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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 110, pp. 1--10.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/110\hfil Positive periodic solutions]
{Positive periodic solutions of  neutral
logistic difference equations with multiple delays}

\author[Y. Li, Q. Chen \hfil EJDE-2007/110\hfilneg]
{Yongkun Li, Qian Chen}  

\address{Department of Mathematics\\
Yunnan University\\
Kunming, Yunnan 650091, China}
\email{yklie@ynu.edu.cn}

\thanks{Submitted May 22, 2007. Published August 14, 2007.}
\thanks{Supported by the National Natural Sciences Foundation of China}
\subjclass[2000]{34K13, 34K40, 92B05}
\keywords{Positive periodic solution;
 neutral functional difference equation; \hfill\break\indent
 strict-set-contraction}

\begin{abstract}
 Using a fixed point theorem of strict-set-contraction, we
 some established the existence of positive periodic
 solutions for the neutral logistic difference equation,
 with multiple delays,
 $$
 \Delta x(n)=x(n)\Big[ a(n) -\sum_{i=1}^{p}a_{i}(n)
 x(n-\tau_{i}(n))  -\sum_{j=1}^{q}c_{j}(n)
 \Delta x(n-\sigma_{j}(n))\Big].
 $$
\end{abstract}

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

\section{Introduction}

Using the continuation theory for $k$-set-contractions
and the Mawhin's continuation theorem, \cite{g1},  the existence of
positive periodic solutions for the following neutral
logistic differential equation with multiple delays
\begin{equation} \label{e1.1}
\frac{{\mathrm{d}} N(t)}{{\mathrm{d}} t}
=N(t)\Big[a(t)-\beta(t)N(t) -\sum_{j=1}^{n}b_{j}(t)
N_j(t-\tau_{j}(t))-\sum_{j=1}^{n}c_{j}(t)
N'_{j}(t-\sigma_{j}(t))\Big]
\end{equation}
are investigated in \cite{l1,l2,y1,y2,y3}. Where
 $a(t)$, $\beta_{j}(t)$, $b_{j}(t)$, $c_{j}(t)$,
$\tau_{j}(t)$, $\gamma_{j}(t)$ are continuous $\omega$-periodic
functions and $a(t)\geq 0$, $\beta_{j}(t)\geq 0$, $b_{j}(t)\geq 0$,
$c_{j}(t)\geq 0$ ($j=1,2,\dots,n$). For the ecological justification
of \eqref{e1.1}, see for example \cite{g2,g3,k1,k2,p1}.


Given $a,b$ be integers and $a<b$, we employ intervals to denote
discrete sets such as $\mathbb{Z}[a,b]=\{a,a+1,\dots,b\}$,
$\mathbb{Z}[a,b)=\{a,\dots,b-1\}$,
$\mathbb{Z}[a,\infty)=\{a,a+1,\dots\}$, etc.
Let $\omega\in \mathbb{Z}[1,\infty)$ be fixed. Throughout this work,
we denote the product of $y(n)$ from $n = a$ to $n = b$ by
$\prod_{n=a}^{n=b}y(n)$ with the understanding that
$ \prod_{n=a}^{n=b} y(n) = 1$ for all $a> b$.

 The main purpose of this paper is to use a fixed point theorem of
strict-set-contraction \cite[Theorem 3]{c1} to
 establish   the  existence  of positive periodic solutions
 of the following neutral logistic difference equation, with multiple delays,
\begin{equation} \label{e1.2}
\Delta x(n)=x(n)\Big[ a(n) -\sum_{i=1}^{p}a_{i}(n)
x(n-\tau_{i}(n)) -\sum_{j=1}^{q}c_{j}(n)
\Delta x(n-\sigma_{j}(n))\Big],
\end{equation}
where  $a,a_{i},c_{j}\in  (\mathbb{Z}(-\infty,\infty),\mathbb{R}^+)$
 and $\tau_i,\sigma_j\in (\mathbb{Z}(-\infty,\infty),
  \mathbb{R})$, $i=1,2,\dots,p$, $j=1,2,\dots,q$ are $\omega$-periodic
  functions.  To the best of our
knowledge, this is
  the first paper to study the existence of periodic solutions of neutral
  logistic difference equations delays.


