数理解析研究所講究録
第 1786 巻 2012 年 45-56

45

Boundedness and Stability of Semi-linear Dynamic
Equations on Time Scales
Jimin Zhang a , Meng Fan ba. School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin,
Heilongjiang, 150080, P.R. Chinab. School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street,

Changchun, Jilin, 130024, P.R. China

Abstract: This paper explores the boundedness and asymptotic stability of a kindof semi-linear dynamic equations on time scales. We derive sufficient criteria for theuniform boundedness and the uniformly ultimate boundedness and establish necessary
and sufficient criterion for the asymptotic stability. The approach is rather nontrivial andis based on the application of the contraction mapping principle.
Keywords: Time scales; Boundedness; Asymptotic stability; Contraction mapping
principle.

1

Introduction

The theory of dynamic equations on time scale [3] has a tremendous potential for
applications and has recently received much attention and seen many progresses in thepast two decades due to the fact that a dynamic equation on time scales is related not
only to the set of real numbers (continuous time scale, differential equations) and the setof integers (discrete time scale, difference equations) but also to more general time scales(an arbitrary nonempty closed subset of the real numbers R). Its history can be traced

back to the calculus on time scales-the foundational work initiated by Stefan Hilger in

his $PhD$ thesis [9] in order to unify continuous and discrete analysis.
Stability plays an important role in the theory of dynamic equations on time scales.
Since the pioneer work of Liapunov more than 100 years ago, Liapunov‘s direct method
has been the primary tool to deal with stability problems in various type of dynamicalsystems such as differential equations[18], difference equations [1] and dynamic equations
on time scales[3, 10]. However, the construction of appropriate Liapunov functions orfunctionals are technical, empirical and are not universally applicable. Criteria deduced
from the direct method usually requires point-wise conditions, while many of the realworld dynamical models call for averages; calculations involved in the direct method are46
very technical and sophisticated. New methods and techniques are needed to address
those difficulties.
Recently, the fixed point theory has been proven to be a powerful tool for dealing
with the stability of differential equations. Burton [5] was among the first who study thestability using fixed point theory. When applicable, by the fixed point theory, one can try
to avoid certain difficulties incurred in applying the Liapunov method but usually achieve
conditions of average type [5]. For the comparison of these two approaches in dealing
with the stability of differential equations, we refer to [4], [5] and [19, 20]. Though there
have been studies of the stability of differential equations using the fixed point theory, it
still remains open whether it can be applied to explore the stability of dynamic equations
on time scales.
In this paper, the main approach is based on the contraction mapping principle (alsoBanach fixed point theorem). The tree of this paper is as following. In Section 2, we
introduce some basic results of the calculus on general time scale. In Section 3, we first
derive sufficient criteria for the uniform boundedness and the uniform ultimate boundedness of (2.2) and then establish necessary and sufficient criterion for the asymptotic
stability of (2.2)

2Preliminaries
be a time scale, i.e., an arbitrary nonempty closed subset of the real numbers.
For more details of the time scale, one can see [3, 9]. To facilitate the discussion below,
Let

$T$

$\mathbb{R}$

we introduce many notations $\sigma=\min\{[0, \infty)\cap T\},$.
Now, we propose some definitions of boundedness and stability of dynamic equations
on time scales. Consider the following nonlinear dynamic equations on time scales$T^{+}=[\sigma, \infty)\cap T,$ $\mathbb{R}^{+}=[0, \infty)$$x^{\Delta}(t)=F(t, x)$

,

(2.1)

where $F$ :is rd-continuous on for all
andis continuous
T. Moreover, for clarity we denote by $x(t, x_{0}, t_{0})$ the solution of (2.1) withon for all
initial values $x(t_{0})=x_{0}$ .
$T\cross \mathbb{R}arrow \mathbb{R},$$\mathbb{R}$$F(\cdot, x)$$T$$x\in \mathbb{R}$$F(t, \cdot)$$t\in$

Definition 2.1. The solutions of (2.1) are uniformly bounded, if for any $\alpha>0$ and, there exists asuch that.
for all$t_{0}\in T^{+}$$\beta_{1}(\alpha)>0$$|x_{0}|\leq\alpha,$ $|x(t, x_{0}, t_{0})|<\beta_{1}$$t\geq t_{0}$Definition 2.2. The solutions of (2.1) are uniformly ultimately bounded for bound ,if there exists a $\beta_{2}>0$ , for any $\alpha>0$ and, there exists a $T(\alpha)>\sigma$ such thatfor all $t\geq t_{0}+T$ .
$\beta_{2}$$t_{0}\in T^{+}$$|x_{0}|\leq\alpha,$$|x(t, x_{0}, t_{0})|<\beta_{2}$47Definition 2.3. [see [10]] The zero solution of (2.1) is said to be stable, if for any, there exists aandsuch thatfor all$t_{0}\in T$$\delta(t_{0}, \epsilon)>0$$|x(t, x_{0}, t_{0})|<\epsilon$$|x_{0}|\leq\delta,$$\epsilon>0$$t\geq t_{0}$.

