\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 100, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/100\hfil Stability and periodicity of solutions] {Stability and periodicity of solutions for delay dynamic systems on time scales} \author[Z.-Q. Zhu, Q.-R. Wang \hfil EJDE-2014/100\hfilneg] {Zhi-Qiang Zhu, Qi-Ru Wang} \address{Zhi-Qiang Zhu \newline Department of Computer Science, Guangdong Polytechnic Normal University, \newline Guangzhou 510665, China} \email{z3825@163.com} \address{Qi-Ru Wang (corresponding author)\newline School of Mathematics and Computational Science, Sun Yat-Sen University, \newline Guangzhou 510275, China} \email{mcswqr@mail.sysu.edu.cn} \thanks{Submitted November 17, 2013. Published April 11, 2014.} \subjclass[2000]{34N05, 34K13, 34K20} \keywords{Delay dynamic system; stability; periodic solution; fixed point;\hfill\break\indent time scale} \begin{abstract} This article concerns the stability and periodicity of solutions to the delay dynamic system $$ x^{\triangle}(t)=A(t) x(t) + F(t, x(t), x(g(t)))+C(t) $$ on a time scale. By the inequality technique for vectors, we obtain some stability criteria for the above system. Then, using the Horn fixed point theorem, we present some conditions under which our system is asymptotically periodic and its periodic solution is unique. In particular, the periodic solution is positive under proper assumptions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\mathbb{R}$ denote the set of real numbers, $\mathbb{R}^{m}$ the $m$-dimensional Euclidean space and $\mathbb{T}$ a time scale, i.e., an nonempty closed subset of $\mathbb{R}$ with the topology and ordering inherited from $\mathbb{R}$. The forward jump operator $\sigma:\mathbb{T}\to\mathbb{T}$ is defined by \[ \sigma(t):=\inf\{s\in\mathbb{T}:s>t\}, \] the backward jump operator $\rho:\mathbb{T}\to\mathbb{T}$ is defined by \[ \rho(t):=\sup\{s\in\mathbb{T}:s0$ and $\int^{\omega}_0b(s) \mathrm{d} s>0$, $h\in C(\mathbb{R},\mathbb{R})$ is $\omega$-periodic, and $f\in C(\mathbb{R},[0,\infty))$. Precisely speaking, Franco et al improved the work in \cite{Cheng} and obtained the following result \cite[Theorem 2]{Liz}. \begin{theorem} \label{thmA} Assume that \begin{gather*} \sigma=e^{\int^{\omega}_0a(s)\mathrm{d} s}, \quad G(t,s)=\frac{\int^{s}_{t}a(r)\mathrm{d} r}{\sigma-1},\\ A=\max_{t\in [0,\omega]}\int^{\omega}_0G(t,s)b(s)\mathrm{d} s,\quad B=\min_{t\in [0,\omega]}\int^{\omega}_0G(t,s)b(s)\mathrm{d}s. \end{gather*} Assume further that $f_0,f_{\infty}\in (0,\infty)$, and \[ \frac{1}{B\max\{f_0,f_{\infty}\}} < \lambda < \frac{1}{A\min\{f_0,f_{\infty}\}}, \] where $f_0$ and $f_{\infty}$ are defined respectively by \[ f_0=\lim_{x\to 0^{+}}\frac{f(x)}{x}, \quad f_{\infty}=\lim_{x\to \infty}\frac{f(x)}{x}. \] Then \eqref{e2} has a positive $\omega$-periodic solution. \end{theorem} In 2003, Zhao \cite{Zhao} studied the asymptotic periodic solutions of \eqref{1} for the case when $\mathbb{T}=[-\tau,\infty)$ and the function $F$ is independent of the second variable. In 2008, Zhu and Cheng \cite{Zhu} considered the periodic solutions of \eqref{1} when $\mathbb{T}=\{-\tau,-\tau+1,-\tau+2,\ldots\}$ with positive integer $\tau$. The aim of this article is to extend the techniques in \cite{Zhu} and establish some criteria for the stability and periodicity of \eqref{1}. As we will see in the sequel, our results will generalize or improve those in \cite{Zhao,Zhu} and, to some extent, possess more comprehensive suitability than \cite{Liz}. \section{Preliminaries} Let $[a,b]_{\mathbb{T}}$ be an interval in $\mathbb{T}$ which is defined by $[a,b]_{\mathbb{T}}=\{t\in \mathbb{T}: a \leq t\leq b\}$. The symbol $[a,\infty)_{\mathbb{T}}$ has a similar meaning. Let $C[a,b]_{\mathbb{T}}$ with $a, b\in\mathbb{T}$ denote the set of all rd-continuous functions mapping $[a, b]_{\mathbb{T}}$ into $\mathbb{R}^{m}$, endowed with the linear structure and the norm $\|\cdot\|$ defined by $\|\varphi\| =\max_{t\in[a, b]_{\mathbb{T}}}\|\varphi(t)\|$, where $\|\varphi(t)\|$ denotes the norm of vector $\varphi(t)$. Then $C[a, b]_{\mathbb{T}}$ is a Banach space. For a given $\varphi \in C[-\tau, 0]_{\mathbb{T}}$, by a solution $x(\varphi)$ of \eqref{1} we mean that $x(\varphi)$ is a function defined on $[-\tau,\infty)_{\mathbb{T}}$ which coincides with $\varphi$ on $[-\tau,0]_{\mathbb{T}}$ and satisfies \eqref{1} on $[0,\infty)_{\mathbb{T}}$. For any $x=(x_1,x_2,\dots,x_m)^{T}, y=(y_1,y_2,\dots,y_m)^{T}\in \mathbb{R}^{m}$, the symbols $x\leq y$ and $x0$, $A_i\in \mathbb{R}^{m\times m}$ with $A_i\geq 0$ for $i\in\{0,1,2\}$, and $C_0\in \mathbb{R}^{m}$ with $C_0>0$ such that \begin{itemize} \item[(H1)] $1-\alpha \mu(t)>0$ for $t\in \mathbb{T}$ and $\lim_{t\to \infty} e_{-\alpha}(t,0)=0$; \item[(H2)] $|A(t)+\alpha I|\leq A_0$ for all $t\in [0,\infty)_{\mathbb{T}}$; \item[(H3)] $|F(t,x_1,x_2)-F(t,y_1,y_2)|\leq A_1|x_1-y_1|+A_2|x_2-y_2|$ and $|C(t)|\leq C_0$ for all $t\in [0,\infty)_{\mathbb{T}}$; \item[(H4)] $\widetilde{\rho}(A_0+A_1+A_2)<\alpha$, where $\widetilde{\rho}$ denotes the spectral radius of square matrices. \end{itemize} We remark that the real number $\alpha>0$ depends on the concrete time scale. For example, let $\mathbb{T}$ be the set of nonnegative integers. Then, $1-\alpha\mu(t)>0 $ when $0<\alpha<1$. In this case we have $e_{-\alpha}(t,0)=(1-\alpha)^{t}$ and hence $\lim_{t\to\infty} e_{-\alpha}(t,0)=0$. While $\mathbb{T}=[0,\infty)$, $e_{-\alpha}(t,0)=e^{-\alpha t}$ and $1-\alpha \mu(t)>0$ for any real number $\alpha>0$. In this case $\lim_{t\to\infty} e_{-\alpha}(t,0)=0$. We remark further that $e_{-\alpha}(t,0)>0$ by the assumption (H1) (see \cite[Theorem 2.44]{Bohner1}). Note that \eqref{1} can be rewritten as \[ x^{\Delta}(t)=-\alpha x(t)+(A(t)+\alpha I)x(t) + F(t,x(t),x(g(t)))+C(t),\quad t\in \mathbb{T}, \] where $I$ is the $m\times m$ identity matrix. Then, similar to \cite[Theorem 5.24]{Bohner1} (see also \cite[Theorems 8.16, 8.24]{Bohner1}), the solution $x(\varphi)$ ($x$ for short) of \eqref{1} is existent and unique on $[-\tau,\infty)_{\mathbb{T}}$, and given by \begin{equation} \begin{gathered} x(t) = \varphi(t), ~t\in[-\tau,0]_{\mathbb{T}}, \\ \begin{aligned} x(t) & = e_{-\alpha}(t,0)\varphi(0)+ \int^{t}_0e_{-\alpha}(t,\sigma(s)) \Big[(A(s)+\alpha I)x(s)\\ &\quad +F(s,x(s),x(g(s)))+C(s)\Big]\Delta s,\quad t\in [0,\infty)_{\mathbb{T}}. \end{aligned} \end{gathered} \label{2} \end{equation} For a subset $\mathbb{S}$ of $\mathbb{R}^{m}$ and $\varepsilon \in \mathbb{R}^{m}$ with $\varepsilon>0$, let $O(\mathbb{S},\varepsilon )$ be defined by \[ O(\mathbb{S},\varepsilon )=\{x\in \mathbb{R}^{m}:\inf_{s \in \mathbb{S}}|x-s|\leq \varepsilon\}. \] \begin{definition} \rm %Def-1 Let $x(\varphi _0)$ be a solution of \eqref{1} and $\varphi \in C[-\tau, 0]_{\mathbb{T}}$. If $\varphi \to \varphi_0$ implies $x(\varphi)(t)\to x(\varphi _0)(t)$ for all $t\in [0,\infty)_{\mathbb{T}}$, then the solution $x(\varphi_0)$ is said to be stable. \end{definition} \begin{definition} \rm %Def-2 System \eqref{1} is said to be equi-bounded if, for any $H\in \mathbb{R}^{m}$ with $H>0$, there exists an $M(H)\in \mathbb{R}^{m}$ such that $\varphi \in C[-\tau, 0]_{\mathbb{T}}$ with $|\varphi (t)|\leq H$ on $[-\tau,0]_{\mathbb{T}}$ implies $|x(\varphi)(t)|\leq M(H)$ for all $t\in [0,\infty)_{\mathbb{T}}$. \end{definition} \begin{definition} \rm %Def-3 A set $\mathbb{S}\subset \mathbb{R}^{m}$ is said to