   For convenience, we introduce the notation
\begin{gather*}
\Theta:=\prod_{k=0}^{\omega-1}(1+a(k)),    \quad
 \Gamma=\sum_{s=0}^{\omega-1}\Big[\Theta^{-1} \sum_{i=1}^{p}a_i(s)-
 \sum_{j=1}^{q}c_j(s)\Big], \\
\Pi=\sum_{s=0}^{\omega-1}\Big[ \sum_{i=1}^{p}a_i(s)+
 \sum_{j=1}^{q}c_j(s)\Big],  \quad
f^M=\max_{n\in \mathbb{Z}[0,\omega-1]}\{f(n)\},\\
f^m=\min_{n\in \mathbb{Z}[0,\omega-1]}\{f(n)\},
\end{gather*}
where $f$ is a continuous $\omega$-periodic function.
In this paper, we assume that
\begin{itemize}
  \item [(H1)]
   $\prod_{k=0}^{\omega-1}(1+a(k))>1$.
  \item [(H2)] $\Theta \sum_{i=1}^{p}a_i(n)-\sum_{j=1}^{q}c_j(n)\geq0$.
  \item [(H3)]
 $(1+a^m) \frac{\Gamma}{\Theta(\Theta-1)}\geq\max_{n\in \mathbb{Z}[0,\omega-1]}
 \Big\{\sum_{i=1}^{p}a_i(n)+\sum_{j=1}^{q}c_j(n)\Big\}$.
  \item [(H4)]
$\frac{\Pi(a^M-1)\Theta}{\Theta-1}\leq\min_{n\in
\mathbb{Z}[0,\omega]}\Big\{\Theta^{-1} \sum_{i=1}^{p}a_i(n)
-\sum_{j=1}^{q}c_j(n)\Big\}$.
  \item [(H5)]
 $\frac{\Theta-1}{\Theta\Gamma}\sum_{j=1}^{q}c_j^M<1$.
\end{itemize}

\section{Preliminaries}

To obtain the existence of a periodic solution of system
\eqref{e1.2}, we first make the following preparations:


 Let $\mathbb{E}$ be
a Banach space and $K$ be a cone in $\mathbb{E}$. The semi-order
induced by the cone $K$ is denoted by ``$\leq$''. That is, $x\leq y$
if and only if $y-x\in K$. In addition, for a bounded subset
$A\subset \mathbb{E}$, let $\alpha_{\mathbb{E}}(A)$ denote the
(Kuratowski) measure of non-compactness defined by
\begin{align*}
\alpha_{E}(A)=\inf\big\{&\delta>0:
\text{ there is a finite number of subsets } A_i\subset A\\
&\text{such that $A=\cup_{i}A_i$ and }
\mathop{\rm diam}(A_i)\leq \delta\big\},
\end{align*}
where $\mathop{\rm diam}(\cdot)$ denotes the diameter of the set.

Let $\mathbb{E}, \mathbb{F}$ be two Banach spaces and  $D\subset
\mathbb{E}$, a continuous and bounded map  $\Phi : \bar{\Omega}
\to \mathbb{F}$ is called
 $k$-set contractive if for any bounded set $S\subset D$ we have
 \[
 \alpha_\mathbb{F}(\Phi(S))\leq k\alpha_\mathbb{E}(S).
 \]
A function $\Phi$  is called strict-set-contractive if it is
$k$-set-contractive for some $0 \leq k < 1$.

The following lemma   is useful for the
proof of our main results of this paper.

\begin{lemma} [\cite{c1,g4}] \label{lem2.1}
Let $K$ be a cone of the real Banach space $\mathbb{E}$ and
$K_{r,R}=\{x\in K: r\leq \|x\|\leq R\}$ with $R>r>0$. Suppose that
$\Phi: K_{r,R}\to K$ is  strict-set-contractive such that
one of the following two conditions is satisfied:
\begin{itemize}
\item [(i)] $\Phi  x \nleq x$, for all $x \in K$, $\|x\|= r$ and
$\Phi x\ngeq x$,  for all $x\in K$, $\|x\|=R$.

\item [(ii)]   $\Phi  x \ngeq x$, for all $x \in K$, $\|x\|= r$ and
$\Phi x\nleq x$, for all $x\in K$, $\|x\|=R$.
\end{itemize}
Then $\Phi$ has at least one  fixed point in $K_{r,R}$.
\end{lemma}

Finding an $\omega$-periodic solution of  \eqref{e1.1} is equivalent to
finding an $\omega$-periodic solution of the equation
\[
x(n)=\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big]
\]
where
\[
G(n,s)=\frac{\prod_{k=s+1}^{n+\omega-1}(1+a(k))}
{\prod_{k=0}^{\omega-1}(1+a(k))-1},\quad
n\leq s\leq n+\omega-1.
\]
It is easy to see that, for $s\in \mathbb{Z}[k,k+\omega-1]$ we have
\[
\frac{1}{\prod_{k=0}^{\omega-1}(1+a(k))-1}\leq
G(k,s)\leq\frac{\prod_{k=0}^{\omega-1}(1+a(k))}{\prod_{k=0}^{\omega-1}(1+a(k))-1}.
\]
Let
\[
\mathbb{X}=\mathbb{Y}=\{x:
\mathbb{Z}(-\infty,\infty)\to\mathbb{R}, x(n+\omega)=x(n),
n\in \mathbb{Z}(-\infty,\infty) \},
\]
$\mathbb{X}$ with the norm $|x|_0=\max_{n\in
\mathbb{Z}[0,\omega-1]}\{|x(n)|\},x\in \mathbb{X}$ and $\mathbb{Y}$
with the norm  $|y|_1=\max_{n\in
\mathbb{Z}[0,\omega-1]}\{|y|_0, |\Delta y|_0\}, y\in \mathbb{Y}$.
Then $\mathbb{X},\mathbb{Y}$ are  Banach spaces. Note that solving
\eqref{e1.2} is equivalent to solving
\begin{equation} \label{e2.1}
    x=\Phi x,
\end{equation}
where
\begin{align*}
(\Phi x)(n)=\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big]
\end{align*}
for $x\in \mathbb{Y}$.

