\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \usepackage{stmaryrd} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 34, 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} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/34\hfil Positive periodic solutions] {Positive periodic solutions for nonautonomous impulsive neutral functional differential systems with time-varying delays on time scales} \author[J. Dai, Y. Li, T. Zhang\hfil EJDE-2010/34\hfilneg] {Jinchun Dai, Yongkun Li, Tianwei Zhang} % in alphabetical order \address{Department of Mathematics, Yunnan University\\ Kunming, Yunnan 650091, China} \email[Jinchun Dai]{daijinchun@163.com} \email[Yongkun Li]{yklie@ynu.edu.cn} \email[Tianwei Zhang]{1200801347@stu.ynu.edu.cn} \thanks{Submitted December 15, 2009. Published March 9, 2010.} \thanks{Supported by grant 10971183 from the National Natural Sciences Foundation of China} \subjclass[2000]{34K13, 34K40, 34K45} \keywords{Periodic solution; neutral functional differential systems; \hfill\break\indent time scale; strict-set-contraction} \begin{abstract} Using a fixed point theorem of strict-set-contraction, we prove the existence of positive periodic solutions for a class of nonautonomous impulsive neutral functional differential system with time-varying delays on time scales. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Yang \cite{y1} studied the dynamic behavior of bounded solutions for the neutral functional differential equation \begin{equation} \label{e1.1} (x(t)-x(t-r))'=-F(x(t))+G(x(t-r)), \end{equation} and obtained the $\omega$ limit set for bounded solutions. Variations of \eqref{e1.1} have received considerable attention in the literature, because they are used as models for phenomena such as population growth, spread of epidemics, dynamics of capital stocks, etc. For more details and references on this subject, we refer the reader to \cite{k2,t1,x1}. As is well known, impulsive differential equations arise naturally in the description of physical and biological phenomena that are subjected to instantaneous changes at some time instants called moments. For a review on this theory, which has seen a significant development over the past decades, we refer the interested reader to the monographs \cite{l1,s1}. It is well known that continuous and discrete systems are very important in applications; also that Stefan Hilger introduce time scales to unify the continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems. To the best of the authors' knowledge, the existence of positive periodic solutions for the following system has not been studied on time scales. \begin{equation} \label{e1.2} \begin{gathered} (x(t)+\lambda c(t)x(t-\tau(t)))^{\Delta} =-\lambda f( t,x(t),(x(t-\tau(t)), \quad t\neq t_k,\; t\in\mathbb{T},\\ x(t_k^{+})=x(t_k^{-})-\lambda I_k(x(t_k)), \end{gathered} \end{equation} where $\lambda>0$, $\mathbb{T}$ is an $\omega$-periodic time scale, $c(t)\in C^{1} (\mathbb{T},\mathbb{R} _{0})$, $\tau(t)\in C (\mathbb{T},\mathbb{R} _{0} )$ are all $\omega$-periodic functions, $I_k(u)\in C (\mathbb{R},\mathbb{R} _{0} )$, $f(t,x,y)\in C (\mathbb{T\times R\times R}, \mathbb{R} _{0} )$ satisfies $f(t+\omega,x,y)=f(t,x,y)$ for all $(t,x,y)\in(\mathbb{T\times R\times R})$, $\omega>0$ is a constant, $\mathbb{R}_{0}=[0,+\infty)$, $\mathbb{R}^{-}=(-\infty,0)$. For each interval $I$ of $\mathbb{R}$, we denote $I_{\mathbb{T}}=I\cap\mathbb{T}$, $x(t_k^{+})$ and $x(t_k^{-})$ represent the right and the left limit of $x(t_k)$ in the sense of time scales, in addition, if $t_k$ is right-scattered, then $x(t_k^{+})=x(t_k)$, whereas, if $t_k$ is left-scattered, then $x(t_k^{-})=x(t_k)$, $k\in\mathbb{Z}$. There exists a positive integer $p$ such that $t_{k+p}=t_k+\omega$, $I_{k+p}=I_k$, $k\in\mathbb{Z}$. Without loss of generality, we also assume that $[0,\omega)_{\mathbb{T}}\cap\{t_k:k\in \mathbb{Z}\} =\{t_{1},t_{2},\ldots,t_{p}\}$. The main purpose of this article is to establish criteria to guarantee the existence of positive periodic solutions of system \eqref{e1.2} by using a fixed point theorem of strict-set-contraction. This article is organized as follows: In Section $2$, we make some preparations. Section $3$, by using a fixed point theorem of strict-set-contraction, we prove the existence of positive periodic solutions of system \eqref{e1.2}. In Section 4, we give an example to illustrate the main results. \section{Preliminaries} In this section, we shall first recall some basic definitions, and lemmas which are used in later. \begin{definition}[\cite{b1}] \label{def2.1} \rm A function $p:\mathbb{T}\to \mathbb{R}$ is said to be regressive provided $1+\mu(t)p(t)\neq0$ for all $t\in \mathbb{T}^{k}$, where $\mu(t)=\sigma(t)-t$ is the graininess function. The set of all regressive rd-continuous functions $f:\mathbb{T}\to \mathbb{R}$ is denoted by $\mathcal{R}$ while the set $\mathcal{R}^{+}$ is given by $\{f\in \mathcal{R}:1+\mu(t)f(t)>0\}$ for all $t\in \mathbb{T}$. Let $p\in\mathcal{R}$. The exponential function is defined by \[ e_{p}(t,s)=\exp\Big(\int_{s}^{t} \xi_{\mu(\tau)}(p(\tau))\Delta\tau\Big), \] where $\xi_{h(z)}$ is the so-called cylinder transformation. \end{definition} \begin{lemma}[\cite{b1}] \label{lem2.1} Let $p,q\in\mathcal{R}$. Then \begin{itemize} \item[(a)] $e_{0}(t,s)\equiv 1$ and $e_{p}(t,t)\equiv 1$; \item[(b)] $\frac{1}{e_{p}(t,s)}=e_{\ominus p}(t,s)$, where $\ominus p(t)=-\frac{p(t)}{1+\mu(t)p(t)}$; \item[(c)] $e_{p}(t,s)=\frac{1}{e_{p}(s,t)}=e_{\ominus p}(s,t)$; \item[(d)] $e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r)$; \item[(e)] $e_{p}^{\Delta}(\cdot,s)=pe_{p}(\cdot,s)$. \end{itemize} \end{lemma} \begin{definition}[\cite{b1}] \label{def2.2} \rm For $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^{k}$, the delta derivative of $f$ at $t$, denoted by $f^{\Delta}(t)$, is the number (provided it exists) with the property that given any $\epsilon>0$, there is a neighborhood $U\subset\mathbb{T}$ of $t$ such that \[ |f(\sigma(t))-f(s)-f^{\Delta}(t)[\sigma(t)-t]| \leq\epsilon|\sigma(t)-s|, \quad \forall s\in U. \] \end{definition} \begin{definition}[\cite{b1}] \label{def2.3} \rm If $F^{\Delta}(t)=f(t)$, then we define the delta integral by \[ \int_{a}^{t}f(s)\Delta s=F(t)-F(a). \] \end{definition} \begin{definition}[\cite{k1}] \label{def2.4} \rm We say that a time scale $\mathbb{T}$ is periodic if there exists $p>0$ such that if $t\in \mathbb{T}$, then $t\pm p\in \mathbb{T}$. For $\mathbb{T}\neq\mathbb{R}$, the smallest positive $p$ is called the period of the time scale. Let $\mathbb{T}\neq\mathbb{R}$ be a periodic time scale with period p. We say that the function $f:\mathbb{T}\to \mathbb{R}$ is periodic with period T if there exists a natural number $n$ such that $T=np,f(t+T)=f(t)$ for all $t\in \mathbb{T}$ and T is the smallest number such that $f(t+T)=f(t)$. If $\mathbb{T}=\mathbb{R}$, we say that $f$ is periodic with period $T>0$ if $T$ is the smallest positive number such that $f(t+T)=f(t)$ for all $t\in \mathbb{T}$. \end{definition} To obtain the existence of a periodic solution of \eqref{e1.2}, we first make the following preparations: Let $E$ be a Banach space and $K$ be a cone in $E$. The semi-order induced by the cone $K$ is defined by $x\leq y$ if $y-x\in K$. In addition, for a bounded subset $A\subset E$, let $\alpha_{E}(A)$ 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 \text{ and }\mathop{\rm diam}(A_i)\leq \delta\Big\}, \end{align*} where $\mathop{\rm diam}(A_i)$ denotes the diameter of the set $A_i$. Let $E, F$ be two Banach spaces and $D\subset E$. A continuous and bounded map $\Phi : \bar{\Omega} \to F$ is called $k$-set contractive if for any bounded set $S\subset D$ we have \[ \alpha_F(\Phi(S))\leq k\alpha_E(S). \] A map $\Phi$ is called strict-set-contractive if it is $k$-set-contractive for some $0 \leq k < 1$. The following lemma, cited from \cite{c1,g1}, is used in the proof of our main results. \begin{lemma}[\cite{c1,g1}] \label{lem2.2} Let $K$ be a cone of the real Banach space $X$ 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} For convenience, we introduce the notation: \[ f^{M}=\max_{t\in[0,\omega]_{\mathbb{T}}}|f(t)|,\quad f^{l}=\min_{t\in[0,\omega]_{\mathbb{T}}}|f(t)|,\quad f^{L}=\max_{t\in[0,\omega]_{\mathbb{T}}}|f^{\Delta}(t)|. \] We will use the following assumptions: \begin{itemize} \item[(H1)] $f(t,0,y)=0$ and there exist positive constants $l_{f}$ and $L_{f}$ such that \[ |f(t,x,y)-f(t,u,v)|\leq L_{f}(|x-u|+|y-v|) \] for all $(t,x,y), (t,u,v)\in (\mathbb{T},\mathbb{R},\mathbb{R})$. \item[(H2)] There exists an $\omega$-periodic function $a(t)\in C(\mathbb{T,\mathbb{R}}^{-})$ such that \[ 1+a(t)\mu(t)\neq 0,\quad - \zeta \lambda^{-1}a(t)- |c^{\Delta}(t)| - \zeta c^{\sigma}(t)\geq 0\quad \text{for all } t\in[0,\omega]_{\mathbb{T}}, \] where $\zeta=\frac{\gamma_{2}\eta_{2}}{\gamma_{1}\eta_{1}}$, $\gamma_{1}=\max_{t\in[0,\omega]_{\mathbb{T}}} \big(e_{\varominus a }(t,t-\omega)-1\big)^{-1}$,\\ $\gamma_{2}=\min_{t\in[0,\omega]_{\mathbb{T}}} \big(e_{\varominus a }(t,t-\omega)-1\big)^{-1}$, $\eta_{1}=\max_{u\in[t-\omega,t]_{\mathbb{T}}} e_{\varominus a }(t,u)$, $\eta_{2}=\min_{u\in[t-\omega,t]_{\mathbb{T}}} e_{\varominus a }(t,u)$. \item[(H3)] $(\varominus a)^{M}\leq 1$. \item[(H4)] \begin{align*} &\max_{t\in[0,\omega]_{\mathbb{T}}}\big \{-\lambda^{-1}a(t)+|c^{\Delta}(t)|+c^{\sigma}(t) +L_{f} \big\}\\ &\leq\frac{\gamma_{2}\eta_{2}}{\gamma_{1}\kappa}\big (1+(\varominus a)^{l}\big) \times\int_{0}^{\omega} \big(-\zeta \lambda^{-1}a(s)-|c^{\Delta }(s) |-\zeta c^{\sigma}(s) \big)\Delta s, \end{align*} where $\kappa=\max_{t\in[0,\omega]_{\mathbb{T}}} \big\{e_{\varominus a }(\sigma(t)+\omega,t)-e_{\varominus a } (\sigma(t),t)\big\}$. \item[(H5)] There exist positive constants $I_k^{l}$, $I_k^{M}$ such that \[ I_k^{l}x^{2}\leq I_k(x)\leq I_k^{M}x^{2} \quad \text{for all }x\in\mathbb{R},\; k\in\mathbb{Z}. \] \item[(H6)] $\gamma_{1}\kappa c^{M}<1$. \item[(H7)] $\gamma_{1}\eta_{1}a^{M}\omega<1$, \[ \lambda<\frac{1-\gamma_{1}\eta_{1}a^{M}\omega} {\gamma_{1}\eta_{1}(c^{L}\omega+\zeta^{-1}c^{M}\omega+L_{f}+I^{M})}, \] where $I^{M}=\sum_{k=1}^{p}I_k^{M}$. \end{itemize} To apply Lemma \ref{lem2.2} to system \eqref{e1.2}, we define \begin{align*} PC(\mathbb{T})=\big\{&x:\mathbb{T} \to\mathbb{R}| x|_{(t_k,t_{k+1})_\mathbb{T}} \in C((t_k,t_{k+1})_\mathbb{T},\mathbb{R}), \\ &\exists x(t_k^{-})=x(t_k), x(t_k^{+}), k\in\mathbb{Z}\big\}, \\ PC^{1}(\mathbb{T})=\big\{&x:\mathbb{T} \to\mathbb{R}| x|_{(t_k,t_{k+1})_\mathbb{T}} \in C((t_k,t_{k+1})_\mathbb{T},\mathbb{R}),\\ & \exists x^{\Delta}(t_k^{-}) =x^{\Delta}(t_k), x(t_k^{+}), k\in\mathbb{Z}\big\}. \end{align*} Set \[ \mathbb{X}=\{x(t)\in PC(\mathbb{T}): x(t+\omega)=x(t)\} \]with the norm $|x|_{0}=\max_{t\in[0,\omega]_{\mathbb{T}}}|x(t)|$, and \[ \mathbb{Y}=\{x(t)\in PC^{1}(\mathbb{T}): x(t+\omega)=x(t)\} \] with the norm $|x|_{1}=\max_{t\in[0,\omega]_{\mathbb{T}}} \{|x|_{0},|x^{\Delta}|_{0}\}$. Then $\mathbb{X}$ and $\mathbb{Y}$ are Banach spaces. Defined the cone $K$ in $\mathbb{Y}$ by \[ K=\{x\in\mathbb{Y}:x(t)\geq\zeta|x|_{1}, \,t\in[0,\omega]_{\mathbb{T}}\}. \] \begin{lemma} \label{lem2.3} A function $x\in \mathbb{Y}$ is a solution of \eqref{e1.2} if and only if \[ x(t)=\int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}} \lambda G(t,t_k) I_k(x(t_k)), \] where \begin{gather*} G(t,s)=\frac{e_{\varominus a }(t,s)} {e_{\varominus a}(t,t-\omega)-1}, \\ \begin{aligned} F(s)&=-\lambda^{-1}a(s)x^{\sigma}(s) + c^{\Delta}(s)x(s-\tau(s)\\ &\quad +c^{\sigma}(s)x^{\Delta}(s-\tau(s)) + f( s,x(s),x(s-\tau(s)),\quad s\in [t-\omega,t]_{\mathbb{T}}, \end{aligned} \end{gather*} and $a(t)$ satisfies {\rm (H2)}. \end{lemma} \begin{proof} Rewrite the first equation of \eqref{e1.2} in the form \[ x^{\Delta}(t) =-\lambda c^{\Delta}(t)x(t-\tau(t)-\lambda c^{\sigma}(t) x^{\Delta}(t-\tau(t))-\lambda f( t,x(t),x(t-\tau(t)), \] then \begin{equation} \label{e2.1} \begin{aligned} x^{\Delta}(t)+ a(t)x^{\sigma}(t) &= \lambda^{-1} a(t)x^{\sigma}(t)- \lambda c^{\Delta}(t)x(t-\tau(t)) -\lambda c^{\sigma}(t)x^{\Delta}(t-\tau(t))\\ &\quad - \lambda f( t,x(t),x(t-\tau(t))\\ &:=-\lambda F(t). \end{aligned} \end{equation} Let $x\in \mathbb{Y}$ be a solution of system \eqref{e1.2}. Multiply both sides of \eqref{e2.1} by $e_{a}(t,0)$ to get \begin{equation} \label{e2.2} \big(x(t)e_{a}(t,0)\big)^{\Delta} =-\lambda e_{a}(t,0)F(t). \end{equation} For any $t\in\mathbb{T}$, there exists $k\in\mathbb{Z}$ such that $t_k$ is the first impulsive point after $t-\omega$. For $s\in[t-\omega,t_k]_{\mathbb{T}}$, we integrate \eqref{e2.2} from $t-\omega$ to $s$ to obtain \[ x(s)e_{a}(s,0)=x(t-\omega)e_{a}(t-\omega,0) -\int_{t-\omega}^{s}\lambda e_{a}(r,0) F(r)\Delta r. \] Then \[ x(t_k)e_{a}(t_k,0)=x(t-\omega)e_{a}(t-\omega,0) -\int_{t-\omega}^{t_k}\lambda e_{a }(r,0) F(r)\Delta r. \] For $s\in(t_k,t_{k+1}]_{\mathbb{T}}$, we integrate \eqref{e2.2} from $t_k$ to $s$ once more to obtain \begin{align*} x(s)e_{a}(s,0) &= x(t_k^{+})e_{a}(t_k,0) -\int_{t_k}^{s}\lambda e_{a }(r,0) F(r)\Delta r\\ &= x(t_k)e_{a}(t_k,0)-\int_{t_k}^{s} \lambda e_{a}(r,0) F(r)\Delta r-\lambda e_{a}(t_k,0)I_k(x(t_k))\\ &= x(t-\omega)e_{a}(t-\omega,0) -\int_{t-\omega}^{s}\lambda e_{a }(r,0) F(r)\Delta r-\lambda e_{a}(t_k,0)I_k(x(t_k)). \end{align*} Repeating the above process for $s\in[t-\omega,t]_{\mathbb{T}}$, we have \begin{align*} x(s)e_{a}(s,0) &=x(t-\omega)e_{a}(t-\omega,0) -\int_{t-\omega}^{s}\lambda e_{a}(r,0) F(r)\Delta r \\ &\quad -\sum_{k:t_k \in[t-\omega,s]_{\mathbb{T}}} \lambda e_{a}(t_k,0)I_k(x(t_k)). \end{align*} Let $s=t$ in the above equality, we get \begin{align*} x(t)e_{a}(t,0) &=x(t-\omega)e_{a}(t-\omega,0) -\int_{t-\omega}^{t}\lambda e_{a}(s,0) F(s)\Delta s\\ &\quad -\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}} \lambda e_{a}(t_k,0)I_k(x(t_k)). \end{align*} Then \begin{align*} x(t)&= \int_{t-\omega}^{t} \frac{e_{\varominus a }(t,s)}{e_{\varominus a}(\omega,0)-1} \lambda F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}}\frac{e_{\varominus a }(t,t_k)} {e_{\varominus a}(\omega,0)-1}\lambda I_k(x(t_k))\\ &= \int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}}\lambda G(t,t_k) I_k(x(t_k)), \end{align*} where we have used Lemma \ref{lem2.1} to simplify the exponentials. The proof is complete. \end{proof} Let the mapping $\Phi$ be defined by \begin{equation} \label{e2.3} (\Phi x)(t)=\int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}}\lambda G(t,t_k) I_k(x(t_k)), \end{equation} where $x\in K$, $t\in \mathbb{T}$, $F(s)$, $G(t,s)$ is given by \eqref{e2.1} and \[ 0<\gamma_{2}\eta_{2}\leq G(t,s) \leq\gamma_{1}\eta_{1}, \quad s\in[t-\omega,t]_{\mathbb{T}}. \] \begin{lemma} \label{lem2.4} Assume that {\rm (H1)--(H4)} hold. Then $\Phi:K\to K$ is well defined. \end{lemma} \begin{proof} For any $x\in K$, we have $\Phi x\in PC^{1}(\mathbb{T})$. In view of \eqref{e2.3}, for $ t\in\mathbb{T}$, we have \begin{equation} \label{e2.4} \begin{aligned} (\Phi x)(t+\omega) &= \frac{1}{e_{\varominus a } (t,t-\omega)-1}\int_{t}^{t+\omega}\lambda e_{\varominus a }(t+\omega,s) F(s)\Delta s \\ &\quad +\sum_{k:t_k \in[t,t+\omega]_{\mathbb{T}}} \lambda G(t+\omega,t_k)I_k(x(t_k)). \end{aligned} \end{equation} Using the periodicity of $a$, $c$, $\tau$, $f$, and letting $u=s-\omega$, by \eqref{e2.4} we obtain \begin{equation} \label{e2.5} \begin{aligned} (\Phi x)(t+\omega) &= \frac{1}{e_{\varominus a }(t,t-\omega)-1} \int_{t-\omega}^{t} \lambda e_{\varominus a }(t+\omega,u+\omega) F(u)\Delta u \\ &\quad +\sum_{k:t_k\in[t-\omega,t]_{\mathbb{T}}} \lambda G(t+\omega,t_k+\omega) I_k (x(t_k+\omega) ). \end{aligned} \end{equation} At the same time, from the definition of $e_{a}(t,s)$ and the periodicity of $a$, we have $e_{\varominus a}(t+\omega,u+\omega)=e_{\varominus a }(t,u)$ and $e_{\varominus a }(t+\omega,u) =e_{\varominus a}(t,u-\omega)$. Thus \eqref{e2.5} becomes $(\Phi x)(t+\omega)=(\Phi x)(t)$. In view of (H2), for $x\in K$, $t_k\in[0,\omega]_{\mathbb{T}}$, we obtain \begin{equation} \label{e2.6} \begin{aligned} F(t)&= -\lambda^{-1}a(t)x^{\sigma}(t) + c^{\Delta}(t)x(t-\tau(t) + c^{\sigma}(t)x^{\Delta}(t-\tau(t)) + f( t,x(t),x(t-\tau(t)) \\ &\geq - \zeta \lambda^{-1}a(t)|x|_{1}-|c^{\Delta}(t)\|x|_{1} - \zeta c^{\sigma}(t) |x|_{1} \\ &= \big( - \zeta \lambda^{-1}a(t)- |c^{\Delta}(t)| - \zeta c^{\sigma}(t) \big)|x|_{1} \geq 0. \end{aligned} \end{equation} For $x\in K$, $t\in[0,\omega]_{\mathbb{T}}$, by \eqref{e2.6} we obtain \begin{align*} |\Phi x|_{0} &= \max_{t\in[0,\omega]_{\mathbb{T}}} \Big\{\int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}} \lambda G(t,t_k) I_k(x(t_k))\Big\}\\ &\leq \lambda \gamma_{1}\eta_{1} \int_{0}^{\omega}F(s)\Delta s +\lambda \gamma_{1}\eta_{1}\sum_{k=1}^{p} I_k(x(t_k)) \end{align*} and \begin{equation} \label{e2.7} \begin{aligned} (\Phi x)(t) &= \int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}}\lambda G(t,t_k) I_k(x(t_k)) \\ &\geq \lambda \gamma_{2}\eta_{2} \int_{0}^{\omega}F(s)\Delta s +\lambda \gamma_{2}\eta_{2}\sum_{k=1}^{p} I_k(x(t_k)) \\ &\geq \frac{\gamma_{1}\eta_{1}}{\gamma_{2}\eta_{2}} |\Phi x|_{0}=\zeta|\Phi x|_{0}. \end{aligned} \end{equation} Form \eqref{e2.3} and (H3), we have \begin{align*} (\Phi x)^{\Delta}(t) &= \Big(\int_{t-\omega}^{\tilde{a}} \lambda G(t,s)F(s)\Delta s+\int_{\tilde{a}}^{t} \lambda G(t,s)F(s)\Delta s\\ &\quad +\sum_{k:t_k\in[t-\omega,t]_{\mathbb{T}}} \lambda G(t,t_k) I_k(x(t_k)) \Big)^{\Delta}\\ &= \frac{e_{\varominus a }(\sigma(t),t) \big(1-e_{a }(t-\omega,t)\big)} {e_{\varominus a }(t,t-\omega)-1}\lambda F(t) +\varominus a(\Phi x)(t)\\ &\leq \varominus a(\Phi x)(t)\leq(\varominus a)^{M} (\Phi x)(t)\leq(\Phi x)(t), \end{align*} where $\tilde{a}\in[t-\omega,t]_{\mathbb{T}}$ is an arbitrary constant. If $(\Phi x)^{\Delta}(t)\geq 0$, we have \begin{equation} \label{e2.8} |\Phi x|_{0}\geq|(\Phi x)^{\Delta}|_{0}. \end{equation} If $(\Phi x)^{\Delta}(t)<0$, by \eqref{e2.7} and (H4), we have \begin{equation} \label{e2.9} \begin{aligned} -(\Phi x)^{\Delta}(t) &=\frac{e_{\varominus a}(\sigma(t)+\omega,t) - e_{\varominus a }(\sigma(t),t) } {e_{\varominus a}(t,t-\omega)-1}\lambda F(t) -\varominus a(\Phi x)(t) \\ &\leq \lambda\gamma_{1}\kappa\big(-\lambda^{-1}a(t)x^{\sigma}(t) + c^{\Delta}(t)x(t-\tau(t)) + c^{\sigma}(t)x^{\Delta}(t-\tau(t)) \\ &\quad + f( t,x(t),x(t-\tau(t)))\big) -(\varominus a)^{l}(\Phi x)(t) \\ &\leq \lambda\gamma_{1}\kappa\big(-\lambda^{-1}a(t) +|c^{\Delta}(t)|+c^{\sigma}(t)+L_{f} \big)|x|_{1}-(\varominus a)^{l}(\Phi x)(t) \\ &\leq \lambda\gamma_{2}\eta_{2}\big(1+(\varominus a)^{l}\big) \int_{0}^{\omega} \big(-\zeta \lambda^{-1}a(s)-|c^{\Delta }(s)|-\zeta c^{\sigma}(s) \big)|x|_{1}\Delta s \\ &\quad -(\varominus a)^{l}(\Phi x)(t) \\ &\leq \lambda\gamma_{2}\eta_{2}\big(1+(\varominus a)^{l}\big) \int_{0}^{\omega}F(s) \Delta s-(\varominus a)^{l}(\Phi x)(t) \\ &\leq \big(1+(\varominus a)^{l}\big)\big(\Phi x)(t) -(\varominus a)^{l}(\Phi x)(t) \\ &= (\Phi x)(t). \end{aligned} \end{equation} Form \eqref{e2.8} and \eqref{e2.9}, we have $|(\Phi x)^{\Delta}|_{0}\leq|(\Phi x)|_{0}$. So $|\Phi x|_{1}=|\Phi x|_{0}$. By \eqref{e2.7} we have $(\Phi x)(t)\geq \zeta |\Phi x|_{1}$. Hence $\Phi x \in K$. The proof is complete. \end{proof} \begin{lemma} \label{lem2.5} Assume that {\rm (H1)-(H6)} hold, then $\Phi:K\cap\Omega_{R}\to K$ is strict-set-contractive, where $\Omega_{R}=\{x\in\mathbb{Y}:|x|_{1}0 \quad \text{for all }t\in[0,\omega]_{\mathbb{T}}. \end{equation} At the same time, for any $t\in[0,\omega]_{\mathbb{T}}$, we have \begin{align*} (\Phi x)(t) &= \int_{t-\omega}^{t}\lambda G(t,s) F(s)\Delta s+\sum_{k:t_k \in[t-\omega,t]_{\mathbb{T}}}\lambda G(t,t_k) I_k(x(t_k))\\ &= \int_{t-\omega}^{t}\lambda G(t,s) \big( -\lambda^{-1}a(s)x^{\sigma}(s)+ c^{\Delta}(s)x(s-\tau(s) + c^{\sigma}(s)x^{\Delta}(s-\tau(s))\\ &\quad + f( s,x(s),x(s-\tau(s))\big) \Delta s+\sum_{k:t_k\in[t-\omega,t]_{\mathbb{T}}} \lambda G(t,t_k) I_k(x(t_k))\\ &\geq \lambda\gamma_{2}\eta_{2} \omega\big(\lambda^{-1}a^{l} \zeta |x|_{1}-c^{L}|x|_{1}-c^{M}|x|_{1} +I^{l}\zeta^{2}|x|_{1}^{2}\big)\\ &\geq \lambda\gamma_{2}\eta_{2} \omega(I^{l}\zeta^{2}R-c^{L}-c^{M})R =R. \end{align*} From \eqref{e3.3} and above inequality, we obtain that $|x|_{0}>|\Phi x|_{0}\geq R$, which is a contradiction. Applying Lemma \ref{lem2.2}, we see that there is at least one nonzero fixed point in $K$. Hence system \eqref{e1.2} has at least one positive $\omega$-periodic solution. The proof is complete. \end{proof} \section{An example} When $\mathbb{T}=\mathbb{R}$, consider the system: \begin{equation} \label{e4.1} \begin{gathered} (x(t)+\lambda \frac{1}{4}x(t-\tau(t)))' =-\lambda(t^{2}+\frac{1}{2}\sin^{2}x(t) +\frac{1}{4}\cos^{2}x(t-1)), \quad t\neq t_k,\quad t\in\mathbb{R},\\ x(t_k^{+})=x(t_k^{-})-\lambda x^{2}(t_k), \quad k\in\mathbb{Z}, \end{gathered} \end{equation} where \[ \lambda<\min\big\{\frac{2}{10001\pi},\frac{4\pi -2(e^{\frac{1}{10001}}-1)e^{\frac{1}{10001}}} {10001\pi(2\pi+5(e^{\frac{1}{10001}}-1)e^{\frac{1}{10001}})}, \frac{20000e^{\frac{1}{10001}}-20002} {10001(\pi e^{\frac{1}{10001}}+42)e^{\frac{1}{10001}}}\big\}, \] $\omega=2\pi$, $p=10$, $L_{f}=1$, $I^{M}=20$. 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