\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 118, pp. 1--16.\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{8mm}} \begin{document} \title[\hfilneg EJDE-2010/118\hfil Three positive periodic solutions] {Existence of three positive periodic solutions for differential systems with feedback controls on time scales} \author[Y. Li, T. Zhang, J. Suo\hfil EJDE-2010/118\hfilneg] {Yongkun Li, Tianwei Zhang, Jianfeng Suo} % in alphabetical order \address{Department of Mathematics, Yunnan University\\ Kunming, Yunnan 650091, China} \email[Yongkun Li]{yklie@ynu.edu.cn} \email[Tianwei Zhang]{1200801347@stu.ynu.edu.cn} \email[Jianfeng Suo]{jfsuo@ynu.edu.cn} \thanks{Submitted March 25, 2010. Published August 20, 2010.} \thanks{Supported by grant 10971183 from the National Natural Sciences Foundation of China} \subjclass[2000]{34K13} \keywords{Positive periodic solutions; feedback controls; time scales; \hfill\break\indent Leggett-Williams multiple fixed point theorem} \begin{abstract} Using the Leggett-Williams multiple fixed point theorem, we establish criteria for the existence of three positive periodic solutions of a class of differential systems with feedback controls on time scales. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Recently, by using the Krasnosel'skii's fixed point theorem for cones, Li and Zhu \cite{l3} studied the existence of positive periodic solutions of the following functional differential systems with feedback controls: $$\label{e1.1} \begin{gathered} \dot{x}(t)=-A(t)x(t)+f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t))), \\ \dot{u}(t)=-B(t)u(t)+C(t)x(h(t,x(t))). \end{gathered}$$ Zeng and Zhou \cite{z1} considered a class of more general functional differential systems with feedback controls of the form $$\label{e1.2} \begin{gathered} \dot{x}(t)=-A(t,x(t))x(t)+f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t))), \\ \dot{u}(t)=-B(t,x(t))u(t)+C(t,x(t))x(h(t,x(t))). \end{gathered}$$ By means of the Krasnosel'skii's fixed point theorem, they obtained some criteria for the existence of two positive periodic solutions of \eqref{e1.2}. Also, by applying the continuation theorem of coincidence degree theory, Li and Zhu \cite{l4} studied the existence of positive periodic solutions to the difference equations with feedback control of the form $$\begin{gathered} N(n+1)=N(n)\exp\Big[r(n)\Big(1-\frac{N(n-m)}{k(n)}-c(n)\mu(n)\Big) \Big], \\ \Delta\mu(n)=-a(n)\mu(n)+b(n)N(n-m), \end{gathered}$$ where $a:\mathbb{Z}\to(0,1), c, k, r, b:\mathbb{Z}\to\mathbb{R^{+}}$ are all $\omega$-periodic functions and $m$ is a positive integer. In the previous ten years, many authors \cite{c2,l2,l5,l10} have argued that the discrete time model governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Consequently, the studies of dynamic systems governed by difference equations have received great attention from more scholars. In fact, continuous and discrete systems are very important in implementing and applications. It is well known that the theory of time scales \cite{b1,b2} has received a lot of attention which was introduced by Stefan Hilger \cite{h1} in order to unify continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems. For the work concerning with the existence of periodic solutions for dynamic systems on time scales, we refer the reader to \cite{a2,a3,c1,l6,l8,l9, l11,l12,z2}. Motivated by above statement, in this paper, we will study the following differential systems with feedback controls on time scales: $$\label{e1.4} \begin{gathered} x^{\Delta}(t)=-A(t,x(t))x(\sigma(t))+\lambda f(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t,x(t)))), \\ u^{\Delta}(t)=-B(t,x(t))u(\sigma(t))+g(t,x_{t},x(h(t,x(t)))),\quad t\in\mathbb{T}, \end{gathered}$$ in which $\mathbb{T}$ is a periodic time scales which has the subspace topology inherited from the standard topology on $\mathbb{R}$, $\lambda>0$ is parameter, \begin{gather*} A(t,x(t))=\operatorname{diag}[a_1(t,x(t)),a_2(t,x(t)),\dots ,a_{n}(t,x(t))],\\ B(t,x(t)) =\operatorname{diag}[b_1(t,x(t)),b_2(t,x(t)),\dots,b_{n} (t,x(t))], \end{gather*} $a_{i}(t,y), b_{i}(t,y), \in C(\mathbb{T}\times \mathbb{R}^{n},\mathbb{R})$ satisfy $a_{i}(t+\omega,y)=a_{i}(t,y)$, $b_{i}(t+\omega,y)=b_{i}(t,y)$ for all $t\in \mathbb{T}, y\in \mathbb{R}^{n}$, $i=1,2,\dots,n$, $t-\tau(t,y)$, $t-\alpha(t,y)$, $h(t,y) \in C(\mathbb{T}\times \mathbb{R}^{n},\mathbb{T})$ satisfy $\tau(t+\omega,y)=\tau(t,y)$, $\alpha(t+\omega,y)=\alpha(t,y),h(t+\omega,y)=h(t,y)$ for all $t\in \mathbb{T}$, $y\in \mathbb{R}^{n}$, $\omega>0$ is a constant, $f$ is a function defined on $\mathbb{T}\times BC\times \mathbb{R}^{n}\times \mathbb{R}^{n}$, satisfying $f(t+\omega,x_{t+\omega},y,z)=f(t,x_{t},y,z)$ for all $t\in \mathbb{T}$, $x\in BC,y,z\in \mathbb{R}^{n}$, where $BC$ denotes the Banach space of all bounded continuous functions $\eta: \mathbb{T}\to \mathbb{R}^{n}$ with the norm $\|\eta\|=\sum_{i=1}^{n}\max_{\theta\in \mathbb{T}}|\eta_{i}(\theta)|$, where $\eta=(\eta_1,\eta_2,\dots,\eta_{n})^{T}$, and $g$ is a function defined on $\mathbb{T}\times BC\times \mathbb{R}^{n}$, satisfying $g(t+\omega,x_{t+\omega},y)=g(t,x_{t},y)$ for all $t\in \mathbb{T}, x\in BC,y\in \mathbb{R}^{n}$. If $x\in BC$, then $x_{t}\in BC$ for any $t\in \mathbb{T}$ is defined by $x_{t}(\theta)=x_{t}(t+\theta)$ for $\theta\in \mathbb{T}$. In the sequel, we denote $f=(f_1,f_2,\dots,f_{n})^{T}, g=(g_1,g_2,\dots,g_{n})^{T}$. Let $\mathbb{R}=(-\infty,+\infty), \mathbb{R}_{+}=[0,+\infty), \mathbb{R}_{+}^{n}=\{(x_1,x_2,\dots,x_{n})^{T}:x_{i}\geq 0,1\leq i\leq n\}$, respectively. For each $x=(x_1,x_2,\dots,x_{n})^{T}\in \mathbb{R}^{n}$, the norm of $x$ is defined as $|x|_{0}=\sum_{i=1}^{n}|x_{i}|$. The main purpose of this paper is to study the existence of at least three nonnegative periodic solutions of \eqref{e1.4} by using the Leggett-Williams multiple fixed point theorem. The organization of this paper is as follows. In Section $2$, we make some preparations. In Section $3$, by using the Leggett-Williams multiple fixed point theorem, we obtain the existence of at least three nonnegative periodic solutions of \eqref{e1.4}. In Section $4$, an example is also provided to illustrate the effectiveness of the main results obtained in Section $3$. \section{Preliminaries} In this section, we shall first recall some basic definitions, lemmas which are used in what follows. \begin{definition}[\cite{b1}] \label{def2.1} \rm A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of the real set $\mathbb{R}$ with the topology and ordering inherited from $\mathbb{R}$. The forward and backward jump operators $\sigma$, $\rho:\mathbb{T}\to\mathbb{T}$ and the graininess $\mu:\mathbb{T}\to\mathbb{R}^{+}$ are defined, respectively, by \sigma(t):=\inf\{s\in\mathbb{T}:s>t\}, \quad\rho(t):=\sup\{s\in\mathbb{T}:st, respectively. Points that are right-dense and left-dense at the same time are called dense. If \mathbb{T} has a left-scattered maximum m, defined \mathbb{T}^{k}=\mathbb{T}-\{m\}; otherwise, set \mathbb{T}^{k}=\mathbb{T}. \end{definition} \begin{definition}[\cite{k1}] \label{def2.2} \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} \begin{definition}[\cite{b1}] \label{def.2.3} \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{def2.4} \rm A function f:\mathbb{T}\to\mathbb{R} is called regulated provided its right-sided limits exist (finite) at all right-dense points in \mathbb{T} and its left-sided limits exist (finite) at all left-dense points in \mathbb{T}. \end{definition} \begin{definition}[\cite{b1}] \label{def2.5} \rm A continuous function F:\mathbb{T}\to\mathbb{R} is called pre-differentiable with (region of differentiation) D, provided D\subset\mathbb{T}^{k}, \mathbb{T}^{k}\backslash D is countable and contains no right-scattered elements of \mathbb{T}, and F is differentiable at each t\in D. \end{definition} \begin{definition}[\cite{b1}] \label{def2.6} \rm Assume that f:\mathbb{T}\to\mathbb{R} is a regulated function. Suppose further that there exists a function F which is pre-differentiable with region of differentiation D such that  F^{\Delta}(t)=f(t)\quad\text{holds for all t\in D}.  We define the Cauchy integral by  \int_{a}^{b}f(s)\Delta s=F(b)-F(a) \quad\text{for all a, b\in \mathbb{T}}.  \end{definition} \begin{definition}[\cite{b1}] \label{def2.7} \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 [(i)] e_{0}(t,s)\equiv 1 and e_{p}(t,t)\equiv 1; \item [(ii)] e_{p}(\sigma(t),s)=(1+\mu(t)p(t))e_{p}(t,s); \item [(iii)] \frac{1}{e_{p}(t,s)}=e_{\ominus p}(t,s), where \ominus p(t)=-\frac{p(t)}{1+\mu(t)p(t)}; \item [(iv)] e_{p}(t,s)=\frac{1}{e_{p}(s,t)}=e_{\ominus p}(s,t); \item [(v)] e_{p}(t,s)e_{p}(s,r)=e_{p}(t,r); \item [(vi)] \frac{e_{p}(t,s)}{e_{q}(t,s)}=e_{p\ominus q}(t,s), where p\ominus q=p\oplus(\ominus q); \item [(vii)] e_{p}^{\Delta}(\cdot,s)=pe_{p}(\cdot,s). \end{itemize} \end{lemma} For convenience, we introduce the following notation: \begin{gather*} r_1^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}} \frac{1}{|1-e_{\ominus a_{i}}(t,t-\omega)|}, \quad r_2^{i}=\inf_{t\in [0,\omega]_{\mathbb{T}}} \frac{1}{|1-e_{\ominus a_{i}}(t,t-\omega)|}, \\ \eta_1^{i}=\sup_{u\in [t-\omega,t]_{\mathbb{T}}} e_{\ominus a_{i}}(t,u),\quad \eta_2^{i}=\inf_{u\in [t-\omega,t]_{\mathbb{T}}} e_{\ominus a_{i}}(t,u), \\ r^{M}=\max_{1\leq i\leq n}\{r_1^{i}\},\quad r^{l} =\min_{1\leq i\leq n}\{r_2^{i}\},\\ \eta^{M}=\max_{1\leq i\leq n}\{\eta_1^{i}\},\quad \eta^{l}=\min_{1\leq i\leq n} \{\eta_2^{i}\}, \\ \gamma^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}} |\ominus a_{i}|,\quad \kappa^{i}=\sup_{t\in [0,\omega]_{\mathbb{T}}}e_{\ominus a_{i}}(\sigma(t),t),\\ \gamma= \max_{1\leq i\leq n}\{\gamma^{i}\}, \quad \kappa=\max_{1\leq i\leq n}\{\kappa^{i}\}. \end{gather*} \begin{lemma} \label{lem2.2} (x(t),u(t))^{T} is an \omega-periodic solution of \eqref{e1.4} if and only if it is an \omega-periodic solution of the system $$\label{e2.1} \begin{gathered} x^{\Delta}(t)=-A(t,x(t))x(\sigma(t))+\lambdaf(t,x_{t},x(t-\tau(t,x(t))),u(t-\alpha(t,x(t)))), \\ u(t)=\int_{t-\omega}^{t}\overline{G}(t,s)g(s,x_{s},x(h(s,x(s))))\Delta s :=(\Phi x)(t), \end{gathered}$$ where  \overline{G}(t,s)=\operatorname{diag}[\overline{G}_1(t,s), \overline{G}_2(t,s),\dots,\overline{G}_{n}(t,s)]  and  \overline{G}_{i}(t,s)=\frac{e_{\ominus b_{i}}(t,s)}{1-e_{\ominus b_{i}}(t,t-\omega)}, \quad s\in[t-\omega,t]_{\mathbb{T}},\; i=1,2,\dots,n.  \end{lemma} \begin{proof} First, assume that (x(t),u(t))^{T} is an \omega-periodic solution of \eqref{e1.4}. From the second equation of \eqref{e1.4}, it follows that $$\label{e2.2} u_{i}^{\Delta}(t)+b_{i}(t,x(t))u_{i}(\sigma(t)) =g_{i}(t,x_{t},x(h(t,x(t)))),\quad i=1,2,\dots,n.$$ Multiply both sides of this equation by e_{b_{i}}(t,0) and then integrate them from t-\omega to t to obtain  \int_{t-\omega}^{t}[e_{b_{i}}(s,0)u_{i}(s)]^{\Delta}\Delta s = \int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i} (s,x_{s},x(h(s,x(s))))\Delta s,  for i=1,2,\dots,n, and then  e_{b_{i}}(t,0)u_{i}(t)-e_{b_{i}}(t-\omega,0)u_{i}(t-\omega) =\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i} (s,x_{s},x(h(s,x(s))))\Delta s,  for i=1,2,\dots,n. Dividing both sides of the above equation by e_{b_{i}}(t,0), we have  u_{i}(t)=\int_{t-\omega}^{t}\frac{e_{\ominus b_{i}}(t,s)} {1-e_{\ominus b_{i}}(t,t-\omega)}g_{i}(s,x_{s}, x(h(s,x(s))))\Delta s ,\quad i=1,2,\dots,n.  So (x(t),u(t))^{T} is an \omega-periodic solution of \eqref{e2.1}. Conversely, assume that (x(t),u(t))^{T} is an \omega-periodic solution of \eqref{e2.1}. Then we have  e_{b_{i}}(t,0)u_{i}(t)-e_{b_{i}}(t-\omega,0)u_{i}(t-\omega) =\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i} (s,x_{s},x(h(s,x(s))))\Delta s,  for i=1,2,\dots,n; that is,  \big(1+e_{b_{i}}(\omega,0)\big)e_{b_{i}}(t,0)u_{i}(t) =\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i} (s,x_{s},x(h(s,x(s))))\Delta s,  for i=1,2,\dots,n. Then \begin{align*} &\big[\big(1+e_{b_{i}}(\omega,0)\big)e_{b_{i}}(t,0)u_{i}(t) \big]^{\Delta}\\ &= e_{b_{i}}(t,0)\big(1+e_{b_{i}}(\omega,0)\big)\big[u_{i}^{\Delta}(t)+b_{i}u_{i}(\sigma(t))\big]\\ &= \Big(\int_{t-\omega}^{t}e_{b_{i}}(s,0)g_{i} (s,x_{s},x(h(s,x(s))))\Delta s\Big)^{\Delta}\\ &= e_{b_{i}}(t,0)g_{i}(t,x_{t},x(h(t,x(t)))) -e_{b_{i}}(t-\omega,0)g_{i} (t-\omega,x_{t-\omega},x(h(t-\omega,x(t-\omega))))\\ &= e_{b_{i}}(t,0)\big(1+e_{b_{i}}(\omega,0)\big) g_{i}(t,x_{t},x(h(t,x(t)))), \end{align*} which implies \begin{align*} u_{i}^{\Delta}(t)+b_{i}(t,x(t))u_{i}(\sigma(t)) =g_{i}(t,x_{t},x(h(t,x(t)))),\quad