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Variationally stable dynamic systems on time scales
Advances in Difference Equations volume 2012, Article number: 129 (2012)
Abstract
In this paper we give a Lyapunov functional characterization of hstability for nonlinear dynamic systems on time scales under the condition of ${u}_{\mathrm{\infty}}$similarity between their variational systems. Furthermore, we give some examples related to the notions of ${u}_{\mathrm{\infty}}$similarity and hstability.
MSC:34N05, 39A30, 34D23, 34K20, 26E70.
1 Introduction
The theory on time scales has been developed as a generalization of both continuous and discrete time theory and applied to many different fields of mathematics [2, 3, 5, 6, 21, 23].
It is widely known that the various types of stability of nonlinear differential equations or difference equations can be characterized by using Lyapunov’s second method, the method of variation of parameters, and inequalities, etc. [1, 21, 22, 24, 38, 40].
Pinto [32] introduced the notion of hstability for differential equations with the intention of obtaining results about stability for weakly stable differential systems under some perturbations. Also, Medina and Pinto [29] applied the hstability to obtain a uniform treatment for the various stability notions in difference systems and extended the study of exponential stability to a variety of reasonable systems called hsystems. Pinto and Medina obtained the important properties about hstability for the various differential systems and difference systems [26–28, 30, 33–35].
Choi et al. [7] investigated hstability for the nonlinear differential systems by means of the notions of Lyapunov functions and ${t}_{\mathrm{\infty}}$similarity introduced by Conti [19]. Trench [39] introduced summable similarity as a discrete analog of Conti’s definition of ${t}_{\mathrm{\infty}}$similarity and investigated the various stabilities of linear difference systems by using summable similarity. Choi and Koo [8] studied the variational stability for nonlinear difference systems by means of ${n}_{\mathrm{\infty}}$similarity. Also, Choi et al. studied the asymptotic property and the hstability of difference systems via discrete similarities and comparison principle [9–11]. For detailed results about the various stabilities including the notions of hstability and strong stability of dynamic systems on time scales, see [12–18, 20, 25, 31].
In this paper we introduce the notion of ${u}_{\mathrm{\infty}}$similarity which extends the continuous ${t}_{\mathrm{\infty}}$similarity [19] and the discrete ${n}_{\mathrm{\infty}}$summable similarity [11]. Then we give a Lyapunov functional characterization of hstability for nonlinear dynamic systems on time scales by assuming the condition of ${u}_{\mathrm{\infty}}$similarity between its variational systems. Furthermore, we give some examples related to the notions of ${u}_{\mathrm{\infty}}$similarity and hstability on time scales.
2 Main results
We refer the reader to Ref. [5] for all the basic definitions and results on time scales necessary for this work (e.g., delta differentiability, rdcontinuity, exponential function and its properties).
It is assumed throughout that a time scale $\mathbb{T}$ will be unbounded above. If $\mathbb{T}$ has a leftscattered maximum m, then ${\mathbb{T}}^{\kappa}=\mathbb{T}\{m\}$. Otherwise, ${\mathbb{T}}^{\kappa}=\mathbb{T}$. Let ${\mathbb{R}}^{n}$ be the ndimensional real Euclidean space. ${C}_{\mathrm{rd}}(\mathbb{T}\times {\mathbb{R}}^{n},{\mathbb{R}}^{n})$ denotes the set of all rdcontinuous functions from $\mathbb{T}\times {\mathbb{R}}^{n}$ to ${\mathbb{R}}^{n}$ and ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$.
We consider the dynamic system
where $f\in {C}_{\mathrm{rd}}(\mathbb{T}\times {\mathbb{R}}^{n},{\mathbb{R}}^{n})$ with $f(t,0)=0$, and ${x}^{\mathrm{\Delta}}$ is the delta derivative of $x:\mathbb{T}\to {\mathbb{R}}^{n}$ with respect to $t\in \mathbb{T}$. We assume that ${f}_{x}=\frac{\partial f}{\partial x}$ exists and is rdcontinuous on $\mathbb{T}\times {\mathbb{R}}^{n}$. Let $x(t)=x(t,{t}_{0},{x}_{0})$ be the unique solution of (2.1) satisfying the initial condition $x({t}_{0},{t}_{0},{x}_{0})={x}_{0}$. For the existence and uniqueness of solutions of nonlinear dynamic system (2.1), see [23].