be a global attractor of \eqref{1} if, for any $\varepsilon\in \mathbb{R}^{m}$ with $\varepsilon>0$ and $\varphi\in C[-\tau, 0]_{\mathbb{T}}$, there exists a positive number $T(\varepsilon, \varphi)\in \mathbb{T}$ such that the solution $x(\varphi )$ of \eqref{1} satisfies $x(\varphi )(t)\in O(\mathbb{S},\varepsilon )$ for all $t\in [T(\varepsilon,\varphi),\infty)_{\mathbb{T}}$. \end{definition} \begin{definition} \rm %Def-4 System \eqref{1} is said to be extremely stable if any two solutions $x(\varphi _1)$ and $x(\varphi _2)$ of \eqref{1} satisfy $x(\varphi _1)(t)-x(\varphi _2)(t)\to 0$ as $t\to\infty$. \end{definition} The subset $B$ of $C[a,b]_{\mathbb{T}}$ is said to be equi-continuous if, for any real number $\varepsilon>0$, there exists a $\delta>0$ such that $\|x(t_1)-x(t_2)\|<\varepsilon$ for all $x\in B$ provided $|t_1-t_2|\leq\delta $ for $t_1,t_2\in [a,b]_{\mathbb{T}}$. In what follows, we will require the time scale version of Arzela-Ascoli theorem. Although a similar result has been given in \cite{Zhu2}, for the completeness we present the following lemma and its proof. \begin{lemma}\label{L:1} If $B\subset C[a,b]_{\mathbb{T}}$ is bounded and equi-continuous, then $B$ is relatively compact. \end{lemma} \begin{proof} Note that $C[a,b]_{\mathbb{T}}$ is a Banach space, we need only to show that $B$ is completely bounded. Since $B$ is equi-continuous, for any real number $\varepsilon>0$, there exists a $\delta>0$ such that when $t_1,t_2\in [a,b]_{\mathbb{T}}$ with $|t_1-t_2|\leq\delta $, it follows that \begin{equation} \|f(t_1)-f(t_2)\|<\varepsilon \quad \text{for all } f\in B. \label{3} \end{equation} For this $\delta$, there exists a partition on $[a,b]_{\mathbb{T}}$ as follows \[ a=t_1\delta$ but $\rho(t_i)=t_{i-1}$ (see \cite[Lemma 5.7]{Bohner2}). Let \[ \widetilde{B}=\{(f(t_1),f(t_2),\dots,f(t_n)):f\in B\}. \] Since $B$ is bounded, $\widetilde{B}\in \mathbb{R}^{m\times n}$ is also bounded and hence is completely bounded. Therefore, there exists an $\varepsilon$-net $\{f_1,f_2,\dots,f_{k}\}$ of $\widetilde{B}$. We assert that it is also an $\varepsilon$-net of $B$. Indeed, for any $f\in B$, $(f(t_1),f(t_2),\dots,f(t_n))\in \widetilde{B}$ implies that there exists some $v\in\{1,2,\dots,k\}$ such that \begin{equation} \|f(t_i)-f_v(t_i)\|<\varepsilon,\quad i=1, 2,\dots, n. \label{4} \end{equation} Note that $t\in[a,b]_{\mathbb{T}}$ implies $t\in[t_i,t_{i+1}]_{\mathbb{T}}$ for some $i\in\{1,2,\dots,n-1\}$. Now there are two cases to consider. Case 1: $t_{i+1}-t_i\leq\delta$. In this case, we have from \eqref{3} and \eqref{4} that \[ \|f(t)-f_v(t)\| \leq \|f(t)-f(t_i)\|+\|f(t_i)-f_v(t_i)\|+\|f_v(t_i)-f_v(t)\| < 3\varepsilon. \] Case 2: $t_{i+1}-t_i>\delta$. In this case, $\rho(t_{i+1})=t_i$ and hence, $t=t_i$ or $t=t_{i+1}$. Thus $\|f(t)-f_v(t)\|<\varepsilon$ by \eqref{4}. To sum up, $\{f_1,f_2,\dots,f_{k}\}$ is an $\varepsilon$-net of $B$. The proof is complete. \end{proof} In the sequel, the following standard results will be imposed. \begin{lemma}[\cite{Horn1,Zhao}] \label{L:2} Let $R\in \mathbb{R}^{m\times m}$ and $\widetilde{\rho} (R)$ be the spectral radius of $R$. If $R\geq 0$ and $\widetilde{\rho} (R)<1$, then $(I-R)^{-1}\geq 0$. \end{lemma} \begin{lemma}[\cite{Horn2}] \label{L:3} Let $\mathbb{X}$ be a Banach space and $u:\mathbb{X}\to \mathbb{X}$ be a completely continuous. If there exists a bounded set $\mathbb{E}$ such that for each $x\in \mathbb{X}$, there exists integer $n=n(x)$ such that $u^{n}(x)\in \mathbb{E},$ then $u$ has a fixed point in $\mathbb{E}$. \end{lemma} \section{Stability analysis} Note that, by assumption {\rm (H4)} and Lemma \ref{L:2}, the matrix $I-\frac{1}{\alpha}(A_0+A_1+A_2)$ is invertible. For