The history of asymptotic stability of dynamic equations on a time scale goes back to Aulbach and Hilger [3]. For a real scalar dynamic equation, stability and instability results were obtained by Gard and Hoffacker [12]. Pötzche [20] provides sufficient conditions for the uniform exponential stability in Banach spaces, as well as spectral stability conditions for time-varying systems on time scales. Doan, Kalauch, and Siegmund [10] established a necessary and sufficient condition for the existence of uniform exponential stability and characterized the uniform exponential stability of a system by the spectrum of its matrix. Properties of exponential stability of a time varying dynamic equation on a time scale have been also investigated recently by Bohner and Martynyuk [7], DaCunha [9], Du and Tien [11], Hoffacker and Tisdell [16], Martynyuk [17], and Peterson and Raffoul [19].

The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 [

14], in order to unify continuous and discrete calculus [

4,

15]. A time scale

$\mathbb{T}$ is a nonempty closed subset of

$\mathbb{R}$. The forward jump operator

$\sigma :\mathbb{T}\u27f6\mathbb{T}$ is defined by

$\sigma (t)=inf\{s\in \mathbb{T}:s>t\}$ (supplemented by

$inf\mathrm{\varnothing}=sup\mathbb{T}$) and the backward jump operator

$\rho :\mathbb{T}\u27f6\mathbb{T}$ is defined by

$\rho (t)=sup\{s\in \mathbb{T}:s<t\}$ (supplemented by

$sup\mathrm{\varnothing}=inf\mathbb{T}$). The graininess function

$\mu :\mathbb{T}\u27f6{\mathbb{R}}^{\ge 0}$ is given by

$\mu (t)=\sigma (t)-t$. A point

$t\in \mathbb{T}$ is said to be right-dense if

$\sigma (t)=t$, right-scattered if

$\sigma (t)>t$, left-dense if

$\rho (t)=t$, left-scattered if

$\rho (t)<t$, isolated if

$\rho (t)<t<\sigma (t)$, and dense if

$\rho (t)=t=\sigma (t)$. A time scale

$\mathbb{T}$ is said to be discrete if t is left-scattered and right-scattered for all

$t\in \mathbb{T}$, and it is called continuous if

*t* is right-dense and left-dense at the same time for all

$t\in \mathbb{T}$. Suppose that

$\mathbb{T}$ has the topology inherited from the standard topology on

$\mathbb{R}$. We define the time scale interval

$[a,b]:=[a,b]\cap \mathbb{T}$. Open intervals and open neighborhoods are defined similarly. A set we need to consider is

${\mathbb{T}}^{k}$ which is defined as

${\mathbb{T}}^{k}=\mathbb{T}\mathrm{\setminus}\{M\}$ if

$\mathbb{T}$ has a left-scattered maximum

*M*, and

${\mathbb{T}}^{k}=\mathbb{T}$ otherwise. A function

$f:\mathbb{T}\u27f6X$ is called right dense continuous, or just rd-continuous, if

- (i)
*f* is continuous at every right-dense point $t\in \mathbb{T}$;

- (ii)
${lim}_{s\u27f6{t}^{-}}f(s)$ exists (finite) for every left-dense point $t\in \mathbb{T}$.

The set of rd-continuous functions $f:\mathbb{T}\u27f6X$ will be denoted by ${C}_{\mathrm{rd}}={C}_{\mathrm{rd}}(\mathbb{T})={C}_{\mathrm{rd}}(\mathbb{T},X)$.

A function

$f:\mathbb{T}\u27f6X$ is called delta differentiable

$(\text{or simply differentiable})$ at

$t\in {\mathbb{T}}^{k}$ provided there exists an

*α* such that for every

$\u03f5>0$ there is a neighborhood

*U* of

*t* with

$\parallel f(\sigma (t))-f(s)-\alpha (\sigma (t)-s)\parallel \le \u03f5|\sigma (t)-s|\phantom{\rule{1em}{0ex}}\text{for all}s\in U.$

In this case, we denote the

*α* by

${f}^{\mathrm{\Delta}}(t)$; and if

*f* is differentiable for every

$t\in {\mathbb{T}}^{k}$, then

*f* is said to be differentiable on

$\mathbb{T}$. If

*f* is differentiable at

$t\in {\mathbb{T}}^{k}$, then it is easy to see that

${f}^{\mathrm{\Delta}}(t)=\{\begin{array}{cc}\frac{f(\sigma (t))-f(t)}{\mu (t)},\hfill & \text{if}\mu (t)0;\hfill \\ {lim}_{s\u27f6t}\frac{f(t)-f(s)}{t-s},\hfill & \text{if}\mu (t)=0.\hfill \end{array}$