Definition 2.4. [see [10]] The zero solution of (2.1) is said to be asymptotically stable, if$x(t, x_{0}, t_{0})arrow$the zero solution is stable and if there exists asuch that if
as.
$\delta(t_{0})>0$$0$$|x_{0}|\leq\delta,$$tarrow\infty$

In this paper, we will explore the boundedness and asymptotic stability of the following
semi-linear equation on general time scale$T$$x^{\Delta}(t)=-a(t)x(\sigma(t))+f(t, x(t))$

.

(2.2)

where
:
is rd-continuous on for all
andiscontinuous on for all
T. Whenor $T=Z,$ $(2.2)$ reduces to the semi-linear
differential equations or difference Equations, whose boundedness and stability have been
extensively studied. For example, see [1, 18].
Carrying out similar arguments as those in Theorem 2.74 in [3], we can easily obtainthe following theorem.
$a\in \mathcal{R}^{+},$$f$$\mathbb{T}\cross \mathbb{R}arrow \mathbb{R},$$t\in$$\mathbb{R}$Theorem 2.1. Suppose$a\in \mathcal{R},$$T$$f(\cdot, x)$$x\in \mathbb{R}$$f(t, \cdot)$$T=\mathbb{R}$$u\in C(T, \mathbb{R})$. Let$t_{0}\in \mathbb{T}$and$x_{0}\in \mathbb{R}$. The unique solutionof the initial value problem$x^{\triangle}(t)=-a(t)x(\sigma(t))+f(t, u(t))$

is given by

3,$x(t)=e_{\ominus a}(t, t_{0})x_{0}+ \int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau$$x(t_{0})=x_{0}$

(2.3)

.

Boundedness and asymptotic stability

In this section, we explore the boundedness and asymptotic stability of the solutions of(2.2). The approach will base on the famous contraction mapping principle (also known
as the Banach fixed point theorem or the contraction mapping theorem).
Lemma 3.1. Let (X, d) be a complete metric space and $P:Xarrow X$ be a contraction
mapping (that is, there exists a constant with $0\leq\lambda<1$ such that $d(P(x), P(y))\leq$$\lambda$$\lambda d(x, y),$ $x,$$y\in X)$ ,

then

$P$

has a unique fixed point in $X$ .

Theorem 3.1. Assume that(i) there is a function$x_{1},$$x_{2}\in \mathbb{R},$$t\in T$;

$b:Tarrow \mathbb{R}^{+}$

such that $|f(t, x_{1})-f(t, x_{2})|\leq b(t)|x_{1}-x_{2}|$ for any

48
(ii)

$\lim_{tarrow\infty}l^{t}\xi_{\mu(\tau)}(a(\tau))\Delta\tau=\infty$

and there are a $0<\chi<1$ and a $M>0$ such that$l^{t}e_{\ominus a}(t, \tau)|f(\tau, 0)|\triangle\tau<M$,$l^{t}e_{\ominus a}(t, \tau)b(\tau)\triangle\tau\leq\chi$,$t\geq\sigma$.

Then the solutions of (2.2) are uniformly bounded.

Proof. According to the condition (ii), for any fixed$t_{0}\geq\sigma$, we have$l_{t_{0}}^{t} \xi_{\mu(\tau)}(a(\tau))\triangle\tau=l^{t}\xi_{\mu(\tau)}(a(\tau))\triangle\tau-\int_{\sigma}^{t_{0}}\xi_{\mu(\tau)}(a(\tau))\triangle\tauarrow\infty$as$tarrow\infty$. Hence, we can find a positive number such
and $e_{ea}(t, t_{0})=1/e_{a}(t, t_{0})arrow 0$ as. For any , let $\beta_{1}=(\alpha_{1}b_{1}+M)/(1-\chi)$ and define
for allthat$tarrow\infty$$t\geq t_{0}$$e_{\ominus a}(t, t_{0})\leq b_{1}$$b_{1}$$\alpha_{1}$$S_{1}=\{u\in C_{rd}(T,$ $\mathbb{R})|u(t_{0})=x_{0}$, and $|u(t)|<\beta_{1}$ for$t\geq t_{0},$$|x_{0}|\leq\alpha_{1}\}$,is a complete metric space endowed withmetric $d(u_{1}, u_{2})= \Vert u_{1}-u_{2}\Vert=\sup_{t\in[t_{0},\infty)}|u_{1}(t)-u_{2}(t)|$. In view of Theorem 2.1, for any
then it is not difficult to show that the set$u\in S_{1}$, we let$u(t_{0})=x_{0}$$S_{1}$$Z_{u}(t)=e_{\ominus a}(t, t_{0})u(t_{0})+ \int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau$and it is easy to show that