i=1,2,\dots,n. \end{align*} So (x(t),u(t))^{T} is an \omega-periodic solution of \eqref{e1.4}. The proof of the lemma is complete. \end{proof} At the same time, from the definition of e_{p}(t,s) and the periodicity of b_{i}, we have e_{\ominus b_{i}}(t+\omega,s+\omega) =e_{\ominus b_{i}}(t,s),\,i=1,2,\dots,n, so it is clear that \overline{G}(t,s)=\overline{G}(t+\omega,s+\omega) for all (t,s)\in\mathbb{T}^{2} and u(t+\omega)=u(t) when x is \omega-periodic solution. Now, \eqref{e2.1} can be reformulated as $$\label{e2.3} x^{\Delta}(t)=-A(t,x(t))x(\sigma(t)) +\lambda f(t,x_{t},x(t-\tau(t,x(t))),(\Phi x)(t-\alpha(t,x(t)))).$$ We proceed from \eqref{e2.3} and obtain $$x(t)=\lambda\int_{t-\omega}^{t}G(t,s)f(s,x_{s},x(s-\tau(s,x(s))), (\Phi x)(s-\alpha(s,x(s))))\Delta s,$$ where  G(t,s)=\operatorname{diag}[G_1(t,s),G_2(t,s),\dots,{G}_{n}(t,s)]  and  G_{i}(t,s)=\frac{e_{\ominus a_{i}}(t,s)}{1-e_{\ominus a_{i}}(t,t-\omega)}, \quad s\in[t-\omega,t]_{\mathbb{T}}, \quad i=1,2,\dots,n.  By the periodicity of a_{i}, i=1,2,\dots,n, it is also obvious that G(t,s)=G(t+\omega,s+\omega) for all (t,s)\in\mathbb{T}^{2}. To obtain our main results, we make the following assumptions throughout this paper. \begin{itemize} \item [(H1)] a_{i}(t,x(t))>0 or a_{i}(t,x(t))<0 for all t\in \mathbb{T}, i=1,2,\dots,n; \item [(H2)] f_{i}(t,\zeta,\xi,\Phi(\eta))a_{i}(t,x(t))\geq0 for all (t,\zeta,\xi,\eta)\in \mathbb{T}\times BC(\mathbb{T}\times\mathbb{R}_{+}^{n})\times\mathbb{R}_{+}^{n}\times\mathbb{R}_{+}^{n},\,i=1,2,\dots,n; \item [(H3)] f(t,\phi_{t},\phi(t-\tau(t,\phi(t))), (\Phi\phi)(t-\alpha(t,\phi(t)))) is a continuous function of t for each \phi\in BC(\mathbb{T}\times\mathbb{R}_{+}^{n}); \item [(H4)] for any L>0 and \epsilon>0, there exists a real number \delta>0 such that \phi,\psi\in BC,\|\phi\|\leq L, \|\psi\|\leq L,\| \phi-\psi\|<\delta,0\leq s\leq \omega imply \begin{align*} &\|f(s,\phi_{s},\phi(s-\tau(s,\phi(s))),(\Phi\phi) (s-\alpha(s,\phi(s))))\\ &-f(s,\psi_{s},\psi(s -\tau(s,\psi(s))), (\Phi\psi)(s-\alpha(s,\psi(s))))\|<\epsilon. \end{align*} \end{itemize} Moreover, for the sake of simplicity, let  f(t,\phi ,\Phi)=f(t,\phi_{t},\phi(t-\tau(t,\phi(t))), (\Phi\phi)(t-\alpha(t,\phi(t))))  and  f(t,x ,\Phi)=f(t,x_{t},x(t-\tau(t,x(t))), (\Phi x)(t-\alpha(t,x(t)))).  Then by (H2), $$\label{e2.5} G_{i}(t,s)f_{i}(t,x ,\Phi)\geq 0 \quad\text{for } (t,s)\in\mathbb{T}^{2},\; i=1,2,\dots,n.$$ Let X be a Banach space and K be a cone in X. A mapping \psi is said to be a concave nonnegative continuous functional on K if \psi:K\to\mathbb{R}_{+} is continuous and  \psi(\mu x+(1-\mu)y)\geq \mu\psi(x)+(1-\mu)\psi(y),\quad x,y\in K, \;\mu\in[0,1].  Let a,b,c>0 be constants with K and X as defined above. Define  K_{a}=\{x\in K:\|x\|0 a constant. Suppose there exists a concave nonnegative continuous function \psi on K with \psi(u)\leq u for u\in\overline{K}_{c_{4}} and let T:\overline{K}_{c_{4}}\to\overline{K}_{c_{4}} be a continuous compact map. Assume that there are numbers c_1,c_2 and c_{3} with 0c_2\}\neq\emptyset and \psi(Tu)>c_2 for all u\in K(\psi,c_2,c_{3}); \item [(ii)] \|Tu\|c_2 for all u\in K(\psi,c_2,c_{4}) with \|Tu\|>c_{3}. \end{itemize} Then T has at least three fixed points u_1,u_2 and u_{3} in \overline{K}_{c_{4}}. Furthermore, e have u_1\in\overline{K}_{c_1}, u_2\in\{u\in K(\psi,c_2,c_{4}):\psi(u)>c_2\}, u_{3}\in\overline{K}_{c_{4}}\backslash\{K(\psi,c_2, c_{4})\cup\overline{K}_{c_1}\}. \end{theorem} Let X=\{x(t)=(x_1,x_2,\dots,x_{n})^{T}\in C(\mathbb{T},\mathbb{R}^{n}):x(t)=x(t+\omega)\} with the norm \|x\|=\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}} |x_{i}(t)|, then X is a Banach space with the norm \|\cdot\|. Define a cone K in X by  K=\{x(t)=(x_1,x_2,\dots,x_{n})^{T}\in X:x_{i}(t)\geq 0,\forall t\in [0,\omega]_{\mathbb{T}},\,i=1,2,\dots,n\}  and an operator T_{\lambda} on X by  (T_{\lambda}x)(t)=\lambda\int_{t-\omega}^{t}G(t,s)f(s,x,\Phi)\Delta s .  