Also, we consider its associated variational systems
and
where $I+\mu (t){f}_{x}(t,x(t))$ is invertible for all $t\in \mathbb{T}$ and I denotes the $n\times n$ identity matrix.
To establish our main results we will use the following lemmas.
Lemma 2.1 [22], Theorem 2.6.4]
Assume that $x(t,{t}_{0},{x}_{0})$ and $x(t,{t}_{0},{y}_{0})$ are the solutions of system (2.1) through $({t}_{0},{x}_{0})$ and $({t}_{0},{y}_{0})$ respectively, which exist for each $t\in {\mathbb{T}}_{0}$ and are such that ${x}_{0}$ and ${y}_{0}$ belong to a convex subset D of ${\mathbb{R}}^{n}$. Then
where Φ is a fundamental matrix of (2.3) and ${\mathbb{T}}_{0}=\mathbb{T}\cap [{t}_{0},\mathrm{\infty})$.
This lemma can be proved in the same manner as that of Theorem 2.6.4 in [22], so we omit the detail.
Lemma 2.2 [23], Lemma 2.7.1]
Let $f\in {C}_{\mathrm{rd}}({\mathbb{T}}^{\kappa}\times D,{\mathbb{R}}^{n})$, where D is an open convex set in ${\mathbb{R}}^{n}$. Suppose that ${f}_{x}$ exists and is rdcontinuous. Then
In order to prove the variation of parameters formula on time scales, we need the following result on differentiability of solutions with respect to initial values.
Lemma 2.3 Assume that $f:{\mathbb{T}}_{0}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ possesses partial derivatives on ${\mathbb{T}}_{0}\times {\mathbb{R}}^{n}$ and ${f}_{x}(t,x(t,{t}_{0},{x}_{0}))$ is rdcontinuous on $\mathbb{T}$. Let $x(t)=x(t,{t}_{0},{x}_{0})$ be the solution of (2.1), which exists for $t\ge {t}_{0}$ and let
Then
exists and is the solution of
The proof of Lemma 2.3 follows simply by differentiating the solution identity
with respect to ${x}_{0}$. It is a special case of [36], Satz 1.2.22].
Remark 2.1 [23], Theorem 2.7.1]
$H(t,{t}_{0},{x}_{0})$ in Lemma 2.3 is given by
It follows from Lemma 2.3 that the fundamental matrix solution $\mathrm{\Phi}(t,{t}_{0},0)$ of (2.2) is given by
and the fundamental matrix solution $\mathrm{\Phi}(t,{t}_{0},{x}_{0})$ of (2.3) is given by
or equivalently
Let ${M}_{n}(\mathbb{R})$ be the set of all $n\times n$ matrices over $\mathbb{R}$. The class of all rdcontinuous functions $A:\mathbb{T}\to {M}_{n}(\mathbb{R})$ is denoted by
Consider the quasilinear dynamic system
where $A\in {C}_{\mathrm{rd}}(\mathbb{T},{M}_{n}(\mathbb{R}))$ and $g:\mathbb{T}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ is rdcontinuous in the first argument with $g(t,0)=0$.
We need the following result which is a slight modification of the variation of constants formula in [24], Theorem 4.6.1].
Lemma 2.4 [36]
The solution $y(t,{t}_{0},{y}_{0})$ of (2.9) satisfies the equation
where Φ is a transition matrix of the linear system
where $A\in {C}_{\mathrm{rd}}({\mathbb{T}}^{\kappa},{M}_{n}(\mathbb{R}))$.