$Z_{u}\in C_{rd}(T, \mathbb{R})$. Obviously,. In addition, for any$Z_{u}(t_{0})=$$t\geq t_{0}$

, we

have
$|Z_{u}(t)| \leq e_{\ominus a}(t, t_{0})|x_{0}|+\int_{t_{0}}^{t}e_{ea}(t, \tau)b(\tau)|u(\tau)|\triangle\tau+\int_{t_{0}}^{t}e_{ea}(t, \tau)|f(\tau, 0)|\triangle\tau$$<b_{1}\alpha_{1}+\chi\beta_{1}+M=\beta_{1}$.

Therefore, we can define a mapping $P$ :(i), for any$u_{1},$$u_{2}\in S_{1}$, we have, for any$S_{1}arrow S_{1}$$t\geq t_{0}$by $(Pu)(t)=Z_{u}(t)$ . By the condition,$|(Pu_{1}-Pu_{2})(t)|=| \int_{t_{0}}^{t}e_{ea}(t, \tau)(f(\tau, u_{1}(\tau))-f(\tau, u_{2}(\tau)))\Delta\tau|$$\leq\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)b(\tau)\Vert u_{1}-u_{2}\Vert\triangle\tau\leq\chi\Vert u_{1}-u_{2}||$

Therefore, $P$ is a contraction mapping and has a unique fixed point inunique solution of (2.2) in$S_{1}$. That is, for any$\alpha_{1}>0$and$t_{0}\in T^{+}$

such that, for any
the solutions of (2.2) satisfies
is, the solutions of (2.2) are uniformly bounded.

$0$$|x_{0}|\leq\alpha_{1)}$$S_{1}$.
, which is the, there exists a$\beta_{1}(\alpha_{1})>$$|x(t, x_{0}, t_{0})|<\beta_{1},$ $t\geq t_{0}$, that$\square$

Theorem 3.2. Assume that the conditions (i) and (ii) in Theorem 3.1 hold. Then thesolutions of (2.2) are uniformly ultimately bounded for bound $\beta_{2}>M/(1-\chi)$ .

49
Proof. By Theorem 3.1, for any
for any
define$|x_{0}|\leq\alpha_{2},$$|x(t, x_{0}, t_{0})|<\beta_{1}$$S_{2}=\{u\in C_{rd}(T, \mathbb{R})|$andfor all$\alpha_{2}>0$$t\geq t_{0}$, there exists asuch that,. In order to prove the conclusion, we$t_{0}\geq\sigma$$d(u(t), B(0,\beta_{3}))arrow 0u(t_{0})=x_{0},|x_{0}|\leq\alpha_{2},$$\beta_{1}(\alpha_{2})>0$

for$|u(t)|<\beta_{1}astarrow\infty$

where $\beta_{3}=M/(1-\chi)(<\beta_{1})$ is a fixed positive constant number and. Then$t\geq t_{0},$$\}$$B(0, \beta_{3})$,is a sphereis a complete metric space endowed with metric$d(u_{1}, u_{2})= \Vert u_{1}-u_{2}\Vert=\sup_{t\in[t_{0},\infty)}|u_{1}(t)-u_{2}(t)|$ .
For any
and $u\in S_{2}$ , there is a.
such that $|u(t)|<\beta_{3}+\epsilon/2$ for any
It follows from the condition (ii) that, for sufficiently large $T_{2}>T_{1},$$t\geq T_{2}$ .
and
For $u\in S_{2}$ , we consider $Z_{u}(t)=e_{\ominus a}(t, t_{0})u(t_{0})+$with center$0$