the sake of convenience, we let \[ U=\big(I-\frac{A_0+A_1+A_2}{\alpha} \big) ^{-1} %\label{5} \] whenever it is defined. \begin{theorem}\label{T:1} For any $\varphi _0\in C[-\tau, 0]_{\mathbb{T}}$, the solution $x(\varphi_0)$ of \eqref{1} is stable. \end{theorem} \begin{proof} Let $\varphi\in C[-\tau, 0]_{\mathbb{T}}$. From \eqref{2} we have \begin{equation} \label{5} \begin{aligned} &|x(\varphi)(t)-x(\varphi _0)(t)| \\ &\leq e_{-\alpha}(t,0)|\varphi(0)-\varphi_0(0)|+ \int^{t}_0e_{-\alpha}(t,\sigma(s))(A_0+A_1)|x(\varphi)(s) -x(\varphi_0)(s)|\Delta s \\ &\quad + \int^{t}_0e_{-\alpha}(t,\sigma(s))A_2|x(\varphi)(g(s)) -x(\varphi_0)(g(s))|\Delta s. \end{aligned} \end{equation} For any $\varepsilon \in \mathbb{R}^{m}$ with $\varepsilon >0$, we assert that $|x(\varphi )(t)-x(\varphi _0)(t)|\leq \frac{1}{\alpha}U \varepsilon$ for all $t\in [0,\infty)_{\mathbb{T}}$ when $|\varphi (t )-\varphi _0(t )|\leq \frac{1}{\alpha}U \varepsilon$ on $[-\tau,0]_{\mathbb{T}}$. Otherwise, there exists a $t_1\in {\mathbb{T}}\cap (0,\infty)$ and a real number $\beta>1$ such that \begin{gather} |x(\varphi)(t)-x(\varphi_0)(t)|\leq \frac{\beta}{\alpha}U\varepsilon,t\in [0,t_1]_{\mathbb{T}}, \label{6} \\ |x_v(\varphi)(t_1)-x_v(\varphi_0)(t_1)|= \big(\frac{\beta}{\alpha}U\varepsilon\big)_v,\label{7} \end{gather} where $x_v(\varphi)(t_1)-x_v(\varphi_0)(t_1)$ stands for some component of the vector $x(\varphi)(t_1)-x(\varphi_0)(t_1)$. Then, we have from \eqref{5} and \eqref{6} that \begin{align*} &|x(\varphi)(t_1)-x(\varphi _0)(t_1)| \\ &\leq e_{-\alpha}(t_1,0)|\varphi(0)-\varphi_0(0)| +\int^{t_1}_0e_{-\alpha}(t_1,\sigma(s))\Delta s \cdot \frac{\beta}{\alpha}(A_0+A_1+A_2)U\varepsilon \\ &< \frac{\beta}{\alpha} e_{-\alpha}(t_1,0)U\varepsilon +\int^{t_1}_0e_{-\alpha}(t_1,\sigma(s))\Delta s \cdot \frac{\beta}{\alpha}[(A_0+A_1+A_2)U\varepsilon+\alpha\varepsilon] \\ &=\frac{\beta}{\alpha} e_{-\alpha}(t_1,0)U\varepsilon+ \frac{1}{\alpha}[1-e_{-\alpha}(t_1,0)]\cdot \frac{\beta}{\alpha}[(A_0+A_1+A_2)U\varepsilon+\alpha\varepsilon]\\ &=\frac{\beta}{\alpha}U\varepsilon, \end{align*} which is in conflict with \eqref{7}, where we have used the formula \[ \int^{t_1}_0e_{-\alpha}(t_1,\sigma(s))\Delta s=-\frac{1}{\alpha}[e_{-\alpha}(t_1,0)-1], \] see \cite[Theorem 2.39]{Bohner1} for the details. Since $\varepsilon >0$ is arbitrary, the solution $x(\varphi _0)$ of \eqref{1} is stable. The proof is complete. \end{proof} \begin{theorem}\label{T:2} System \eqref{1} is equi-bounded and the set \[ \mathbb{S}=\big\{ s\in \mathbb{R}^{m}:|s|\leq \frac{1}{\alpha}UC_0\big\} \] is a global attractor of \eqref{1}. \end{theorem} %T:2 \begin{proof} For the first part, we note that $\frac{1}{\alpha}UC_0>0$. Then, for any $H\in \mathbb{R}^{m}$ with $H>0$, there exists a real number $\beta_0 \geq 1$ such that $H\leq \frac{\beta_0}{\alpha}UC_0$. Now, for any given $\varphi \in C[-\tau, 0]_{\mathbb{T}}$ with $|\varphi (t)|\leq H$ on $[-\tau,0]_{\mathbb{T}}$, we assert that the solution $x(\varphi)$ of \eqref{1} satisfies $|x(\varphi)(t)|\leq \frac{\beta_0}{\alpha}UC_0$ on $[0,\infty)_{{\mathbb{T}}}$. Otherwise, similar to the proof of Theorem \ref{T:1}, there exist a $t_1\in {\mathbb{T}}\cap (0,\infty)$ and a real number $\beta_1>1$ such that \begin{gather} |x(\varphi)(t)|\leq \frac{\beta_1\beta_0}{\alpha}UC_0,t\in [0,t_1]_{\mathbb{T}} , \label{8}\\ |x_v(\varphi)(t_1)|= \big(\frac{\beta_1\beta_0}{\alpha}UC_0\big)_v.