And let \[ T_{\lambda}x=(T_{\lambda}^{1}x,T_{\lambda}^{2}x, \dots,T_{\lambda}^{n}x)^{T}. \begin{lemma} \label{lem2.3} $T_{\lambda}(K)\subset K$ and $T_{\lambda}:K\to K$ is well-defined. \end{lemma} \begin{proof} For each $x\in K$, by $(H_3)$, we have $T_{\lambda}x\in C(\mathbb{T},\mathbb{R}^{n})$, with the periodicity of $G(t,s)$ and $(\Phi x)(t)$, then \begin{align*} (T_{\lambda}x)(t+\omega) &= \lambda\int_{t}^{t+\omega}G(t+\omega,s)f(s,x,\Phi)\Delta s\\ &= \lambda\int_{t-\omega}^{t}G(t+\omega,s+\omega) f(s+\omega,x,\Phi)\Delta s\\ &= \lambda\int_{t-\omega}^{t}G(t,s)f(s,x,\Phi)\Delta s\\ &= (T_{\lambda}x)(t) , \end{align*} and by \eqref{e2.5}, $T_{\lambda}x\in K$. The proof is complete. \end{proof} \begin{lemma} \label{lem2.4} $T_{\lambda}:K\to K$ is completely continuous. \end{lemma} \begin{proof} We first show that $T_{\lambda}$ is continuous. By (H4), for any $L>0$ and $\epsilon>0$, there exists a $\delta>0$ such that $\phi,\psi\in BC$, $\|\phi\|\leq L$, $\|\psi\|\leq L$, $\| \phi-\psi\|<\delta$, $0\leq s\leq \omega$ imply \begin{align*} &\|f(s,\phi_{s},\phi(s-\tau(s,\phi(s))),(\Phi\phi)(s-\alpha(s,\phi(s))))\\ & -f(s,\psi_{s},\psi(s-\tau(s,\psi(s))), (\Phi\psi)(s-\alpha(s,\psi(s))))\|<\frac{\epsilon} {\lambda r^{M}\eta^{M}}. \end{align*} If $x,y\in K$ with $\|x\| \leq L$, $\|y\| \leq L$, and $\| x-y\|<\delta$, then \begin{align*} &\|(T_{\lambda}x)(t)-(T_{\lambda}y)(t)\|\\ &\leq\lambda\int_{t-\omega}^{t}\max_{1\leq i\leq n} |G_{i}(t,s)|\|f(s,x_{s},x(s-\tau(s,x(s))), (\Phi x)(s-\alpha(s,x(s))))\\ &\quad -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|\Delta s\\ &\leq \lambda r^{M}\eta^{M}\int_{0}^{\omega}\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\\ &\quad -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|\Delta s\\ &< \lambda r^{M}\eta^{M}\frac{\epsilon}{\lambda r^{M}\eta^{M}} = \epsilon \end{align*} for all $t\in[0,\omega]_{\mathbb{T}}$. This yields $\|T_{\lambda}x -T_{\lambda}y\|<\epsilon$. Thus, $T_{\lambda}$ is continuous. Next, we show that $T_{\lambda}$ maps any bounded sets in $K$ into relatively compact sets. Now, we first prove that $f$ maps bounded sets into bounded sets. Indeed, let $\epsilon=1$. By (H4), for any $\mu>0$, there exists $\delta>0$ such that $x, y\in BC$, $\|x\|\leq \mu$, $\|y\|\leq \mu$, $\|x-y\|<\delta$, $0\leq s\leq \omega$ imply \begin{align*} &\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\\ & -f(s,y_{s},y(s-\tau(s,y(s))),(\Phi y)(s-\alpha(s,y(s))))\|<1. \end{align*} Choose a positive integer $N$ such that $\frac{\mu}{N} < \delta$. Let $x\in BC$ and define $x^{k}(t)=\frac{k}{N}x(t)$ for $k=0,1,2,\dots,N$. If $\|x\|\leq\mu$, then \begin{align*} \|x^{k}-x^{k-1}\|=\|\frac{k}{N}x(t)-\frac{k-1}{N}x(t)\|\leq\frac{1}{N} \|x\|\leq\frac{\mu}{N} <\delta. \end{align*} Thus, \begin{align*} &\|f(s,x_{s}^{k},x^{k}(s-\tau(s,x^{k}(s))),(\Phi x^{k}) (s-\alpha(s,x^{k}(s))))\\ & -f(s,x_{s}^{k-1},x^{k-1}(s-\tau(s,x^{k-1}(s))), (\Phi x^{k-1})(s-\alpha(s,x^{k-1}(s))))\|<1 \end{align*} for all $s\in [0,\omega]_{\mathbb{T}}$. This yields \begin{align*} &\|f(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s,x(s))))\|\\ &=\|f(s,x_{s}^{N},x^{N}(s-\tau(s,x^{N}(s))),(\Phi x^{N}) (s-\alpha(s,x^{N}(s))))\|\\ &\leq \sum_{k=1}^{N}\|f(s,x_{s}^{k},x^{k}(s-\tau(s,x^{k}(s))), (\Phi x^{k})(s-\alpha(s,x^{k}(s))))\\ &\quad -f(s,x_{s}^{k-1},x^{k-1}(s-\tau(s,x^{k-1}(s))), (\Phi x^{k-1})(s-\alpha(s,x^{k-1}(s))))\|\\ &\quad +\|f(s,0,0,0\|\\ &< N+\|f(s,0,0,0\|=:Q. \end{align*} For $t\in [0,\omega]_{\mathbb{T}}$, we have \begin{align*} \|T_{\lambda}x\| &= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|(T_{\lambda}^{i}x)(t)|\\ &\leq \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x_{s},x(s-\tau(s,x(s))),(\Phi x)(s-\alpha(s, x(s))))|\Delta s\\ &\leq \lambda r^{M}\eta^{M}\omega Q. \end{align*} Finally, for $t\in \mathbb{T}$, \begin{align*} (T_{\lambda}^{i}x)^{\Delta}(t) &= [\lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s ]^{\Delta}\\ &= [\lambda\int_{t-\omega}^{t}\frac{e_{\ominus a_{i}}(t,s)} {1-e_{\ominus a_{i}}(t, t-\omega)}f_{i}(s,x,\Phi)\Delta s]^{\Delta}\\ &= \lambda\frac{e_{\ominus a_{i}}(\sigma(t),t)-e_{\ominus a_{i}}(\sigma(t),t-\omega)} {1-e_{\ominus a_{i}}(t,t-\omega)}f_{i}(t,x,\Phi)\\ &\quad +\lambda\frac{1}{1-e_{\ominus a_{i}}(t,t-\omega)}\ominus a_{i} \int_{\tilde{a}}^{t} e_{\ominus a_{i}}(t,s)f_{i}(s,x,\Phi)\Delta s\\ &\quad -\lambda\frac{1}{1-e_{\ominus a_{i}}(t,t-\omega)}\ominus a_{i} \int_{\tilde{a}}^{t-\omega}e_{\ominus a_{i}}(t,s)f_{i}(s,x,\Phi) \Delta s\\ &= \ominus a_{i}(T_{\lambda}^{i}x)(t) +\lambda e_{\ominus a_{i}}(\sigma(t),t) f_{i}(t,x,\Phi), \end{align*} where $\tilde{a}\in [0,\omega]_{\mathbb{T}}$ is an arbitrary constant, $i=1,2,\dots,n$. So we obtain $|(T_{\lambda}^{i}x)^{\Delta}(t)|\leq\gamma^{i}|T_{\lambda}^{i}x| +\lambda\kappa^{i}|f_{i}(t,x,\Phi)|,\quad i=1,2,\dots,n.