and radius$\beta_{3}$$\epsilon>0$$S_{2}$$t\geq T_{1}$$T_{1}$$\alpha_{2}e_{ea}(t, t_{0})<\epsilon/4$$\beta_{1}\chi e_{\ominus a}(t, T_{1})<\epsilon/4,$$\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau$. Obviously, $Z_{u}(t_{0})=u(t_{0})=x_{0}$ andover, by the proof of Theorem 3.1, we have $|Z_{u}(t)|<\beta_{1}$ for any
have$Z_{u}(t)\in C_{rd}(T, \mathbb{R})$$t\geq t_{0}$. If$t\geq T_{2}$. More-, then we$|Z_{u}(t)| \leq e_{\ominus a}(t, t_{0})|u(t_{0})|+|\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau|$$=e_{\ominus a}(t, t_{0})|u(t_{0})|+ \int_{t_{0}}^{T_{1}}e_{\ominus a}(t, \tau)b(\tau)|u(\tau)|\triangle\tau$$+ \int_{T_{1}}^{t}e_{\ominus a}(t, \tau)b(\tau)|u(\tau)|\triangle\tau+\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)|f(\tau, 0)|\triangle\tau$$\leq\frac{\epsilon}{4}+\beta_{1}e_{\ominus a}(t, T_{1})\int_{t_{0}}^{T_{1}}e_{\ominus a}(T_{1}, \tau)b(\tau)\triangle\tau$$+( \beta_{3}+\frac{\epsilon}{2})\int_{T_{1}}^{t}e_{\ominus a}(t, \tau)b(\tau)\triangle\tau+\beta_{3}(1-\chi)$$< \frac{\epsilon}{4}+\beta_{1}\chi e_{\ominus a}(t, T_{1})+(\beta_{3}+\frac{\epsilon}{2})\chi+\beta_{3}(1-\chi)$$< \frac{\epsilon}{2}+\beta_{3}+\frac{\epsilon}{2}=\epsilon+\beta_{3}$.

Since is arbitrary, we can conclude thatapproaches. Now weasdefine a mapping $P:S_{2}arrow S_{2}$ by $(Pu)(t)=Z_{u}(t)$ . It is not difficult to show that isa contraction mapping by the same arguments as those in Theorem 3.1. Hence, hasa unique fixed point in , which is a solution of (2.2). Therefore, for any fixed positive
number , we can choose
as the bound of uniform ultimate boundedness.
$Z_{u}(t)$$\epsilon$$B(0, \beta_{3})$$tarrow\infty$

$P$

$P$

$S_{2}$$c$$\beta_{2}=\beta_{3}+c$$\square$

Theorem 3.3. Assume that(i)$\lim tarrow\infty\inf l^{t}\xi_{\mu(\tau)}(a(\tau))\triangle\tau>-\infty$

and $f(t, 0)=0$ for all$t\in \mathbb{T}$;50(ii) there is a function$b(t)|x_{1}-x_{2}|,$$|x_{1}|,$:$b$and an $N>0$ such that $|f(t, x_{1})-f(t, x_{2})|\leq$$Tarrow \mathbb{R}^{+}$$|x_{2}|\leq N,$$t\in T$(iii) there is a $0<\chi<1$ such that;$l^{t}e_{ea}(t, \tau)b(\tau)\triangle\tau\leq\chi$for$t\geq\sigma$.

Then the zero solution of (2.2) is asymptotically stable if and only if
(iv)

as

$l^{t}\xi_{\mu(\tau)}(a(\tau))\triangle\tauarrow\infty$$tarrow\infty$.

Proof. (Sufficiency) If the condition (iv) is satisfied, then there is a $b_{2}>0$ such that,for any$t_{0}\geq\sigma,$$\delta_{1}b_{2}+\chi N\leq N$$|e_{\ominus a}(t, t_{0})|\leq b_{2}$. For any

$S_{3}=\{u\in C_{rd}(T,$$|x_{0}|\leq\delta_{1}$$\mathbb{R})|,$

It is easy to show that

$S_{3}$

for all $t\geq t_{0}$ . Next, we choose a, define$u(t_{0})=x_{0},$ $|u(t)|\leq N$

for$t\geq t_{0},$$u(t)arrow 0$$\delta_{1}>0$

assuch that$tarrow\infty\}$.

is a complete metric space endowed with metric $d(u_{1}, u_{2})=$$\Vert u_{1}-u_{2}\Vert=\sup_{t\in[t_{0},\infty)}|u_{1}(t)-u_{2}(t)|$.