To apply Lemma \ref{lem2.1} to  \eqref{e1.2}, we define the cone $K$ in
$\mathbb{Y}$ by
 \begin{equation} \label{e2.2}
K=\{x \in \mathbb{Y}: x(n)\geq 0 \,\, \text{and}\,\, x(n)\geq
\Theta^{-1} |x|_1, n\in \mathbb{Z}[0,\omega-1]\}.
 \end{equation}
In what follows, we will give some lemmas concerning $K$ and $\Phi$
defined by \eqref{e2.1} and \eqref{e2.2}, respectively.

\begin{lemma} \label{lem2.2} Assume that {\rm (H1)--(H3)} hold.
\begin{itemize}
  \item [(i)] If $a^M\leq 1$, then $\Phi: K\to K$ is well defined.
  \item [(ii)] If (H4) holds and $a^M>1$, then $\Phi: K\to K$ is well defined.
\end{itemize}
\end{lemma}

\begin{proof}
For each $x\in K$, it is clear that $\Phi x\in \mathbb{Y}$. In view
of \eqref{e2.2}, for $n\in \mathbb{Z}(-\infty,\infty)$, we obtain
\begin{align*}
&(\Phi x)(n+\omega)\\
&= \sum_{s=n+\omega}^{n+2\omega-1}G(n+\omega,s)x(s)
\Big[ \sum_{i=1}^{p}a_i(s) x(s-\tau_i(s)) +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big]\\
&= \sum_{u=n}^{n+\omega-1}G(n+\omega,u+\omega)x(u+\omega)
\Big[ \sum_{i=1}^{p}a_i(u+\omega) x(u+\omega-\tau_i(u+\omega))\\
 &\quad +\sum_{j=1}^{q}c_j(u+\omega)
\Delta x(u+\omega-\sigma_j(u+\omega))\Big]\\
 &= (\Phi x)(n).
\end{align*}
 That is, $(\Phi x)(n+\omega)=(\Phi x)(n)$,
$n\in \mathbb{Z}(-\infty,\infty)$. So $\Phi x \in \mathbb{Y}$. In view of
(H2), for $x \in K, n\in\mathbb{Z}(-\infty,\infty)$, we have
\begin{equation} \label{e2.3}
\begin{aligned}
& \sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s)) +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big] \\
&\geq \sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))- \sum_{j=1}^{q}c_j(s)
|\Delta x(s-\sigma_j(s))|\Big] \\
&\geq \sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s)\Theta^{-1} |x|_1- \sum_{j=1}^{q}c_j(s)
|x|_1\Big]
\geq  0.
\end{aligned}
\end{equation}
Therefore, for $x\in K, n\in \mathbb{Z}[0,\omega-1]$.
We find that
\[
   |\Phi x|_0 \leq \frac{\Theta}{\Theta-1}
\sum_{s=n}^{n+\omega-1}x(s)\Big[ \sum_{i=1}^{p}a_i(s)x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big]
\]
and
\begin{equation} \label{e2.4}
\begin{aligned}
(\Phi x)(n)&\geq \frac{1}{\Theta-1}\sum_{s=0}^{\omega-1}x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big] \\
&= \frac{1}{\Theta-1}\sum_{s=0}^{\omega-1}x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big] \\
&\geq \Theta^{-1}|\Phi x|_0.
\end{aligned}
\end{equation}
Now, we show that $(\Phi x)(n)\geq \Theta^{-1} |(\Phi x)|_1, n\in
\mathbb{Z}[0,\omega-1]$. From \eqref{e2.2}, we have
 \begin{align}
 &\Delta(\Phi x)(n) \nonumber\\
  &= \Delta(\Phi x)(n+1)-\Delta(\Phi x)(n) \nonumber \\
  &= \sum_{s=n+1}^{n+\omega}G(n+1,s)x(s)\Big[ \sum_{i=1}^{p}a_i(s)
x(s-\tau_i(s))  +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big] \nonumber \\