$ Then \begin{align*} \|(T_{\lambda}x)^{\Delta}(t)\| &= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}| (T_{\lambda}^{i}x)^{\Delta}(t)|\\ &\leq \gamma\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|T_{\lambda}^{i}x| +\lambda\kappa\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}}|f_{i}(t,x,\Phi)|\\ &\leq \gamma\|T_{\lambda}x\|+\lambda\kappa\|f(t,x,\Phi)\|\\ &\leq \lambda\gamma r^{M}\eta^{M}\omega Q+\lambda\kappa Q. \end{align*} Hence $\{T_{\lambda}x: x\in K,\|x\|\leq\mu\}$ is a family of uniformly bounded and equicontinuous functions on $[0,\omega]_{\mathbb{T}}$. Applying Arzela-Ascoli theorem on time scales \cite{a1}, the function $T_{\lambda}$ is completely continuous. The proof is complete. \end{proof} \begin{lemma} \label{lem2.5} Existence of nonnegative periodic solutions of \eqref{e1.4} is equivalent to the existence of fixed point problem of $T_{\lambda}$ in $K$. \end{lemma} The proof of the above lemma is straight forward and we will omit it. \section{Main results} Let $$f_{i}^{h}=\limsup_{\phi_{i}\to h} \max_{t\in[0,\omega]_{\mathbb{T}}} \frac{|f_{i}(t,\phi,\Phi)|}{|a_{i}(t,\phi)|\phi_{i}},\quad i=1,2,\dots,n,$$ where $\phi(t)=(\phi_1(t),\phi_2(t),\dots,\phi_{n}(t))^{T}\in C(\mathbb{T},\mathbb{R}^{n})$, $|\phi|_1=\sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}|\phi_{i}(t)|$ and $\delta=\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}$. From the definitions of $\eta^{M}$ and $\eta^{l}$, it is obvious that $\delta>1$. \begin{theorem} \label{thm3.1} Assume that {\rm (H1)-(H4)} hold, there are constants $01$; \item [(H6)] $f_{i}^{\infty}<\omega$, $i=1,2,\dots,n$; \item [(H7)] $\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \geq\delta c_2\omega$ for $c_2\leq|x|_1\leq\|x\|\leq\delta c_2$; \item [(H8)] $\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \leq\frac{c_1\omega}{r^{M}\eta^{M}}$ for $0\leq|x|_1\leq \|x\|\leq c_1$ \end{itemize} Then \eqref{e1.4} has at least three nonnegative $\omega$-periodic solutions for $$\frac{1}{r^{M}\eta^{M}\omega}<\lambda<\frac{1}{\omega}.$$ \end{theorem} \begin{proof} Since $f_{i}^{\infty}<\omega$ holds for $1\leq i\leq n$, there exist $\varepsilon\in (0,\omega)$ and $\theta>0$ such that $|f_{i}(t,x,\Phi)|\leq \varepsilon|a_{i}(t,x)|x_{i}$ for $x_{i}\geq \theta$, $t\in [0,\omega]_{\mathbb{T}}$, $i=1,2,\dots,n$. Let $$\xi_{i}=\max_{0\leq x_{i}\leq \theta,0\leq t\leq \omega}|f_{i}(t,x,\Phi)|, \quad i=1,2,\dots,n, \quad \xi=\sum_{i=1}^{n}\xi_{i}.$$ Then $|f_{i}(t,x,\Phi)|\leq \varepsilon|a_{i}(t,x)|x_{i}+\xi_{i}$ for $x_{i}\geq 0$, $t\in [0,\omega]_{\mathbb{T}}$, $i=1,2,\dots,n$. Choose $$c_{4}>\max\Big\{\frac{r^{M}\eta^{M}\xi\omega}{\omega-\varepsilon}, \delta c_2\Big\}.$$ Then for $x\in \overline{K}_{c_{4}}$, we have \begin{align*} \|T_{\lambda}x\| &= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}|T_{\lambda}^{i}x|\\ &= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\ &= \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}\lambda \int_{t-\omega}^{t}|G_{i}(t,s)||f_{i}(s,x,\Phi)|\Delta s\\ &\leq \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}|G_{i}(t,s)|(\varepsilon|a_{i}(s,x)|x_{i}+\xi_{i})\Delta s\\ &\leq \sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}|G_{i}(t,s)|(\varepsilon|a_{i}(s,x)||x_{i}|+\xi_{i})\Delta s\\ &\leq \lambda[\varepsilon \max_{0\leq i\leq n} \{\int_{t-\omega}^{t}G_{i}(t,s)a_{i}(s,x)\Delta s\}\sum_{i=1}^{n} \max_{s\in[0,\omega]_{\mathbb{T}}}|x_{i}(s)|\\ &\quad +\sum_{i=1}^{n}\max_{t\in[0,\omega]_{\mathbb{T}}}\int_{t-\omega}^{t} |G_{i}(t,s)|\xi_{i}\Delta s]\\ &\leq \lambda[\varepsilon\|x\|+\sum_{i=1}^{n}r_1^{i}\eta_1^{i}\xi_{i}\omega]\\ &\leq \lambda[\varepsilon c_{4}+r^{M}\eta^{M}\xi\omega ]\\ &< \frac{1}{\omega}[\varepsilon c_{4}+ r^{M}\eta^{M}\xi\omega] < c_{4}. \end{align*} Hence $T_{\lambda}:\overline{K}_{c_{4}}\to\overline{K}_{c_{4}}$. Next, we define a concave nonnegative continuous function $\psi$ on $K$ by $\psi(x)=\sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}}\\ |x_{i}(t)|$, then $\psi(x)\leq \|x\|$. Let $c_{3}=\delta c_2 =\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}c_2$ and $\phi_{0}(t)=\{\phi_{0},0,\dots,0\}^{T}, \phi_{0}$ is any given number satisfying $c_2<\phi_{0}c_2\}\neq\emptyset$. Further, for $x\in K(\psi,c_2,c_{3})$, by (H7) \begin{aligned} \psi(T_{\lambda}x) &= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}} |(T_{\lambda}^{i}x)(t)| \\ &= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\ &\geq \lambda r^{l}\eta^{l}\sum_{i=1}^{n}\int_{0}^{\omega} |f_{i}(s,x,\Phi)|\Delta s \\ &= \lambda r^{l}\eta^{l}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \\ &\geq \lambda r^{l}\eta^{l}\delta c_2\omega \\ &= \lambda r^{l}\eta^{l}\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}} c_2\omega \\ &\geq\lambda r^{M}\eta^{M} c_2\omega > c_2, \end{aligned} \label{e3.6} so condition (i) of Theorem \ref{thm2.1} holds. Now, let $x\in\overline{K}_{c_1}$, by (H8) \begin{align*} \|T_{\lambda}x\| &= \sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s\\ &\leq \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x,\Phi)|\Delta s \\ &\leq \lambda r^{M}\eta^{M}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s\\ &< \frac{1}{\omega} r^{M}\eta^{M}\frac{c_1\omega}{r^{M}\eta^{M}}\ = c_1, \end{align*} then $T_{\lambda}x\in\overline{K}_{c_1}$. Finally, for $x\in K(\psi,c_2,c_{4})$ and $\|T_{\lambda}x\|>c_{3}$, so \begin{align*} c_{3}<\|T_{\lambda}x\| &\leq \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\max_{t\in [0,\omega]_{\mathbb{T}}} \int_{t-\omega}^{t}|f_{i}(s,x,\Phi)|\Delta s\\ &= \lambda r^{M}\eta^{M}\sum_{i=1}^{n}\int_{0}^{\omega} |f_{i}(s,x,\Phi)|\Delta s \\ &= \lambda r^{M}\eta^{M}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s, \end{align*} which implies \begin{aligned} \psi(T_{\lambda}x) &= \sum_{i=1}^{n}\min_{t\in [0,\omega]_{\mathbb{T}}} \lambda\int_{t-\omega}^{t}G_{i}(t,s)f_{i}(s,x,\Phi)\Delta s \\ &\geq \lambda r^{l}\eta^{l}\sum_{i=1}^{n}\int_{0}^{\omega}|f_{i}(s,x,\Phi)|\Delta s \\ &= \lambda r^{l}\eta^{l}\int_{0}^{\omega}|f(s,x,\Phi)|_{0}\Delta s \\ &> \lambda r^{l}\eta^{l}\frac{c_{3}}{\lambda r^{M}\eta^{M}} \\ &= \frac{c_{3}}{\delta} = c_2. \end{aligned} \label{e3.7} So all the conditions of Theorem \ref{thm2.1} are satisfied. Consequently, \eqref{e1.4} has at least three nonnegative $\omega$-periodic solutions. This completes the proof. \end{proof} \begin{theorem} \label{thm3.2} Let $f_{i}^{0}<\omega,\,i=1,2,\dots,n$. Assume that there exists a constant $c_2>0$ such that {\rm (H1)-(H7)} holds, then \eqref{e1.4} has at least three nonnegative $\omega$-periodic solutions for $$\frac{1}{r^{M}\eta^{M}\omega}<\lambda<\frac{1}{\omega}.$$ \end{theorem} \begin{proof} Since $f_{i}^{0}<\omega$ holds for $1\leq i\leq n$, there exist $\rho$, $\zeta$, $0<\rho<\omega$ and 0<\zeta24c_2\omega >\delta c_2\omega; \end{align*} that is, (H7) holds. For0\leq|x|_1\leq \|x\|\leq c_1, \begin{align*} \int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\mathrm{d}s &= \int_{0}^{\omega}\frac{e^{8}}{2\pi}[x_1(t+\theta)+x_2(t)]^{2} e^{-x_1(t-\theta)x_2(t)}|2+\sin u_1(t)|\mathrm{d} s\\ &\quad +\int_{0}^{\omega}\frac{e^{8}}{3\pi}[x_1(t)+x_2(t+\theta)]^{2} e^{-x_1(t)x_2(t-\theta)}|2+\cos u_2(t)|\mathrm{d} s\\ &\leq 6e^{8}c_1^{2} < 0.0000006\\ &< \frac{c_1\omega}{r^{M}\eta^{M}} = 0.000006066488\pi, \end{align*} hence (H8) holds, it is obvious that (H1)-(H4) hold. By Theorem \ref{thm3.1}, \eqref{e4.1} has at least three nonnegative2\pi$-periodic solutions. \end{proof} \begin{example} \label{exa4.2} \rm When$\mathbb{T}=\mathbb{Z}$, the following system has at least three nonnegative$2$-periodic solutions for$1/4<\lambda<1/2$: $$\label{e4.2} \begin{gathered} \Delta x(n)=(1-e^{\sin n\pi-\ln\frac{\sqrt{2}}{2}}) x(n+1) +\lambda \frac{18x^{2}(n)(3+\sin(x(n-1))+\cos(u(n)))}{e^{x(n)}}, \\ \Delta u(n) =-(0.85-0.05\sin n\pi)u(n)+0.001x(n),\quad n\in\mathbb{Z} \end{gathered}$$ \end{example} \begin{proof} Corresponding