For any
and $u\in S_{3}$ , we can easily find a $T_{3}>t_{0}$ such that $|u(t)|<\epsilon/3$
for all $t\geq T_{3}$ . It follows from the condition (iv) that there is a $T_{4}>T_{3}$ such that. Define $Z_{u}(t)=e_{ea}(t, t_{0})u(t_{0})+$forand$\epsilon>0$$N\chi e_{\ominus a}(t, T_{3})<\epsilon/3$$\delta_{1}e_{\ominus a}(t, t_{0})<\epsilon/3$$\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau$$t\geq T_{4}$, then $Z_{u}(t_{0})=u(t_{0})=x_{0}$ and $Z_{u}(t)\in C_{rd}(T, \mathbb{R})$ . If $t\geq T_{4}$ , then$|Z_{u}(t)| \leq e_{\ominus a}(t, t_{0})|u(t_{0})|+|\int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, u(\tau))\triangle\tau|$$=e_{ea}(t, t_{0})|u(t_{0})|+ \int_{t_{0}}^{T_{3}}e_{\ominus a}(t, \tau)b(\tau)|u(\tau)|\triangle\tau+\int_{T_{3}}^{t}e_{\ominus a}(t, \tau)b(\tau)|u(\tau)|\triangle\tau$$\leq\frac{\epsilon}{3}+Ne_{\ominus a}(t, T_{3})\int_{t_{0}}^{T_{3}}e_{\ominus a}(T_{3}, \tau)b(\tau)\triangle\tau+\frac{\epsilon}{3}\int_{T_{3}}^{t}e_{\ominus a}(t, \tau)b(\tau)\triangle\tau$$< \frac{\epsilon}{3}+N\chi e_{ea}(t, T_{3})+\frac{\epsilon}{3}\chi<\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon$.

by $(Pu)(t)=Z_{u}(t)$ . Then $P$ is a contraction:
We now define a mapping, which is a solution of (2.2) andmapping. Thus, $P$ has a unique fixed point in$x(t)=x(t, t_{0}, x_{0})arrow 0$ as.
, choose
Now we show that the zero solution of (2.2) is stable. For any. To prove the conclusion, we will show that,asuch thatsuch. Assume that there exists a
for any
for any
thatandfor $t_{0}\leq\tau<t^{*}$ . From Theorem 2.1, the solutions of (2.2)$P$$S_{3}arrow S_{3}$$S_{3}$$tarrow\infty$$\epsilon>0(\epsilon<b_{2})$$\delta_{2}>0(\delta_{2}<\epsilon)$$|x_{0}|<\delta_{2},$$\delta_{2}b_{2}+\chi\epsilon<\epsilon$$|x(t, x_{0}, t_{0})|<\epsilon$$|x(t^{*})|=\epsilon$

can be expressed as$|x(\tau)|<\epsilon$$t\geq t_{0}$$t^{*}>t_{0}$51$x(t)=e_{\ominus a}(t, t_{0})x(t_{0})+ \int_{t_{0}}^{t}e_{\ominus a}(t, \tau)f(\tau, x(\tau))\triangle\tau$Hence, we have

(3.1)

.

$|x(t^{*})| \leq\delta_{2}e_{\ominus a}(t^{*}, t_{0})+\int_{t_{0}}^{t^{*}}e_{\ominus a}(t^{*}, \tau)b(\tau)|x(\tau)|\triangle\tau\leq\delta_{2}b_{2}+\chi\epsilon<\epsilon$, whichcontradicts to the definition of , then the zero solution of (2.2) is stable. Therefore, if(iv) is satisfied, then the zero solution of (2.2) is asymptotically stable.
$t^{*}$(Necessity) If (iv) fails, then there exist a sequence $\{t_{n}\}(t_{n}arrow\infty as narrow\infty)$ andsome real number$m_{1}$

such that

there is a positive constant$n=1,2,$$\cdots$$L$$\lim_{narrow\infty}l^{t_{n}}\xi_{\mu(\tau)}(a(\tau))\triangle\tau=m_{1}$such that. It is easy to show that$|l^{t_{n}}\xi_{\mu(\tau)}(a(\tau))\triangle\tau|\leq L$and$e_{a}(t_{n}, \sigma)\leq e^{L}$

for all. Therefore, it follows from the condition (iii) that.