&\quad -\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s))\Big] \nonumber \\
  &= G(n+1,n+\omega)x(n+\omega)\Big[ \sum_{i=1}^{p}a_i(n+\omega)
x(n+\omega-\tau_i(n+\omega)) \nonumber \\
&\quad  +\sum_{j=1}^{q}c_j(n+\omega)
 \Delta x(n+\omega-\sigma_j(n+\omega))\Big] \nonumber \\
&\quad +\sum_{s=n}^{n+\omega-1}G(n+1,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s) \Delta x(s-\sigma_j(s))\Big] \nonumber \\
&\quad -G(n+1,n)x(n)\Big[ \sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 +\sum_{j=1}^{q}c_j(n)
 \Delta x(n-\sigma_j(n))\Big] \nonumber \\
&\quad -\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s) \Delta x(s-\sigma_j(s))\Big] \nonumber \\
&= [G(n+1,n+\omega)-G(n+1,n)]x(n)\Big[ \sum_{i=1}^{p}a_i(n) x(n-\tau_i(n)) \nonumber \\
&\quad  +\sum_{j=1}^{q}c_j(n)
 \Delta x(n-\sigma_j(n))\Big] \nonumber \\
&\quad +\sum_{s=n}^{n+\omega-1}[G(n+1,s)-G(n,s)]x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s)) \nonumber \\
&\quad  +\sum_{j=1}^{q}c_j(s) \Delta x(s-\sigma_j(s))\Big] \nonumber \\
&= a(n)\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s) \Delta x(s-\sigma_j(s))\Big] \nonumber \\
&\quad -x(n)\Big[ \sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 +\sum_{j=1}^{q}c_j(n) \Delta x(n-\sigma_j(n))\Big] \nonumber \\
&= a(n)(\Phi x)(n)-x(n)\Big[ \sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 +\sum_{j=1}^{q}c_j(n) \Delta x(n-\sigma_j(n))\Big]. \label{e2.5}
\end{align}
It follows from the above inequality and \eqref{e2.3}  that
if $(\Phi x)(n)\geq 0$, then
\begin{equation} \label{e2.6}
(\Delta\Phi x)(n)\leq a(n)(\Phi x)(n)\leq a^M(\Phi x)(n)\leq (\Phi
x)(n).
\end{equation}
On the other hand, from \eqref{e2.5} and (H3), if $(\Delta\Phi x)(n) <0$,
then
\begin{equation} \label{e2.7}
\begin{aligned}
&- \Delta(\Phi x)(n) \\
&= x(n)\Big[ \sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 +\sum_{j=1}^{q}c_j(n)
\Delta x(n-\sigma_j(n))\Big]-a(n)(\Phi x)(n)
 \\
&\leq |x|_1^2\Big[ \sum_{i=1}^{p}a_i(n)
 +\sum_{j=1}^{q}c_j(n)\Big]-a^m(\Phi x)(n)
 \\
&\leq (1+a^m)\frac{1}{\Theta(\Theta-1)}|x|_1^2\sum_{s=0}^{\omega-1}\Big[\Theta^{-1}
\sum_{i=1}^{p}a_i(s)-\sum_{j=1}^{q}c_j(s)\Big]
 -a^m(\Phi x)(n) \\
&= (1+a^m)\sum_{s=n}^{n+\omega-1}
\frac{1}{\Theta-1}\Theta^{-1}|x|_1\Big[\Theta^{-1}|x|_1
\sum_{i=1}^{p}a_i(s)-|x|_1\sum_{j=1}^{q}c_j(s)\Big]-a^m(\Phi x)(n) \\
&\leq (1+a^m)\sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[a(s)x(s-\tau_i(s))
-\sum_{j=1}^{q}c_j(s)|\Delta x(s-\sigma_j(s))|\Big] \\
&\quad -a^m(\Phi x)(n) \\
&= (1+a^m)(\Phi x)(n)-a^m(\Phi x)(n) \\
&= (\Phi x)(n).
\end{aligned}
\end{equation}
It follows from \eqref{e2.6} and \eqref{e2.7} that
$|(\Delta\Phi x)|_0\leq |\Phi x|_0$. So $|\Phi x|_1 = |\Phi x|_0$.
 By \eqref{e2.4} we have
$(\Phi x)(n) \geq \Theta^{-1} |\Phi x|_1$. Hence, $\Phi  x \in K$.
The proof of part (i) is complete.

