end{equation}
by choosing the $2\pi$-periodic time scale
\[
\mathbb{T}=\bigcup_{s\in \mathbb{Z}}[2(s-1)\pi ,2s\pi ].
\]
Then, system (\ref{d1}) has at least two $2\pi$-periodic solutions.

Direct calculations lead to
$$
\kappa =\min \{\mathbb{R^+\cap\mathbb{T}}\}=0,
$$
$I_\omega =[\kappa , \kappa +\omega ]\cap \mathbb{T}=[0,2\pi]$,
 ${\bar a_1}=1/60, {\bar a_2}=1$,
 ${\bar b_1}=1/20, {\bar b_2}=1/3$,
$\exp \{{\bar b_1}+ {|\bar b_1}|\}<1.135$,
$l_{+}<2.302$. It is straight forward to check that
\begin{gather*}
{\bar a_2} = 1.000>0.757>{\bar b_2}\big(1+2\sqrt{a/n}\big)
\exp \{({|\bar b_1|}+{\bar b_1})\omega \},
\\
{\bar b_1} = 0.050>0.045>{\bar a_1}l_{+}\exp \{({|\bar
b_1|}+{\bar b_1})\omega \},
\end{gather*}
which show that all the conditions in Theorem \ref{thm3.1} are fulfilled. By
Theorem \ref{thm3.1} we derive that (\ref{d1}) has at least two
$2\pi$-periodic solutions.

\begin{remark} \label{rmk 4.1} \rm
System (\ref{d1}) models two populations (one
is the predator and the other is the prey) that are both continuous
in one period of the year, die out in other period of the year,
and their offspring are renascent after incubating or dormant
in another period of the year, both of them giving rise to
non-overlapping populations.
\end{remark}

\subsection*{Acknowledgements}
This work is supported by a grant 3777501 from the Natural Science
Development Foundation of CUIT of China.

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\end{document}