$l^{t_{n}}e_{a}(\tau, \sigma)b(\tau)\triangle\tau=l^{t_{n}}e_{a}(t_{n}, \sigma)e_{\ominus a}(t_{n}, \tau)b(\tau)\triangle\tau\leq\chi e_{a}(t_{n}, \sigma)\leq e^{L}$

This implies that there exists a convergent subsequence. Without loss of generality, we
still assume that it isconstant$m_{2}$$\{t_{n}\}$such that$\lim_{narrow\infty}l^{t_{n}}e_{a}(\tau, \sigma)b(\tau)\triangle\tau=m_{2}$. Hence, we can find sufficiently large$k^{*}$

such thatfor some positive$\int_{t_{k^{*}}}^{t_{n}}e_{a}(\tau, \sigma)b(\tau)\triangle\tau<$. Since the zero solution of (2.2) is
asymptotically stable, for given a real number $B>0$ , there exits a $\delta_{0}>0(\delta_{0}<B)$ suchthat $|x(t, x(t_{k^{*}}), t_{k^{*}})|<B$ for,. For allwith the initial value$(1-\chi)/(2Q^{2})$for$n\geq k^{*}$, where$Q= \sup_{t\geq\sigma}e_{\ominus a}(t, \sigma)$$|x(t_{k^{*}})|=\delta_{0}$$t\geq t_{k^{*}}$

we have$t\geq t_{k^{*}}$. This$|x(t)| \leq x(t_{k}\cdot)e_{\ominus a}(t_{n}, t_{k^{*}})+\int_{t_{k^{*}}}^{t}e_{\ominus a}(t, \tau)|f(\tau, x(\tau))|\triangle\tau\leq\delta_{0}Q+\chi\sup_{t\geq t_{k^{*}}}|x(t)|$shows that $|x(t)|\leq\delta_{0}Q/(1-\chi)$ for all$t\geq t_{k^{*}}$. Meanwhile, it is easy to show that$|x(t_{n})| \geq\delta_{0}e_{\ominus a}(t_{n}, t_{k^{*}})-\int_{t_{k}}^{t_{n}}.e_{\ominus a}(t_{n}, \tau)b(\tau)|x(\tau)|\triangle\tau$$\geq\delta_{0}e_{\ominus a}(t_{n}, t_{k^{*}})-\frac{\delta_{0}Q}{1-\chi}e_{\ominus a}(t_{n}, \sigma)\int_{t_{k^{*}}}^{t_{n}}e_{a}(\tau, \sigma)b(\tau)\triangle\tau$$\geq e_{\ominus a}(t_{n}, t_{k^{*}})[\delta_{0}-\frac{\delta_{0}Q}{1-\chi}Q\int_{t_{k^{*}}}^{t_{n}}e_{a}(\tau, \sigma)b(\tau)\triangle\tau]$$\geq\frac{1}{2}\delta_{0}e_{\ominus a}(t_{n}, t_{k^{*}})\geq\frac{1}{2}\delta_{0}e^{-2L}$for. This implies that $x(t)\prec\div 0$ as, which is contradiction. That is, (iv)
is necessary for asymptotically stable of the zero solution of (2.2). This completes the$n\geq k^{*}$$tarrow\infty$

proof.

$\square$Next, we consider the existence and stability of periodic solutions of (2.2) by the

famous contraction mapping principle. Therefore, the time scale$\mathbb{T}$is supposed to be52-periodic, i.e., $t\in$ I implies $t\pm\omega\in$ T. This implies that the graininessperiodic, that is, $\mu(t\pm\omega)=\mu(t)$ . Some examples of such time scales are $Z,$$\omega$. Moreover, in (2.2), we assume thatare both -periodic in .
$\omega$$a\in C_{rd}(T, \mathbb{R}^{+})$$\omega-$$2k+$$\bigcup_{k\in Z}[2k,$$\mathbb{R},$$1],$ $\bigcup_{k\in Z}\bigcup_{n\in N}\{k+\frac{1}{n}\}$is also$\mu$, and $a(t),$ $f(t, x)$$t$Lemma 3.2. $x(t)$ is an -periodic solution of (2.2) if and only if $x(t)$ is a solution of$\omega$$x(t)= \int_{t}^{t+\omega}\frac{e_{a}(\tau,t)}{e_{a}(\sigma+\omega,\sigma)-1}f(\tau, x(\tau))\triangle\tau$

(3.2)

.

Let $x(t)$ be an -periodic solution of (2.2), then one has $x^{\Delta}(t)+a(t)x(\sigma(t))=$$f(t, x(t))$ . Multiplying both sides of the above equation byleads to$e_{a}(t, \sigma)f(t, x(t))$ . Integrating from, we have $x(t+\omega)e_{a}(t+\omega, \sigma)-x(t)e_{a}(t, \sigma)=$to

Proof.