Part (ii): In view of the proof of (i), we only need to prove that
$(\Delta\Phi x)(n) \geq 0$ implies
\[
(\Delta\Phi x)(n) \leq (\Phi x)(n).
\]
 From \eqref{e2.3}, \eqref{e2.5}, (H2) and (H4), we
obtain
\begin{align*}
&(\Delta\Phi x)(n) \\
&\leq a(n)(\Phi x)(n)-\Theta^{-1}|x|_1\Big[
\sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 -\sum_{j=1}^{q}c_j(n)|\Delta x(n-\sigma_j(n))|\Big]
 \\
&\leq a(n)(\Phi x)(n)-\Theta^{-1}|x|_1^2\Big[
\sum_{i=1}^{p}\Theta^{-1}a_i(n) -\sum_{j=1}^{q}c_j(n)
\Big] \\
&\leq a^M(\Phi
x)(n)-|x|_1^2\frac{a^M-1}{\Theta-1}\sum_{s=n}^{n+\omega-1}\Big[\sum_{i=1}^{p}a_i(s)
+\sum_{j=1}^{q}c_j(s)
\Big] \\
&\leq a^M(\Phi x)(n)-(a^M-1)\sum_{s=n}^{n+\omega-1}
\frac{\Theta}{\Theta-1}\Theta^{-1}|x|_1\Big[\sum_{i=1}^{p}a_i(s)|x|_1
+\sum_{j=1}^{q}c_j(s)|x|_1
\Big]\\
&\leq a^M(\Phi x)(n)-(a^M-1) \sum_{s=n}^{n+\omega-1}G(n,s)x(s)
\Big[\sum_{i=1}^{p}a_i(s)x(s-\tau_i(s))\\
&\quad +\sum_{j=1}^{q}c_j(s)|\Delta x(s-\sigma_j(s))|\Big] \\
&\leq a^M(\Phi x)(n)-(a^M-1)
\sum_{s=n}^{n+\omega-1}G(n,s)x(s)
\Big[\sum_{i=1}^{p}a_i(s)x(s-\tau_i(s))\\
&\quad +\sum_{j=1}^{q}c_j(s)\Delta x(s-\sigma_j(s))\Big]\\
&= a^M(\Phi x)(n)-(a^M-1)(\Phi x)(n)\\
&= (\Phi x)(n).
\end{align*}
The proof of (ii) is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
$\Phi(S)$ is precompact in $\mathbb{X}$ for any bounded
set $S$ in $\mathbb{Y}$.
\end{lemma}

\begin{proof}
Let $d$ be a constant and
    $S=\{x\in \mathbb{Y}: \|x\|_1 <d \}$ is a bounded
   set. We prove that $\overline{\Phi S}$ is compact. To do this, we
   must show that any sequence in $\Phi S$ contains a convergent
   subsequence. Thus, let $\{x_{m},m\in \mathbb{Z}[1,\infty)\}$ be a sequence in $S$. Let
   us show  that $\{\Phi x_{m},m\in \mathbb{Z}[1,\infty)\}$ has a convergent
 subsequence.   For convenience, we set
   \[
M(x(n))=x(n)\Big[\sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
 +\sum_{j=1}^{q}c_j(n)
\Delta x(n-\sigma_j(n))\Big],
   \]
then
\[
(\Phi x)(n)=\sum_{s=n}^{n+\omega-1}G(n,s)x(s)M(x(s)).
\]
It is obvious that  the sequence $\{M(x_{m}(0))\}$,
$m\in \mathbb{Z}[1,\infty)$, is bounded;
so the sequence $\{M(x_{m}(0))\}$ contains a convergent
subsequence. So let the sequence $\{x_{m, 0}\}$,
be a subsequence of   $\{x_{m}\}$ such that $\{M(x_{m, 0}(0))\}$ 
is  convergent.

Again, $\{M(x_{m, 0}(1))\}$, $m\in \mathbb{Z}[1,\infty)$, contains a
convergent subsequence. So,
let $\{x_{m, 1}\}$ be a subsequence of $\{x_{m, 0}$,
such that $\{M(x_{m, 1}(1))\}$ is convergent. Observe that
 $\{x_{m, 1}\}$ is a subsequence of $\{x_{m}\}$ and
 that $\{M(x_{m, 1}(0))\}$ and $\{M(x_{m, 1}(1))\}$
 are  convergent.

Now, $\{M(x_{m, 1}(-1)\}$ contains a convergent subsequence.
Let $\{x_{-1, m, 1}\}$ be a subsequence of
$\{x_{m, 1}\}$ such that $\{M(x_{-1, m, 1}(-1))\}$
is convergent. Also, $\{x_{-1, m, 1}\}$  is a subsequence of
$\{x_{m}\}$    and $\{M(x_{-1, m, 1}(-1))\}$,
$\{M(x_{-1, m, 1}(0))\}$, $\{M(x_{-1, m, 1}(1))\}$ are
   convergent.