\label{9} \end{gather} By the same method as above, from \eqref{2} and \eqref{8} we have \[ |x(\varphi)(t_1)|<\frac{\beta_1\beta_0}{\alpha}UC_0, \] which contradicts \eqref{9}. Next we show the second part. By the equi-boundedness, it follows that the solution $x(\varphi)$ of \eqref{1} is bounded for any $\varphi \in C[-\tau, 0]_{\mathbb{T}}$ and hence we may assume that \[ \limsup_{t\to\infty}|x(\varphi)(t)|=\frac{1}{\alpha}UC_0+\beta, \] where $\beta \in \mathbb{R}^{m}$. Therefore, for any given $\varepsilon\in\mathbb{R}^{m}$ with $\varepsilon>0$, there exists a large $t_1\in [0,\infty)_{\mathbb{T}}$ such that \begin{equation} |x(\varphi)(t)|\leq \frac{1}{\alpha}UC_0+\beta+\varepsilon \quad \text{for all }t\in[t_1-\tau,\infty)_{\mathbb{T}}.\label{10} \end{equation} Let \[ U(\varepsilon)=\frac{1}{\alpha}UC_0+\beta+\varepsilon \] and $x(t)$ for $x(\varphi)(t)$. According to \eqref{2} we see that when $t\in[t_1,\infty)_{\mathbb{T}}$, \begin{align*} x(t)&=e_{-\alpha}(t,t_1)x(t_1)\\ &\quad +\int^{t}_{t_1}e_{-\alpha}(t,\sigma(s))[(A(s)+\alpha I)x(s)+F(s,x(s),x(g(s))))+C(s)]\Delta s, \end{align*} and this, together with \eqref{10}, yields \begin{align*} |x(t)|&\leq e_{-\alpha}(t,t_1)|x(t_1)|+\int^{t}_{t_1}e_{-\alpha}(t,\sigma(s))\Delta s [(A_0+A_1+A_2)U(\varepsilon)+C_0] \\ &= e_{-\alpha}(t,t_1)|x(t_1)|+ \frac{1}{\alpha}[1-e_{-\alpha}(t,t_1)][(A_0+A_1+A_2)U(\varepsilon)+C_0], \end{align*} which, with the aid of $e_{-\alpha}(t,t_1)=e_{-\alpha}(t,0)e_{-\alpha}(0,t_1)\to 0 $ as $t\to \infty$ (see assumption (H1)), results in \[ \frac{1}{\alpha}UC_0+\beta \leq \frac{1}{\alpha}[(A_0+A_1+A_2)U(\varepsilon)+C_0], \] namely \begin{equation}\label{11} \big(1-\frac{A_0+A_1+A_2}{\alpha}\big)\beta \leq \frac{A_0+A_0+A_2}{\alpha}\varepsilon. \end{equation} Note that assumption (H4) and Lemma \ref{L:2} imply \[ \big(1-\frac{A_0+A_1+A_2}{\alpha}\big)^{-1}\geq 0. \] Consequently, by \eqref{11} we see that \[ \beta\leq \big(I-\frac{A_0+A_1+A_2}{\alpha}\big)^{-1}\frac{A_0+A_1+A_2}{\alpha}\varepsilon, \] which yields $\beta \leq 0$. Now from \eqref{10} we show that $\mathbb{S} $ is a global attractor of \eqref{1}. The proof is complete. \end{proof} \begin{remark} \rm The proof of the first part in Theorem \ref{T:2} shows that the solution $x(\varphi)$ of \eqref{1} satisfies that \[ |x(\varphi)(t)|\leq \mathbb{S}\quad \text{ for all } t\in [0,\infty)_{\mathbb{T}} \] when $\varphi \in \mathbb{S}$, where $\mathbb{S}$ is defined as in Theorem \ref{T:2}. In other words, $\mathbb{S}$ is an invariant set of \eqref{1}. Accordingly, our Theorem \ref{T:2} covers the conclusions in \cite[Theorems 1 and 2 ]{Zhao}. \end{remark} Note that if the functions $A(t)$, $C(t)$ and $F(t,x, y)$ satisfy $A(t)+\alpha I\geq 0$, $C(t)\geq 0 $ and $F(t,x,y) \geq 0 $ for all $t\in \mathbb{T}$ and $x,y\in \mathbb{R}^{m}$, then from \eqref{2} we have \[ x(\varphi)(t) \geq e_{-\alpha}(t,0)\varphi(0)\to 0 \text{ as }t\to \infty. \] Hence, by \eqref{10} we see that, for any given positive vector $\varepsilon \in \mathbb{R}^{m}$ there exists a positive number $T\in \mathbb{T}$ such that \[ -\varepsilon \leq x(\varphi)(t) \leq \frac{1}{\alpha}UC_0+\varepsilon \quad\text{for } t \in[T,\infty)_{\mathbb{T}}. \] Now the following conclusion is clear. \begin{theorem}\label{T:3} Assume that in \eqref{1}, the functions $A(t)$, $C(t)$ and $F(t,x, y)$ satisfy $A(t)+\alpha I\geq 0$, $C(t)\geq 0 $ and $F(t,x,y) \geq 0 $ for all $t\in \mathbb{T}$ and $x,y\in \mathbb{R}^{m}$. Then the set \[ \mathbb{S}_0=\big\{ s\in \mathbb{R}^{m}:0\leq s\leq \frac{1}{\alpha}UC_0\big\} \] is a global attractor of \eqref{1}. \end{theorem} \begin{theorem} \label{T:4} System \eqref{1} is extremely stable. \end{theorem} \begin{proof} Let $x_1=x(\varphi_1)$ and $x_2=x(\varphi_2)$ be any two solutions of \eqref{1} and $z(t)=x_1(t)-x_2(t)$. Then, Theorem \ref{T:2} implies that $z(t)$ is bounded. Thus we may assume that \begin{equation} \limsup_{t\to\infty}|z(t)|=\widetilde{z}.