$\omega$$(e_{a}(t, \sigma)x(t))^{\Delta}=$$e_{a}(t, \sigma)$$t$$t+\omega$. Since $x(t)$ is -periodic, one can easily reach (3.2). Therefore,$\int_{t}^{t+\omega}e_{a}(\tau, \sigma)f(\tau, x(\tau))\triangle\tau$$\omega$

the necessity of the claim is valid. The proof of the sufficiency is trivial.

$\square$

Theorem 3.4. Assume that the conditions (ii), (iii) and (iv) of Theorem 3.3 are satisfied.

If A$l^{\sigma+\omega}b(\tau)\triangle\tau<1$holds, where $A=e_{a}(\sigma+\omega, \sigma)/(e_{a}(\sigma+\omega, \sigma)-1)$ , then (2.2) has aunique asymptotically stable periodic solution.

Proof. Define$S_{4}=\{u\in C(T,$ $\mathbb{R})|u(t+\omega)=u(t)$It is not difficult to show that$(S_{4}, \Vert\cdot\Vert)$

for all $t\in T\}$ ,$\Vert u\Vert=\max_{t\in I_{\omega}}|u(t)|$for $u\in S_{4}$ .

is a Banach space. Define a mapping$Tu$ $(t)=l^{t+\omega} \frac{e_{a}(\tau,t)}{e_{a}(\sigma+\omega,\sigma)-1}f(\tau, u(\tau))\triangle\tau$Obviously, $T:S_{4}arrow S_{4}$ . Meanwhile, for any$u_{1},$$u_{2}\in S_{4}$$T$

as follows:.

, we have$|Tu_{1}( \tau)-Tu_{2}(\tau)|=\int_{t}^{t+\omega}\frac{e_{a}(\tau,t)}{e_{a}(\sigma+\omega,\sigma)-1}|f(\tau, u_{1}(\tau))-f(\tau, u_{2}(\tau))|\triangle\tau$$\leq Al^{\sigma+\omega_{b(\tau)\triangle\tau\Vert u_{1}-u_{2}\Vert}}$.

Then, $T$ is a contraction mapping with a unique fixed point in$S_{4}$, which is a uniqueperiodic solution of (2.2) by Lemma 3.2. Proceeding the same as those in the proofof Theorem 3.3, we can easily show that the periodic solution is asymptotically stable.
Therefore, (2.2) has a unique asymptotically stable periodic solution.

$\square$53

4

Application
Now, we turn to some concrete dynamic equations on time scales, which incorporate

many mathematical models in real-world applications when the time scale reduces toor Z.

$\mathbb{R}$

Example 4.1. Consider the following autonomous dynamic equation(4.1)$x^{\triangle}(t)=-rx(\sigma(t))+\eta e^{-\gamma x(t)}$

where

$r,$

$\eta,$$\gamma$are all positive constants and the initial values of (4.1) are positive.

Theorem 4.1. If
ultimately bounded.

$\eta\gamma<r$, then the solutions of (4.1) are uniformly bounded and uniformly

Proof. By Theorem 2.1, it is not difficult to show that the solutions of (4.1) are always
positive for all$t\geq\sigma$

and for any$x_{2}\in \mathbb{R}^{+}$$x_{1},$

if the initial values are positive. Obviously,, we have$|\eta e^{-\gamma x_{1}}-\eta e^{-\gamma x_{2}}|\leq\eta\gamma|x_{1}-x_{2}|$$\eta\gamma l^{t}e_{\ominus r}(t, \tau)\triangle\tau=\frac{\eta\gamma}{r}(1-e_{\ominus r}(t, \sigma))<\frac{\eta\gamma}{r}<1$,$\lim_{tarrow\infty}l^{t}\xi_{\mu(\tau)}(r)\Delta\tau=\infty$. Moreover,$\eta l^{t}e_{\ominus r}(t, \tau)\triangle\tau\leq\frac{\eta}{r}$,$t\geq\sigma$.

Theorem 3.1 and Theorem 3.2 imply the claims.

$\square$Remark 1. Let
or $T=$ Z. Then (4.1) can been reformulated as continuous ordiscrete Lasota-Wazewska model without delay. This kind of models with delay have been
extensively discussed in differential equations [6, 16] and difference equations [11].
$\mathbb{T}=\mathbb{R}$

Example 4.2. Consider the following nonautonomous dynamic equation$x^{\triangle}(t)=-a(t)x(\sigma(t))+b(t)$where$a,$ $b\in C_{rd}(T, \mathbb{R}^{+})$

and

$b$

,

(4.2)

is bounded on T.