A function

$F:\mathbb{T}\u27f6X$ is called an antiderivative of

$f:\mathbb{T}\u27f6X$ if

${F}^{\mathrm{\Delta}}(t)=f(t)$,

$t\in {\mathbb{T}}^{k}$. The Cauchy integral is defined by

${\int}_{s}^{t}f(\tau )\mathrm{\Delta}\tau =F(t)-F(s),\phantom{\rule{1em}{0ex}}s,t\in \mathbb{T},$

where *F* is an antiderivative of *f*. Every rd-continuous function $f:\mathbb{T}\u27f6X$ has an antiderivative and $F(t)={\int}_{s}^{t}f(\tau )\mathrm{\Delta}\tau $ is an antiderivative of *f*, *i.e.*, ${F}^{\mathrm{\Delta}}(t)=f(t)$, $t\in {\mathbb{T}}^{k}$. Equations which include Δ-derivatives are called dynamic equations. We refer the reader to the very interesting monographs of Bohner and Peterson [5, 6].

**Definition 1.1** A mapping

$A:\mathbb{T}\u27f6L(X)$ is called regressive if

$I+\mu (t)A(t)$ is invertible for every

$t\in \mathbb{T}$, and we say that

${x}^{\mathrm{\Delta}}(t)=A(t)x(t),\phantom{\rule{1em}{0ex}}t\in \mathbb{T}$

is regressive if *A* is regressive. We say that a real valued function $p(t)$ on $\mathbb{T}$ is regressive (resp. positively regressive) if $1+\mu (t)p(t)\ne 0$ (resp. $1+\mu (t)p(t)>0$), $t\in \mathbb{T}$. The family of all regressive functions (resp. positively regressive functions) is denoted by $\mathcal{R}$ (resp. ${\mathcal{R}}^{+}$).

It is well known that if

$A\in B{C}_{\mathrm{rd}}\mathcal{R}(\mathbb{T},L(X))$, the space of all right dense continuous and regressive bounded functions from

$\mathbb{T}$ to

$L(X)$, then the initial value problem (IVP)

${x}^{\mathrm{\Delta}}(t)=A(t)x(t),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},\phantom{\rule{2em}{0ex}}x(s)={x}_{s}\in X$

(1.1)

has the unique solution

$x(t)={e}_{A}(t,s){x}_{s}.$

Here,

${e}_{A}(t,s)$ is the exponential operator function. For more details, see [

2]. When

$X=\mathbb{R}$ and

$A(t)=p(t)$ is a real valued function, Eq. (

1.1) yields

${x}^{\mathrm{\Delta}}(t)=p(t)x(t),\phantom{\rule{1em}{0ex}}t\in \mathbb{T},\phantom{\rule{2em}{0ex}}x(s)=1,$

(1.2)

whose solution has the closed form

$x(t)={e}_{p}(t,s)=exp{\int}_{s}^{t}{\xi}_{\mu (\tau )}(p(\tau ))\mathrm{\Delta}\tau ,$

(1.3)

where

${\xi}_{\mu}(z)=\{\begin{array}{cc}\frac{1}{\mu}Log(1+\mu z),\hfill & \text{if}\mu 0;\hfill \\ z,\hfill & \text{if}\mu =0,\hfill \end{array}$

(1.4)

and

$Logz=log|z|+iargz$,

$-\pi <argz\le \pi $ is the principal logarithm function. It is evident that when

$p(\tau )\ge 0$,

$\tau \in \mathbb{T}$, then

${\xi}_{\mu (\tau )}(p(\tau ))=\{\begin{array}{cc}\frac{1}{\mu (\tau )}log(1+\mu (\tau )p(\tau )),\hfill & \text{if}\mu (\tau )0;\hfill \\ p(\tau ),\hfill & \text{if}\mu (\tau )=0\hfill \end{array}$

(1.5)

$=\underset{u\u27f6\mu {(\tau )}^{+}}{lim}\frac{log(1+up(\tau ))}{u}.$

(1.6)

It can be seen that for

$\lambda >0$ with

$-\lambda \in {\mathcal{R}}^{+}$, the following claim is true

${e}_{-\lambda}(t,\tau )\le {e}^{-\lambda (t-\tau )},\phantom{\rule{1em}{0ex}}t\ge \tau ,t,\tau \in \mathbb{T}.$

(1.7)

Indeed, by taking

$p(\tau )=-\lambda $ in Eqs. (1.5) and (1.6), we have

$\begin{array}{rl}{\xi}_{\mu (s)}(-\lambda )& =\{\begin{array}{cc}\frac{1}{\mu (s)}ln(1-\lambda \mu (s)),\hfill & \text{for}\mu (s)0;\hfill \\ -\lambda ,\hfill & \text{for}\mu (s)=0,\hfill \end{array}\phantom{\rule{1em}{0ex}}s\in \mathbb{T}\\ =\underset{u\u27f6\mu {(s)}^{+}}{lim}\frac{log(1-\lambda u)}{u}\end{array}$

(1.8)

This implies that the claim is true.

