\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 $$\label{e1.1} \frac{{\mathrm{d}} N(t)}{{\mathrm{d}} t} =N(t)\Big[a(t)-\beta(t)N(t) -\sum_{j=1}^{n}b_{j}(t) N_j(t-\tau_{j}(t))-\sum_{j=1}^{n}c_{j}(t) N'_{j}(t-\sigma_{j}(t))\Big]$$ 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 $a b$. 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, $$\label{e1.2} \Delta x(n)=x(n)\Big[ a(n) -\sum_{i=1}^{p}a_{i}(n) x(n-\tau_{i}(n)) -\sum_{j=1}^{q}c_{j}(n) \Delta x(n-\sigma_{j}(n))\Big],$$ 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 $$\label{e2.1} x=\Phi x,$$ 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 $$\label{e2.2} K=\{x \in \mathbb{Y}: x(n)\geq 0 \,\, \text{and}\,\, x(n)\geq \Theta^{-1} |x|_1, n\in \mathbb{Z}[0,\omega-1]\}.$$ 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 \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} 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 \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} 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 $$\label{e2.6} (\Delta\Phi x)(n)\leq a(n)(\Phi x)(n)\leq a^M(\Phi x)(n)\leq (\Phi x)(n).$$ On the other hand, from \eqref{e2.5} and (H3), if $(\Delta\Phi x)(n) <0$, then \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} 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 $S=\{x\in \mathbb{Y}: \|x\|_1 0$ 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 $\Omega_R=\{x\in \mathbb{Y}: |x|_1 1$, 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 $$\label{e3.1} (\Phi x)(n) - x(n) \geq \Theta^{-1} |\Phi x - x |_1 \geq0 \quad \text{for all } t \in [0,\omega].$$ 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 $$\label{e3.3} x(n)-(\Phi x)(n)\geq \Theta^{-1}|x-\Phi x|_1>0.$$ 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} \begin{thebibliography}{00} \bibitem{c1} N. P. C\'ac, J. A. 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