Continuing in this fashion we find, for each
$l\in \mathbb{Z}(-\infty,\infty)$, a
subsequence $\{x_{-(l+1), m, (l+1)}\}$,
$m\in \mathbb{Z}[1,\infty)\}$ of $\{x_{-l, m, l},
   m\in \mathbb{Z}[1,\infty)\}$ such  that\\
 $\{M(x_{-(l+1), m, (l+1)}(l+1)),m\in \mathbb{Z}[1,\infty)\}$ and
   $\{M(x_{-(l+1), m, (l+1)}(-(l+1))),m\in \mathbb{Z}[1,\infty)\}$
    is convergent. Observe that
   also the sequences $\{M(x_{-(l+1), m, l+1}(-l)),m\in \mathbb{Z}[1,\infty)\}$,
   \dots,  $\{M(x_{-(l+1), m, l+1}(0)),m\in \mathbb{Z}[1,\infty)\}$,
   \dots, $\{M(x_{-(l+1), m, l+1}(l)),m\in \mathbb{Z}[1,\infty)\}$ are convergent.

Consider now the sequence $\{x_{-u, u, u},u\in \mathbb{Z}[1,\infty)\}$. Observe that it
   is a subsequence of $\{x_{m},m\in \mathbb{Z}[1,\infty)\}$, and also that $\{M(x_{-u, u, u}(l)),u\in
   \mathbb{Z}[1,\infty)\}$ is convergent for all $l\in \mathbb{Z}(-\infty,\infty)$.
    Let us show that $\{\Phi x_{-u, u, u},u\in
   \mathbb{Z}[1,\infty)\}$ is a Cauchy sequence.

Since $\{M(x_{-u, u, u}(l)),u\in \mathbb{Z}[1,\infty)\}$ is convergent for all $l\in \mathbb{Z}(-\infty,\infty)$,
   let $\epsilon>0$ be given, there exists $u_{0}\in \mathbb{Z}[1,\infty)$ such that
   $e,g\in \mathbb{Z}[1,\infty)$ with $e,g\geq u_{0}$,
   \[
     \sup_{n\in\mathbb{Z}[0,\omega-1]}|M(x_{-e, e, e}(n))-M(x_{-g, g, g}(n))|
     <\frac{\varepsilon(\Theta-1)}{\Theta}.
   \]
 Therefore, if $e,g\in \mathbb{Z}[1,\infty)$
with $e,g\geq u_{0}$, for all $n\in Z$, we have

\begin{align*}
\Vert (\Phi x_{-e, e, e})(n)-(\Phi x_{-g, g, g})(n)\Vert
      &\leq \sum_{u=n}^{n+\omega-1}G(n,u)
               |M(x_{-e, e, e}(u))-M(x_{-g, g ,g}(u))|\\
      &\leq  \frac{\Theta}{\Theta-1}\sup_{n\in [0,T-1]}|M(x_{-e, e, e}(n))
                 -M(x_{-g, g, g}(n))|\\
      &< \varepsilon.
\end{align*}
This proves that $\{\Phi x_{-u, u, u},u\in \mathbb{Z}[1,\infty)\}$
is a Cauchy sequence in $\mathbb{X}$, and with this, the proof of
Lemma \ref{lem2.3} is complete.
\end{proof}

\begin{lemma} \label{lem2.4}
Assume that {\rm(H1)--(H3)} hold and $R\sum_{j=1}^{q}c_j^M <1$.
\begin{itemize}
  \item [(i)] If $a^M\leq 1$, then $\Phi : K \bigcap\bar{\Omega}_R \to K$
  is strict-set-contractive,
  \item [(ii)] If (H4) holds and $a^M> 1$, then $\Phi : K
\bigcap\bar{\Omega}_R \to K$ is strict-set-contractive,
\end{itemize}
where $\Omega_R=\{x\in \mathbb{Y}: |x|_1<R\}$.
\end{lemma}