In fact, (4.2) is general to incorporate many single species models as special cases. For
example, if we let
and $x(t)=1/N(t)$ , then (4.2) reduces to the famous Verhulst$T=\mathbb{R}$logistic equation $\dot{N}(t)=N(t)(a(t)-b(t)N(t))$ . If $T=Z$ and $x(t)=1/N(t)$ , then (4.2)
reduces to the famous Beverton-Holt equation [2, 13], $N(t+1)=(1+a(t))N(t)/[1+$$b(t)N(t)]$ . If $b(t)=a(t)\ln(c(t))$ and $x(t)=\ln(N(t))$ , then (4.2) reduces to the continuous
Gompertz single species model [7, 14], $\dot{N}(t)=a(t)N(t)\ln(c(t)/N(t))$ . Whenordiscrete Gompertz single species model [15],when $T=$ Z.
$\mathbb{T}=\mathbb{R}$$N(t+1)=N(t)_{C(t)}^{\frac{1}{1+a(t)}\frac{a(t)}{1+a(t)}}$Applying those theorems in Section 3 to (4.2), one can easily reach the following
claims.

54, then the solutions of (4.2) are uniformly bounded andTheorem 4.2. If
are -periodic, then (4.2) has a unique
uniformly ultimately bounded. Moreover, if$\overline{a}=\inf_{t\in T}(a(t))>0$

$a,$

$b$

$\omega$

peri odic solution $x(t)= \int_{-\infty}^{t}e_{ea}(t, \tau)b(\tau)\triangle\tau$ , which is asymptotically stable.

Example 4.3. Consider$x^{\Delta}(t)=-a(t)x( \sigma(t))+\frac{b(t)}{1+x^{2}(t)}$where
and is bounded on T. Whenof physiological control systems [12, 17].
$b$$a,$ $b\in C_{rd}(T, \mathbb{R}^{+})$

Theorem 4.3. Let$\overline{a}=\inf_{t\in T}(a(t))>0$

and

(4.3)

,

$T=\mathbb{R},$$\Vert b\Vert=\sup_{t\in T}(b(t))$. If$(4.3)$$\Vert b\Vert<\overline{a}$is a particular case, then the solutionsof (4.3) are uniformly bounded and uniformly ultimately bounded.
Proof. Obviously,$\lim_{tarrow\infty}l^{t}\xi_{\mu(\tau)}(a(\tau))\triangle\tau=$$x_{1}^{2})-1/(1+x_{2}^{2})|\leq|x_{1}-x_{2}|$oo. For. In addition, for any$\Vert b\Vert/\overline{a}(1-e_{\ominus a}(t, \sigma))<\Vert b\Vert/\overline{a}<1$$x_{1},$$x_{2}\in \mathbb{R}$$t\geq\sigma$, one can reach $|1/(1+$, we have$l^{t}e_{\ominus a}(t, \tau)b(\tau)\triangle\tau=$. It follows from Theorem 3.1 and Theorem 3.2 that thesolutions of (4.3) are uniformly bounded and uniformly ultimately bounded.

$\square$

By Theorem 3.4, we can easily get

Theorem 4.4. Assume that

hold. If

$a,$

$b$

are both w-periodic and the conditions of Theorem 4.3$\frac{e_{a}(\sigma+\omega,\sigma)}{e_{a}(\sigma+\omega,\sigma)-1}l^{\sigma+\omega}b(\tau)\triangle\tau<1$, then (4.3) has a unique asymptotically stable

periodic solution.

Example 4.4. Consider the dynamic equation$x^{\Delta}(t)=-a(t)x(\sigma(t))+b(t)\tanh(x(t))+\gamma(t)$where$a,$ $b,$ $\gamma\in C_{rd}(T, \mathbb{R}^{+}),$$b,$ $r$(4.4)

are both bounded on T.

$(4.4)$ reduces to a single artificial effective neuron with dissipation [7, 8].
When. Applying Theorem 3.1,
It is clear that $|\tanh(x_{1})-\tanh(x_{2})|\leq|x_{1}-x_{2}|$ for
Theorem 3.2 and Theorem 3.4 to (4.4), one can easily find that Theorem 4.3 and Theorem
4.4 hold for (4.4).
$T=\mathbb{R},$$x_{1},$$x_{2}\in \mathbb{R}$Acknowledgments
M. Fan was supported by NSFC-10971022, NCET-08-0755 and FRFCU-10JCXK003, J.
Zhang was supported by NSFC-11126269(TianYuan), the Foundation of Heilongjiang
Education Committee (12511413) and QL201001.

55

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