For the Lyapunovlike function $V\in {C}_{\mathrm{rd}}(\mathbb{T}\times {\mathbb{R}}^{n},{\mathbb{R}}_{+})$, we recall the following definition.
Definition 2.2 [23], Definition 3.1.1]
We define the generalized derivative ${D}^{+}{V}_{(\text{2.1})}^{\mathrm{\Delta}}(t,x(t))$ of $V(t,x)$ relative to (2.1) as follows: given $\epsilon >0$, there exists a neighborhood $N(\epsilon )$ of $t\in \mathbb{T}$ such that
where $x(t)$ is any solution of (2.1) and the upper right Dini derivative ${V}_{\ast}^{\mathrm{\Delta}}(t)$ of ${V}_{\ast}$ is given by
where ${V}_{\ast}(t)=V(t,x(t))$.
Then it is well known that
if $V(t,x)$ is Lipschitzian in x for each $t\in \mathbb{T}$ [22, 40].
In case $t\in \mathbb{T}$ is right dense, we have
In case $t\in \mathbb{T}$ is right scattered and $V(t,x(t))$ is continuous at t, we have
In fact, if $x(t)$ is a solution of (2.1), we have
by the chain rule of a differentiable function $V(t,x(t))$ [37], Theorem 1].
We note that the total difference of the function V along the solutions x of (2.1) is given by
Choi et al. [7] investigated hstability for nonlinear differential systems using the notions of ${t}_{\mathrm{\infty}}$similarity and Lyapunov functions. Also, Choi et al. [11] introduced the notion of ${n}_{\mathrm{\infty}}$summable similarity which is the corresponding ${t}_{\mathrm{\infty}}$similarity for the discrete case and then characterized hstability in variation and asymptotic equilibrium in variation for nonlinear difference system via ${n}_{\mathrm{\infty}}$summable similarity and comparison principle.
Now, we define ${u}_{\mathrm{\infty}}$similarity on time scales in order to unify (continuous ) ${t}_{\mathrm{\infty}}$similarity and (discrete) ${n}_{\mathrm{\infty}}$similarity for matrixvalued functions.
Let ${\mathfrak{M}}_{n}(\mathbb{R})$ be the set of all $n\times n$ invertible matrices over $\mathbb{R}$, and ${C}_{\mathrm{rd}}^{1}({\mathbb{T}}^{\kappa},{\mathfrak{M}}_{n}(\mathbb{R}))$ be the set of all rdcontinuous differentiable functions S from ${\mathbb{T}}^{\kappa}$ to ${\mathfrak{M}}_{n}(\mathbb{R})$ such that S and ${S}^{1}$ are bounded on ${\mathbb{T}}_{0}$.
Definition 2.3 A function $A:{\mathbb{T}}^{\kappa}\to {M}_{n}(\mathbb{R})$ is called regressive if for each $t\in {\mathbb{T}}^{\kappa}$ the $n\times n$ matrix $I+\mu (t)A(t)$ is invertible.
The class of all rdcontinuous and regressive functions from ${\mathbb{T}}^{\kappa}$ to ${M}_{n}(\mathbb{R})$ is denoted by
Definition 2.4 [17]
Let $A,B\in {C}_{\mathrm{rd}}\mathcal{R}({\mathbb{T}}^{\kappa},{M}_{n}(\mathbb{R}))$ and ${t}_{0}\in \mathbb{T}$. A function A is ${u}_{\mathrm{\infty}}$similar to a function B if there exists an absolutely integrable function $F\in {C}_{\mathrm{rd}}(\mathbb{T},{M}_{n}(\mathbb{R}))$, i.e., ${\int}_{{t}_{0}}^{\mathrm{\infty}}F(t)\mathrm{\Delta}t<\mathrm{\infty}$, such that
for some $S\in {C}_{\mathrm{rd}}^{1}({\mathbb{T}}^{\kappa},{\mathfrak{M}}_{n}(\mathbb{R}))$.