In the sequel, we denote by

$\mathbb{T}\subseteq {\mathbb{R}}^{\ge 0}$ for a time scale which is an additive semigroup with the property that

$a-b\in \mathbb{T}$ for any

$a,b\in \mathbb{T}$ such that

$a>b$. In this case,

$\mathbb{T}$ is called a semigroup time scale. We assume

*X* is a Banach space. Finally, we assume that

$T=\{T(t):t\in \mathbb{T}\}\subset L(X)$ is a

${C}_{0}$-semigroup on

$\mathbb{T}$, that is, it satisfies

- (i)
$T(t+s)=T(t)T(s)$ for every $t,s\in \mathbb{T}$ (the semigroup property).

- (ii)
$T(0)=I$ (*I* is the identity operator on *X*).

- (iii)
${lim}_{t\u27f6{0}^{+}}T(t)x=x$ (*i.e.*, $T(\cdot )x:\mathbb{T}\u27f6X$ is continuous at 0) for each $x\in X$.

If in addition

${lim}_{t\u27f6{0}^{+}}\parallel T(t)-I\parallel =0$, then

*T* is called a uniformly continuous semigroup. A linear operator

*A* is called the generator [

1] of a

${C}_{0}$-semigroup

*T* if

$Ax=\underset{s\u27f6{0}^{+}}{lim}\frac{T(\mu (t))x-T(s)x}{\mu (t)-s},\phantom{\rule{1em}{0ex}}x\in D(A),$

(1.10)

where the domain $D(A)$ of *A* is the set of all $x\in X$ for which the above limit exists uniformly in *t*. Clearly, when $\mathbb{T}={\mathbb{R}}^{\ge 0}$, the concept of the generator defined by relation (1.10) coincides with the classical definition by Hille. See [13].

In Section 2 of this paper we present some results from [

1] that we need in our study. One of them is that an abstract Cauchy problem

${x}^{\mathrm{\Delta}}(t)=Ax(t),\phantom{\rule{1em}{0ex}}t>\tau ,t,\tau \in \mathbb{T},\phantom{\rule{2em}{0ex}}x(\tau )={x}_{\tau}\in D(A)$

(1.11)

has the unique solution

$x(t)=T(t-\tau ){x}_{\tau},\phantom{\rule{1em}{0ex}}t\in \mathbb{T},t\ge \tau $

when *A* is the generator of the ${C}_{0}$-semigroup *T*. When $\mathbb{T}={\mathbb{R}}^{\ge 0}$, we get the classical existence and uniqueness theorem of the abstract Cauchy problem (1.11); see [21]. The other results include some properties of *T* and its generator *A*, which we use in the subsequent sections. The solution $x(t)=x(t,\tau ,{x}_{\tau})$ is a function of the variables *t*, *τ* and the initial value ${x}_{\tau}$. Generally, we consider *τ* and ${x}_{\tau}$ as parameters. Therefore, when we investigate the asymptotic behavior of $x(t,\tau ,{x}_{\tau})$ with respect to $\mathbb{T}$, we must investigate whether or not the asymptotic behavior uniformly depends on *τ* or ${x}_{\tau}$. Accordingly, there are many types of stability which we give in Section 3.

S. K. Choi, D. M. Im, and N. Koo in [

8], Theorem 3.5] proved that the stability of the time variant abstract Cauchy problem

${x}^{\mathrm{\Delta}}(t)=A(t)x(t),\phantom{\rule{2em}{0ex}}x({t}_{0})={x}_{0},\phantom{\rule{1em}{0ex}}{t}_{0}\in \mathbb{T},$

(1.12)

where $A\in {C}_{\mathrm{rd}}\mathcal{R}(\mathbb{T},{M}_{n}(\mathbb{R}))$, $n\in \mathbb{N}$ and ${M}_{n}(\mathbb{R})$ is the family of all $n\times n$ real matrices is equivalent to the boundedness of all its solutions. DaCunha in [9] defined the concepts of uniform stability and uniform exponential stability. These two concepts involve the boundedness of the solutions of the regressive time varying linear dynamic Eq. (1.12). He established a characterization of uniform stability and uniform exponential stability in terms of the transition matrix for system (1.12). Also, he illustrated the relationship between the uniform asymptotic stability and the uniform exponential stability.

In Section 4, we extend these results for the case where *A* is the generator of *T* and we prove that the concepts of stability and uniform stability are same.

Sections 5 and 6 are devoted to establishing characterizations for many other types of stability, like asymptotic stability, uniform asymptotic stability, global asymptotic stability, exponential stability, and uniform exponential stability for the abstract Cauchy problem (1.11).

We end this paper with a new illustrative example including non-regressive dynamic equation and we investigate its stability.