\label{12} \end{equation} This means that, for any positive vector $\varepsilon\in\mathbb{R}^{m}$, there exists a large $t_1\in[0,\infty)_{\mathbb{T}}$ such that \begin{equation} |z(t)|\leq\widetilde{z}+\varepsilon, \quad t\in[t_1-\tau,\infty)_{\mathbb{T}}.\label{13} \end{equation} In addition, $z(t)$ must satisfy \[ z^{\Delta}(t)=A(t)z(t)+F(t,x_1(t),x_1(g(t)))-F(t,x_2(t),x_2(g(t))). \] Similar to \eqref{2}, it follows that when $t\in[ t_1,\infty)_{\mathbb{T}}$, \begin{align*} z(t)&=e_{-\alpha}(t,t_1)z(t_1) +\int^{t}_{t_1}e_{-\alpha}(t,\sigma(s)) (A(s)+\alpha I)z(s)\Delta s \\ &\quad +\int^{t}_{t_1}e_{-\alpha}(t,\sigma(s))[F(s,x_1(s),x_1(g(s)))-F(s,x_2(s),x_2(g(s)))]\Delta s, \end{align*} which, associated with \eqref{13}, implies that when $t\in[t_1,\infty)_{\mathbb{T}}$, \begin{equation} |z(t)|\leq e_{-\alpha}(t,t_1)|z(t_1)|+ \frac{1}{\alpha}[1-e_{-\alpha}(t,t_1)](A_0+A_1+A_2)(\widetilde{z}+\varepsilon). \label{14} \end{equation} Invoking \eqref{12}, from \eqref{14} we have \[ \widetilde{z}\leq \frac{A_0+A_1+A_2}{\alpha}(\widetilde{z}+\varepsilon) \] and then \begin{equation} \widetilde{z}\leq \Big(I-\frac{A_0+A_1+A_2}{\alpha}\Big)^{-1}\frac{A_0+A_1+A_2}{\alpha}\varepsilon. \label{15} \end{equation} Since $\varepsilon>0$ is arbitrary, by \eqref{12} and \eqref{15} we see that $\lim_{t\to\infty}z(t)=0$, which completes our proof. \end{proof} \section{Existence of periodic solutions}\label{s4} In this section, we consider the existence of periodic solutions for \eqref{1}. To do this, we set \begin{equation}\label{16} G(t,x(t))=A(t)x(t)+F(t,x(t),x(g(t)))+C(t). \end{equation} For a solution $x(\varphi)$ of \eqref{1} and $t\in [0,\infty)_{\mathbb{T}}$, the delay function $x_{t}(\varphi):[-\tau,0]_{\mathbb{T}}\to \mathbb{R}^{m}$ is defined by $x_{t}(\varphi)(\theta)=x(\varphi)(t+\theta)$ for $\theta\in[-\tau,0]_{\mathbb{T}}$. \begin{theorem}\label{T:5} Suppose that $\omega\geq \tau$, $A(t)=A(t+\omega)$, $C(t)=C(t+\omega)$, $F(t,\cdot,\cdot)=F(t+\omega,\cdot,\cdot)$ and $g(t+\omega)=g(t)+\omega$ for all $t\in [0,\infty)_{\mathbb{T}}$. Then \eqref{1} has a unique $\omega$-periodic solution. \end{theorem} \begin{proof} Let $u: C[-\tau, 0]_{\mathbb{T}}\to C[-\tau, 0]_{\mathbb{T}}$ be defined by \[ u(\varphi)=x_{\omega}(\varphi), \varphi\in C[-\tau, 0]_{\mathbb{T}}. \] Then, Theorem \ref{T:1} implies that $u$ is continuous on $C[-\tau, 0]_{\mathbb{T}}$. Since the solution of \eqref{1} is existent and unique, by the periodicity of $A$, $C$ and $F$ with respect to $t$ we obtain that \begin{equation} x(\varphi)(t+\omega)=x(x_{\omega}(\varphi))(t), t\in[0,\infty)_{\mathbb{T}},\label{17} \end{equation} where $x(x_{\omega}(\varphi))$ represents the solution of \eqref{1} through $x_{\omega}(\varphi)$. From \eqref{17} we have $x_{\omega}(x_{\omega}(\varphi))=x_{2\omega}(\varphi)$ and hence $u^{2}(\varphi)=x_{2\omega}(\varphi)$. In general, for any positive integer $n$, it follows from the mathematical induction that $u^{n}(\varphi)=x_{n\omega}(\varphi)$. (i) First, we show that $u$ maps any bounded subset $\mathbb{E}\subset C[-\tau, 0]_{\mathbb{T}}$ into a relatively compact set. We note first that Theorem \ref{T:2} implies that there exists $M_1\in\mathbb{R}^{m}$ such that for any $\varphi\in\mathbb{E}$, the solution $x(\varphi)$ of \eqref{1} satisfies $|x(\varphi)(t)|\leq M_1$ for all $t\in \mathbb{T}$. Then, by assumption (H3), there exists $M_2\in\mathbb{R}^{m}$ so that $|G(t,x(\varphi)(t))|\leq M_2$ for all $t\in \mathbb{T}$ and $\varphi\in\mathbb{E}$, where $G$ is defined by \eqref{16}. Subsequently, for all $\varphi\in\mathbb{E}$ we have \begin{equation} \label{18} \begin{aligned} |u^{\Delta}(\varphi)(\theta)| &= |x^{\Delta}(\varphi)(\omega+\theta)| \\ &= |G(\omega+\theta, x(\varphi)(\omega+\theta))|\leq