Remark 2.5 If $\mathbb{T}=\mathbb{R}$, then ${u}_{\mathrm{\infty}}$similarity becomes ${t}_{\mathrm{\infty}}$similarity and if $\mathbb{T}=\mathbb{Z}$, then ${u}_{\mathrm{\infty}}$similarity becomes ${n}_{\mathrm{\infty}}$similarity. Also if A and B are ${u}_{\mathrm{\infty}}$similar with $F=0$ defined on $\mathbb{T}$, then they are kinematically similar [4].
Let $\mathbb{N}({n}_{0})=\{{n}_{0},{n}_{0}+1,\dots ,{n}_{0}+k,\dots \}$, where ${n}_{0}$ is a nonnegative integer and $\mathfrak{M}$ denote the set of all $s\times s$ invertible matrixvalued functions defined on $\mathbb{N}({n}_{0})$.
Remark 2.6 [11], Definition 2.5]
A matrix function $A\in \mathfrak{M}$ is ${n}_{\mathrm{\infty}}$summably similar to a matrix function $B\in \mathfrak{M}$ if there exists an absolutely summable $s\times s$ matrix $F(n)$ over $\mathbb{N}({n}_{0})$, that is,
such that
for some $S\in \mathfrak{S}$.
For the example of ${n}_{\mathrm{\infty}}$summable similarity, see [11].
Remark 2.7 We can easily show that the ${n}_{\mathrm{\infty}}$summable similarity is an equivalence relation in the similar manner of Trench in [39]. Also, if A and B are ${n}_{\mathrm{\infty}}$summably similar with $F(n)=0$, then we say that they are kinematically similar.
Pinto [32] introduced the notion of hstability which is an extension of the notions of exponential stability and uniform stability of the solutions of differential equations. The symbol $\cdot $ will be used to denote any convenient vector norm in ${\mathbb{R}}^{n}$. We recall the notions of hstability for dynamic systems on time scales in [14].
Definition 2.8 System (2.1) is called an hsystem if there exist a positive rdcontinuous function $h:\mathbb{T}\to \mathbb{R}$, a constant $c\ge 1$ and $\delta >0$ such that
for ${x}_{0}<\delta $ (here $h{(t)}^{1}=\frac{1}{h(t)}$).
Moreover, system (2.1) is said to be
(h S) hstable if h is a bounded function in the definition of hsystem,
(Gh S) globally hstable if system (2.1) is h S for every ${x}_{0}\in D$, where $D\subset {\mathbb{R}}^{n}$ is a region which includes the origin,
(h SV) hstable in variation if system (2.3) is h S,
(Gh SV) globally hstable in variation if system (2.3) is Gh S.
For the various definitions of stability, we refer to [20] and we obtain the following possible implications for system (2.1) among the various types of stability:
as in [7, 34]. See [8, 26, 28, 29] for stability of nonlinear difference systems.
We consider two linear dynamic systems
and
where $A,B\in {C}_{\mathrm{rd}}\mathcal{R}({\mathbb{T}}^{\kappa},{M}_{n}(\mathbb{R}))$.
We say that if A and B are ${u}_{\mathrm{\infty}}$similar, then systems (2.15) and (2.16) are ${u}_{\mathrm{\infty}}$similar.
Lemma 2.5 [14], Lemma 2.3]
System (2.15) is an hsystem if and only if there exist a positive rdcontinuous function h defined on $\mathbb{T}$ and a constant $c\ge 1$ such that
where ${\mathrm{\Phi}}_{A}$ is a fundamental matrix solution of (2.15).
We obtain the following result from Lemma 2.3 in [17].
Lemma 2.6 Assume that A and B are ${u}_{\mathrm{\infty}}$similar. Then
where ${\mathrm{\Phi}}_{A}$ and ${\mathrm{\Phi}}_{B}$ are the matrix exponential functions of (2.15) and (2.16) respectively.
Medina and Pinto [29], Theorem 3] showed that h SV implies h S. Also, they proved the converse when the condition
for ${x}_{0}\le \delta $ holds [29], Theorem 14].