to system \eqref{e1.4}, we have$a(n)=e^{\sin n\pi-\log\frac{\sqrt{2}}{2}}-1$,$b(n)=0.85-0.05\sin n\pi, g(n)=0.001x(n)$,$f(n,x,\Phi)=\frac{18x^{2}(n)(3+\sin(x(n-1))+\cos(u(n)))}{e^{x(n)}}$,$\omega=2. So we obtain \begin{align*} e_{\ominus a}(n,n-\omega) &= \exp\Big\{\int_{n-\omega}^{n}\text{Log}\Big(1-\frac{a(\tau)}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{\int_{n-2}^{n}\log\Big(\frac{1}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{-\int_{n-2}^{n}\Big(\sin \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{-\sum_{\tau=n-2}^{n-1}\Big(\sin \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\Big\}\\ &= \exp\big\{\log\frac{1}{2}\big\} = \frac{1}{2}. \end{align*} Thenr^{M}=r^{l}=\frac{1}{1-\frac{1}{2}}=2$. In a similar argument as the above process, it is not difficult to calculate that$\eta^{M}=1$,$\eta^{l}=\frac{1}{2}$. Then$r^{M}\eta^{M}=2, \delta=\frac{r^{M}\eta^{M}}{r^{l}\eta^{l}}=\frac{2}{2\times \frac{1}{2}}=2$. Furthermore,$f^{\infty}=0<2$and$f^{0}=0<2\pi$hold, so conditions (H5) and (H6) are satisfied. Choose$c_2=1$, then$c_{3}=\delta c_2=2$. For$c_2\leq\|x\|\leq2 c_2, we obtain \begin{align*} \int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\Delta s &= \int_{0}^{2}\Big|\frac{18x^{2}(s)(3+\sin(x(s-1))+\cos(u(s)))}{e^{x(s)}}\Big|\,\Delta s\\ &= \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)(3+\sin(x(s-1))+\cos(u(s)))}{e^{x(s)}}\Big|\\ &\geq \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)}{e^{x(s)}}\Big|\\ &\geq \sum_{s=0}^{1}\frac{18c_2^{2}}{e^{2c_2}} = \frac{36}{e^{2}}\\ &\geq 4 = c_2\delta\omega; \end{align*} that is, (H7) holds. In addition, it is obvious that (H1)-(H4) hold. By Theorem \ref{thm3.2}, \eqref{e4.2} has at least three nonnegative2$-periodic solutions. \end{proof} \begin{example} When$\mathbb{T}=\mathbb{Z}$, the following system has at least three nonnegative$2$-periodic solutions for$\frac{1}{4}<\lambda<\frac{1}{2}$: $$\label{e4.3} \begin{gathered} \Delta x(n)=(1-e^{\cos n\pi-\ln\frac{\sqrt{2}}{2}}) x(n+1) +\lambda \frac{18x^{2}(n)(3+\cos(x(n-1))+\sin^{2}(u(n)))}{1+x^{2}(n)}, \\ \Delta u(n) =-(0.85-0.05\cos n\pi)u(n)+0.001x(n),\quad n\in\mathbb{Z}\,. \end{gathered}$$ \end{example} \begin{proof} Corresponding to system \eqref{e1.4}, we have$a(n)=e^{\cos n\pi-\log\frac{\sqrt{2}}{2}}-1, b(n)=0.85-0.05\cos n\pi, g(n)=0.001x(n)$,$f(n,x,\Phi)=\frac{18x^{2}(n) (3+\cos(x(n-1))+\sin^{2}(u(n)))}{1+x^{2}(n)}$,$\omega=2. So we obtain \begin{align*} e_{\ominus a}(n,n-\omega) &= \exp\Big\{\int_{n-\omega}^{n}\text{Log}\Big(1-\frac{a(\tau)}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{\int_{n-2}^{n}\log\Big(\frac{1}{1+a(\tau)}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{-\int_{n-2}^{n}\Big(\cos \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\,\Delta\tau\Big\}\\ &= \exp\Big\{-\sum_{\tau=n-2}^{n-1}\Big(\cos \tau\pi-\log\frac{\sqrt{2}}{2}\Big)\Big\}\\ &= \exp\big\{\log\frac{1}{2}\big\} = \frac{1}{2}. \end{align*} In a similar argument as Example \ref{exa4.2}, it is not difficult to get thatr^{M}\eta^{M}= \delta=2$. Furthermore,$f^{\infty}=0<2$holds, so conditions (H5) and (H6) are satisfied. Choose$c_2=1$, then$c_{3}=\delta c_2=2$. For$c_2\leq\|x\|\leq2 c_2, we obtain \begin{align*} \int_{0}^{\omega}|f(s,x,\Phi)|_{0}\,\Delta s &= \int_{0}^{2}\Big|\frac{18x^{2}(s)(3+\cos(x(s-1)) +\sin^{2}(u(s)))}{1+x^{2}(s)}\Big|\,\Delta s\\ &= \sum_{s=0}^{1}\Big|\frac{18x^{2}(s)(3+\cos(x(s-1)) +\sin^{2}(u(s)))}{1+x^{2}(s)}\Big|\\ &\geq\sum_{s=0}^{1}\Big|\frac{36}{5}\Big|\\ &\geq 8 = 2c_2\delta\omega; \end{align*} that is, (H9) holds. Choosec_1=0.01$. For$0\leq\|x\|\leq c_1$, we have$\|f(n,x,\Phi)\|\leq 90\|x\|^{2}\leq\|x\|$; that is, (H10) holds. In addition, it is obvious that (H1)-(H4) hold. By Theorem \ref{thm3.3}, \eqref{e4.3} has at least three nonnegative$2$-periodic solutions. \end{proof} \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, M. Bohner, P. Rehak; \emph{Half-linear dynamic equations, nonlinear analysis and applications}: to V. Lakshmikantham on his$80th\$ birthday, vol. 1. Dordrecht: Kluwer Academic Publishers; (2003) 1-57. \bibitem{a2} D.R. Anderson and J. 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