\begin{proof}
 We  prove only (i), since the proof of
(ii) is similar. It is easy to see that $\Phi $ is continuous and
bounded. Now we prove that $\alpha _{\mathbb{Y}} (\Phi(S)) \leq
\big(R\sum_{j=1}^{q}c_j^M\big)\alpha_{\mathbb{Y}}(S)$ for any
bounded set $S\subset\bar{\Omega}_R$. Let $\eta = \alpha
_{\mathbb{Y}} (S)$. Then, for any positive number $\varepsilon
<\big(R\sum_{j=1}^{q}c_j^M\big)\eta$, there is a finite family of
subsets $\{S_i\}$ satisfying $S = \bigcup_ iS_i$ with diam($S_i)
\leq \eta + \varepsilon$. Therefore
\begin{equation} \label{e2.8}
| x - y|_1 \leq \eta + \varepsilon \quad
\text{for all } x,y \in S_{i}.
\end{equation}
Since $S$ and $S_i$ are precompact in $\mathbb{X}$, it follows that
there is a finite family of subsets $\{S_{ij}\}$ of $S_i$ such that
$S_i = \bigcup_j S_{ij}$ and
\begin{equation} \label{e2.9}
|x - y|_0 \leq \varepsilon \quad \text{for all } x,y \in S_{ij}.
\end{equation}
By Lemma \ref{lem2.3}, we know that $\Phi(S)$ is precompact in $\mathbb{X}$.
Then, there is a finite family of subsets $\{S_{ijk}\}$ of $S_{ij}$
such that $S_{ij} = \bigcup_kS_{ijk}$ and
\begin{equation} \label{e2.10}
|\Phi x - \Phi y|_0\leq \varepsilon \quad \text{for all }
x, y \in S_{ijk}.
\end{equation}
  From \eqref{e2.3}, \eqref{e2.5} and (2.8)-\eqref{e2.10} and (H2), for any
$x, y \in S_{ijk}$, we obtain
\begin{align*} %2.11
& |(\Delta\Phi x)-(\Delta\Phi y)|_0 \\
&= \max_{n\in
\mathbb{Z}[0,\omega-1]}\Big\{\Big|a(n)(\Phi x)(n)-a(n)(\Phi y)(n) \\
&\quad -x(n)\Big[\sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
+\sum_{j=1}^{q}c_j(n) \Delta x(n-\sigma_j(n))
\Big] \\
&\quad +y(n)\Big[\sum_{i=1}^{p}a_i(n) y(n-\tau_i(n))
+\sum_{j=1}^{q}c_j(n) \Delta y(n-\sigma_j(n))
\Big]\Big|\Big\} \\
&\leq \max_{n\in
\mathbb{Z}[0,\omega-1]}\{|a(n)[(\Phi x)(n)-(\Phi y)(n)]|\} \\
&\quad +\max_{n\in \mathbb{Z}[0,\omega-1]}\Big\{\Big|x(n)\Big[
\sum_{i=1}^{p}a_i(n) x(n-\tau_i(n)) +\sum_{j=1}^{q}c_j(n) \Delta x(n-\sigma_j(n))\Big] \\
&\quad -y(n)\Big[\sum_{i=1}^{p}a_i(n) y(n-\tau_i(n))
+\sum_{j=1}^{q}c_j(n)
 \Delta y(n-\sigma_j(n))\Big]\Big|\Big\} \\
&\leq a^M|(\Phi x)-(\Phi y)|_0 \\
&\quad +\max_{n\in
\mathbb{Z}[0,\omega-1]}\Big\{\Big|x(n)\Big[\Big(\sum_{i=1}^{p}a_i(n)
x(n-\tau_i(n)) +\sum_{j=1}^{q}c_j(n) \Delta x(n-\sigma_j(n))\Big) \\
&\quad -\Big(\sum_{i=1}^{p}a_i(n) x(n-\tau_i(n))
+\sum_{j=1}^{q}c_j(n)
\Delta x(n-\sigma_j(n))\Big)\Big]\Big|\Big\} \\
&\quad +\max_{n\in
\mathbb{Z}[0,\omega-1]}\Big\{\Big|y(n)\Big[\sum_{i=1}^{p}a_i(n)
y(n-\tau_i(n)) \\
&\quad
+\sum_{j=1}^{q}c_j(n) \Delta y(n-\sigma_j(n))\Big][x(n)-y(n)]\Big|\Big\} \\
&\leq a^M\varepsilon +R\max_{n\in
\mathbb{Z}[0,\omega-1]}\Big\{\sum_{i=1}^{p}a_i(n)
|x(n-\tau_i(n))-y(n-\tau_i(n))| \\
&\quad  +\sum_{j=1}^{q}c_j(n) |\Delta x(n-\sigma_j(n))-\Delta
y(n-\sigma_j(n))|
\Big\} \\
&\quad +\varepsilon\max_{n\in
\mathbb{Z}[0,\omega-1]}\Big\{\sum_{i=1}^{p}a_i(n)
y(n-\tau_i(n)) +\sum_{j=1}^{q}c_j(n) |\Delta y(n-\sigma_j(n))|\Big\} \\
&\leq a^M\varepsilon +R\varepsilon\Big(b^M \Big)
+R(\eta+\varepsilon)\Big(\sum_{j=1}^{q}c_j^M
\Big)+R\varepsilon\Big(b^M +
\sum_{j=1}^{q}c_j^M\Big) \\
&= \Big(R\eta \sum_{j=1}^{q}c_j^M\Big)+\hat{H}\varepsilon,
\end{align*}
where $\hat{H}=a^M+2Rb^M+2R\sum_{j=1}^{q}c_j^M$.
 From the above inequality and \eqref{e2.10}, we have
\[
|\Phi  x - \Phi y|_1 \leq
\Big(R\sum_{j=1}^{q}c_j^M\Big)\eta+\hat{H}\varepsilon\quad
\text{for all } x, y \in S_{ijk}.
\]
 Since $\varepsilon$ is arbitrary small, it follows that
\[
 \alpha_{C_{\omega}^1}(\Phi  (S))\leq
 \Big(R\sum_{j=1}^{q}c_j^M\Big)\alpha_{C_{\omega}^1}(S).
\]
 Therefore, $\Phi $ is strict-set-contractive. The proof of
Lemma \ref{lem2.4} is complete.
\end{proof}