In order to establish our main results, we will introduce the following condition.
(H): ${f}_{x}(t,0)$ and ${f}_{x}(t,x(t,{t}_{0},{x}_{0}))$ are ${u}_{\mathrm{\infty}}$similar for $t\ge {t}_{0}$ and ${x}_{0}\le \delta $ for some constant $\delta >0$ and ${\int}_{{t}_{0}}^{\mathrm{\infty}}\frac{h(t)}{h(\sigma (t))}F(t)\mathrm{\Delta}t<\mathrm{\infty}$ with the positive rdcontinuous function $h(t)$ defined on $\mathbb{T}$.
Lemma 2.7 [12], Theorem 3.4]
Assume that condition (H) is satisfied. Then variational system (2.2) is also an hsystem if and only if variational system (2.3) is an hsystem.
We can obtain the same result about Lemma 2.7 by assuming that ${f}_{x}(t,0)$ and ${f}_{x}(t,x(t,{t}_{0},{x}_{0}))$ are ${u}_{\mathrm{\infty}}$quasisimilar for $t\ge {t}_{0}$ instead of the condition (H) in Lemma 2.7 [18], Theorem 3.3].
For nonlinear dynamic system (2.1), we can show that
by using the concept of ${u}_{\mathrm{\infty}}$similarity.
We study the relation of hstability between two systems (2.1) and (2.3) by assuming the condition (H) is satisfied.
Theorem 2.8 [29], Theorem 2]
Suppose that condition (H) is satisfied. If $x=0$ of (2.1) is hstable, then $v=0$ of (2.2) is hstable.
We obtain the following result from (2.8).
Theorem 2.9 If $z=0$ of (2.3) is hstable, then $x=0$ of (2.1) is hstable.
We can obtain the following result by using Lemma 2.7 and Theorem 2.8.
Theorem 2.10 Assume that condition (H) is satisfied. If $x=0$ of (2.1) is hstable, then $z=0$ of (2.1) is hstable in variation.
Remark 2.9 For nonlinear dynamic system (2.1), we show that two concepts of hstability and hstability in variation are equivalent under the condition that two variational systems (2.2) and (2.3) are ${u}_{\mathrm{\infty}}$similar.
Choi et al. investigated Massera type converse theorems for the nonlinear difference system $x(n+1)=f(n,x(n))$ via ${n}_{\mathrm{\infty}}$similarity in [8], Theorem 5] and [9], Theorem 2.1]. Furthermore, they characterized hstability in variation for the nonlinear difference system by using the notion of ${n}_{\mathrm{\infty}}$summable similarity in [11].
We need the following lemma to prove our main theorem.
Lemma 2.11 [36], Korollar 2.1.13]
If the delta differentiable function $h:\mathbb{T}\to \mathbb{R}$ is positive, then $\frac{{h}^{\mathrm{\Delta}}(t)}{h(t)}$ is positively regressive, and ${e}_{p}(t,{t}_{0})$ satisfies
where $p(t)=\frac{{h}^{\mathrm{\Delta}}(t)}{h(t)}$.
We can obtain the following result that characterizes hstability for nonlinear dynamic system (2.1) via the notions of Lyapunov functions and ${u}_{\mathrm{\infty}}$similarity. It is adapted from Theorem 3.6.1 in [22] and Theorem 3.1 in [25].