M_2, ~\theta\in[-\tau,0]_{\mathbb{T}}, \end{aligned} \end{equation} where $\Delta$ denotes the derivative with respect to $\theta$, and the hypothesis $\omega\geq \tau$ has been imposed. Now invoking the Mean Value Theorem \cite[Theorem 1.14]{Bohner2}, we have from \eqref{18} that \[ |u(\varphi)(\theta_1)-u(\varphi)(\theta_2)|\leq M_2|\theta_1-\theta_2| \text{ for all } \varphi\in\mathbb{S}, \] which means that $u(\mathbb{E})$ is equi-continuous. Since $u(\mathbb{E})$ is bounded, Lemma \ref{L:1} implies that $u(\mathbb{E})$ is relatively compact and hence $u$ is completely continuous. (ii) Next, we show that $u$ has a fixed point. To this end, we recall that, by Theorem \ref{T:2}, \[ \big\{s\in\mathbb{R}^{m}:|s|\leq\frac{1}{\alpha}U C_0 \big\} \] is a global attractor of \eqref{1}. Then, for a given $\varepsilon_0\in\mathbb{R}^{m}$ with $\varepsilon_0>0$, and any $\varphi\in C[-\tau, 0]_{\mathbb{T}}$, there exists $\widetilde{T}=\widetilde{T}(\varepsilon_0,\varphi)$ such that \begin{equation} |x(\varphi)(t)|\leq \frac{1}{\alpha}U C_0+\varepsilon_0 \text{ for all } t\in[\widetilde{T},\infty)_{\mathbb{T}}.\label{19} \end{equation} Taking an integer $n=n(\varepsilon_0,\varphi)$ with $n \omega\geq \widetilde{T}+\tau$, it follows from \eqref{19} that \begin{equation} |x(\varphi)(n \omega+\theta)|\leq \frac{1}{\alpha}U C_0+\varepsilon_0 \quad \text{for all } \theta\in[-\tau,0]_{\mathbb{T}}.\label{20} \end{equation} Let \begin{equation}\label{21} \mathbb{E}=\big\{\varphi\in C[-\tau, 0]_{\mathbb{T}}:|\varphi(\theta)|\leq\frac{1}{\alpha}U C_0+\varepsilon_0,\theta\in[-\tau,0]_{\mathbb{T}}\big\}. \end{equation} Then \eqref{20} implies $x_{n\omega}(\varphi)\in \mathbb{E}$ and hence $u^{n}(\varphi)\in \mathbb{E}$. By Lemma \ref{L:3} we see that $u$ has a fixed point $\varphi_0\in \mathbb{E}$. (iii) Note that the solution $x(\varphi_0)$ of \eqref{1} through $\varphi_0$ satisfies \[ x(\varphi_0)(t)= x(u(\varphi_0))(t)=x(x_{\omega}(\varphi_0))(t)=x(\varphi_0)(t+\omega)\quad \text{for all }t\in[0,\infty)_{\mathbb{T}}, \] where we have invoked \eqref{17}. Hence $x(\varphi_0)$ is $\omega$-periodic. Further, Theorem \ref{T:4} implies that this periodic solution of \eqref{1} is unique. The proof is complete. \end{proof} Note that under the conditions in Theorem \ref{T:3}, the set \[ \big\{ s\in \mathbb{R}^{m}:0\leq s\leq \frac{1}{\alpha}UC_0\big\} \] is a global attractor of \eqref{1}. Then, if we replace $\mathbb{E}$ as in \eqref{21} by \[ \mathbb{E}_0=\big\{\varphi\in C[-\tau, 0]_{\mathbb{T}}: -\varepsilon_0\leq \varphi(\theta) \leq \frac{1}{\alpha}UC_0 + \varepsilon_0, \theta\in[-\tau,0]_{\mathbb{T}}\big\}, \] then, in a similar way as in the proof of Theorem \ref{T:5} we see that $u$ has a fixed point $\varphi_0\in \mathbb{E}_0$ for which $x(\varphi_0)(t)$ is a unique $\omega$-periodic solution of \eqref{1}. Since $\varepsilon_0>0$ is arbitrary, $\varphi_0(\theta)\geq 0$ for all $\theta\in[-\tau,0]_{\mathbb{T}}$. Therefore, from \eqref{2} we have \[ x(\varphi_0)(t)\geq 0 ~\text{ and }~x(\varphi_0)(t)~\text{ is not identically zero on }~ \mathbb{T} \] provided $C(t)\geq 0$ is not identically zero on $\mathbb{T}$. In this case we refer to the periodic solution $x(\varphi_0)$ as positive. Now we extract the essence and obtain the following result. \begin{theorem}\label{T:6} Under the conditions in Theorems \ref{T:3} and \ref{T:5}, if $C(t)$ is not identically zero on $\mathbb{T}$, then system \eqref{1} has a unique positive $\omega$-periodic solution. \end{theorem} Now we see that under the conditions in Theorem \ref{T:5}, system \eqref{1} possesses a unique periodic solution $x(\varphi_0)$. On the other hand, by Theorem \ref{T:4} we have \[ x(\varphi)(t)-x(\varphi_0)(t)\to 0 \quad \text{as }t\to\infty \] for any solution $x(\varphi)$ of \eqref{1}. That is, $x(\varphi)$ is an asymptotical periodic solution of \eqref{1}. Furthermore, $x(\varphi_0)(t)\equiv 0$ in case $C(t)\equiv 0$ on $\mathbb{T}$. As a consequence, the following result is clear. \begin{theorem}\label{T:7} Under the conditions in Theorem \ref{T:5}, each solution of \eqref{1} is asymptotically periodic. Specially, if $C(t)\equiv 0$ on $\mathbb{T}$, then each solution $x(t)$ of \eqref{1} converges to zero as $t\to \infty$. \end{theorem} \begin{remark} \rm As a final remark, our results for periodicity cover those in \cite[Theorem 3]{Zhao} and improve the conclusion in \cite[Theorem 14]{Zhu}. Indeed, the authors in \cite{Zhu} only obtained an existence criterion of periodic solutions and ignored the uniqueness and asymptotical property. \end{remark} \section{Examples}\label{s5} Let us consider the following two examples to better understand our results. \begin{example} \rm % Example-1 Let $\mathbb{T}=[-1,\infty)\subset \mathbb{R}$ and consider the equation of Nicholson blowflies type \cite[Example 1]{Liz} \begin{equation}\label{23} x'(t)=-\frac{1}{2}x(t)+\frac{\lambda}{2}|x(t-1)|\big(\alpha+\beta e^{-x(t-1)}\big)+c(t), \quad t\in [-1,\infty), \end{equation} where $\alpha,\beta$ and $\lambda$ are positive constants, and $c(t)\geq 0$ is a $1$-periodic function. \end{example} (i) For the case $c(t)\equiv 0$, to invoke Theorem \ref{thmA} we first calculate \[ f_0=\alpha+\beta, \quad f_{\infty}=\alpha, \quad A=1, \quad B=e^{-1/2}. \] Then, for any $\alpha$ and $\beta$ satisfying \[ \frac{1}{B(\alpha+\beta)}\geq \frac{1}{A\alpha}, \] Theorem \ref{thmA} is invalid since $(\frac{1}{B(\alpha+\beta)},\frac{1}{A\alpha})$ is empty. Now we turn to impose our criteria under the same conditions as above. To this end, we take $F(t,x_1,x_2)=\frac{\lambda}{2}|x_2|(\alpha+\beta e^{-x_2})$ and \[ A_1=0,\quad A_2=\frac{\lambda(\alpha+\beta)}{2}. \] Then \[ |F(t,x_1,x_2)-F(t,y_1,y_2)|\leq A_2|x_2-y_2| \] and hence, for \begin{equation}\label{24} \lambda < \frac{1}{\alpha+\beta}, \end{equation} Theorem \ref{T:7} implies that each solution $x(t)$ of \eqref{23} converges to zero as $t\to \infty$. (ii) For the case $c(t)\geq 0$ and not identically zero, as long as the positive constants $\alpha,\beta$ and $\lambda$ fulfill the relation \eqref{24}, our Theorems \ref{T:6} and \ref{T:7} imply that \eqref{23} has a unique positive $1$-periodic solution and others are asymptotically periodic. \begin{example} \rm % Example-2 Let $\mathbb{T}=\cup_{k=1}^{\infty}[4k-8,4k-6]$ and $\alpha=1/3$. Then $1-\alpha\mu(t)>0$ for all $t\in\mathbb{T}$, $\mathbb{T}$ is a $4$-periodic time scale and $e_{-\alpha}(t,0)\to 0$ as $t\to \infty$. Now we consider the delay dynamic system \begin{equation}\label{22} x^{\Delta}(t)=\begin{pmatrix} -1/3& 0 \\ 0 & -1/3\\ \end{pmatrix} x(t) + \begin{pmatrix} 1/4& 1 \\ 0 & 1/4 \end{pmatrix} x(t-4)+\begin{pmatrix} \sin(\pi t/2)\\ \cos(\pi t/2) \end{pmatrix}, \quad t\in \mathbb{T} \end{equation} and take \[ A_0=A_1=0, \quad A_2=\begin{pmatrix} 1/4& 1 \\ 0 & 1/4 \end{pmatrix}, \quad C_0=\begin{pmatrix} 1\\ 1 \end{pmatrix}. \] \end{example} Then, assumptions (H1)--(H4) are satisfied. Now by Theorems \ref{T:5} and \ref{T:7}, system \eqref{22} has a unique $4$-periodic solutions and others are asymptotically periodic. \subsection*{Acknowledgments} This research was supported by the NNSF of China, under grants 11271379 and 11071238. \begin{thebibliography}{99} \bibitem{Agarwal} R. Agarwal, M. Bohner, D. O'Regan, A. Peterson; \emph{Dynamic equations on time scales: a survey}, J. Comput. Appl. 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