\section{Main Result}

Our main result of this paper is as follows.

\begin{theorem} \label{thm3.1}
Assume that {\rm (H1)--(H3), (H5)} hold.
\begin{itemize}
  \item [(i)] If $a^M \leq 1$, then system \eqref{e1.2} has at least
 one positive $\omega$-periodic solution.
  \item [(ii)] If (H4) holds and $a^M > 1$, then system \eqref{e1.2}
 has at least one positive $\omega$-periodic solution.
\end{itemize}
\end{theorem}

\begin{proof}
We only need to prove (i), since the proof of (ii) is similar.
Let $R=\frac{\Theta(\Theta-1)}{\Gamma}$ and
$0 < r < \frac{(\Theta-1)}{\Theta\Pi}$. Then we have $0< r< R$.
From Lemmas \ref{lem2.2} and \ref{lem2.4}, we know that $\Phi$ is strict-set-contractive on
$K_{r,R}$. In view of \eqref{e2.5}, we see that if there exists $x^*\in K$
such that $\Phi x^* = x^*$, then $x^*$ is one positive
$\omega$-periodic solution of system \eqref{e1.2}. Now, we shall prove that
condition (ii) of Lemma \ref{lem2.1} hold.

First, we prove that $\Phi  x \ngeq x$, for all $x \in K$,
$|x|_1=r$. Otherwise, there exists $x \in K$, $|x|_1= r$
 such that $\Phi x\geq x$. So $|x| > 0$ and $\Phi  x - x \in K$, which
implies that
\begin{equation} \label{e3.1}
(\Phi  x)(n) - x(n) \geq \Theta^{-1} |\Phi x - x |_1 \geq0 \quad
\text{for all }  t \in [0,\omega].
\end{equation}
Moreover, for $t \in [0,\omega]$, we have
\begin{align*}
  (\Phi x)(n)&= \sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[ \sum_{i=1}^{p}a_i(s)
x(s-\tau_i(s))
 +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s)) \\
&\leq \frac{\Theta}{\Theta-1}r|x|_0
\sum_{s=0}^{\omega-1}\Big[\sum_{i=1}^{p}a_i(s)+
\sum_{j=1}^{q}c_j(s)\Big] \\
&= \frac{r}{\Theta-1}\Pi|x|_0 \\
&<\Theta^{-1}|x|_0.
\end{align*} %3.2
In view of the above inequality and \eqref{e3.1}, we have
\[
|x|_0\leq |\Phi x|<\Theta^{-1}|x|_0<|x|_0,
\]
which is a contradiction.

Finally, we prove that $\Phi x\nleq x$, for all $x\in K$, $|x|_1=R$ holds.
For this case, we need to prove that
\[
\Phi  x \nless x \quad   x \in K,\, |x|_1= R.
\]
Suppose, for the sake of contradiction, that there exists $x \in K$
and $|x|_1= R$ such that $\Phi  x < x$. Thus $x - \Phi x \in K
\setminus \{0\}$. Furthermore, for any $t \in [0,\omega]$, we have
\begin{equation} \label{e3.3}
    x(n)-(\Phi x)(n)\geq \Theta^{-1}|x-\Phi x|_1>0.
\end{equation}
In addition, for any $t \in [0,\omega]$, we find
\begin{align*}
(\Phi  x)(n)
&= \sum_{s=n}^{n+\omega-1}G(n,s)x(s)\Big[
\sum_{i=1}^{p}a_i(s) x(s-\tau_i(s)) +\sum_{j=1}^{q}c_j(s)
\Delta x(s-\sigma_j(s)) \\
&\geq \frac{1}{\Theta-1}\Theta^{-1}|x|_1^2\sum_{s=0}^{\omega-1}
\Big[\Theta^{-1} \sum_{i=1}^{p}a_i(s) -\sum_{j=1}^{q}c_j(s)\Big] \\
&= \frac{1}{\Theta(\Theta-1)} \Gamma R^2
= R.
\end{align*}
 From this inequality and \eqref{e3.3}, we obtain
\[
|x|>|\Phi  x|_0 \geq R,
\]
which is a contradiction. Therefore, conditions (i) and (ii)
hold. By Lemma \ref{lem2.1}, we see that $\Phi$ has at least one nonzero
fixed point in $K$. Therefore, system \eqref{e1.2} has at least one
positive $\omega$-periodic solution. The proof of Theorem \ref{thm3.1}
 is complete.
\end{proof}


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\end{document}