Theorem 2.12 Assume that condition (H) is satisfied. Suppose further that ${h}^{\mathrm{\Delta}}(t)$ exists and is rdcontinuous on $\mathbb{T}$. Then system (2.1) is GhS if and only if there exists a function $V(t,x)$ defined on $\mathbb{T}\times {\mathbb{R}}^{n}$ such that the following properties hold:

(i)
$x\le V(t,x)\le cx$ for $(t,x)\in \mathbb{T}\times {\mathbb{R}}^{n}$ and a constant $c\ge 1$;

(ii)
$V(t,{x}_{1})V(t,{x}_{2})\le c{x}_{1}{x}_{2}$ for $t\in \mathbb{T}$ and ${x}_{1},{x}_{2}\in {\mathbb{R}}^{n}$;

(iii)
${V}^{\mathrm{\Delta}}(t,x)\le \frac{{h}^{\mathrm{\Delta}}(t)}{h(t)}V(t,x)$ for $(t,x)\in \mathbb{T}\times {\mathbb{R}}^{n}$;

(iv)
$V(t,x)$ is continuous on $\mathbb{T}\times {\mathbb{R}}^{n}$;
$$\underset{(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})\to (t,x),\stackrel{\u02c6}{t}\ge t}{lim}V(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})V(t,x)=0.$$
Proof Necessity: Suppose that system (2.1) is Gh S. Then system (2.1) is Gh SV by Theorem 2.10, i.e., there exist a constant $c\ge 1$ and a positive rdcontinuous bounded function h defined on $\mathbb{T}$ such that for each $x\in {\mathbb{R}}^{n}$
where Φ is a fundamental matrix solution of (2.3).
Fix $t\in \mathbb{T}$. Let ${A}_{t}:=\{\tau \in {\mathbb{R}}_{+}:t+\tau \in \mathbb{T}\}$. Then we note that ${A}_{t}$ is nonempty from $0\in {A}_{t}$.
Define the function $V:\mathbb{T}\times {\mathbb{R}}^{n}\to {\mathbb{R}}_{+}$ by
where $x(t+\tau ,t,x)$ is a unique solution of system (2.1) for $(t,x)\in \mathbb{T}\times {\mathbb{R}}^{n}$ with the initial value $x(t,t,x)=x$. From Gh S of (2.1) we have
Furthermore, we obtain
Thus V satisfies property (i).
Let $(t,{x}_{1}),(t,{x}_{2})\in \mathbb{T}\times {\mathbb{R}}^{n}$. Then we have
It follows from Lemma 2.2 that for each ${x}_{1}$ and ${x}_{2}$ in a convex subset D of ${\mathbb{R}}^{n}$
In view of (2.18), (2.19) and (2.20), we have
This implies that $V(t,x)$ is globally Lipschitzian in x for fixed $t\in \mathbb{T}$.
Next, we will prove property (iii). Let $x(t,{t}_{0},{x}_{0})$ be a unique solution of system (2.1) for each initial point $({t}_{0},{x}_{0})\in \mathbb{T}\times {\mathbb{R}}^{n}$. We will consider two cases, $\sigma (t)=t$ and $\sigma (t)>t$, in the proof.
Suppose that $\sigma (t)=t$ and let $\delta \in {A}_{t}$. By the uniqueness of solutions of (2.1) and the definition of h S, we have
Suppose that $\sigma (t)>t$. Then it follows from the definition of ${V}^{\mathrm{\Delta}}(t)$ that
Since the solution of (2.1) is unique, we have the following derivative:
Thus property (iii) was satisfied for two cases.
The continuity of $V(t,x)$ can be proved in a similar manner of Theorem 3.6.1 in [22] and Theorem 3.1 in [25]. It remains to show that V is continuous in the sense of (iv): let $t\in \mathbb{T}$, $x\in {\mathbb{R}}^{n}$ be fixed and choose $\epsilon >0$ arbitrary. Then ${\delta}_{1}>0$ and ${\delta}_{2}>0$ must be found such that
holds for all
and all $\stackrel{\u02c6}{x}\in {B}_{{\delta}_{2}}(x)$, where ${B}_{{\delta}_{2}}(x)$ is an open ball centered on x of radius ${\delta}_{2}$.
If $t\in \mathbb{T}$ is right scattered, then we can always choose a suitable ${\delta}_{1}$ such that $\stackrel{\u02c6}{t}=t$ is the only point satisfying condition (2.21) (see [22], Theorem 3.6.1]). Thus $V(t,x)$ is continuous in $(t,x)\in \mathbb{T}\times {\mathbb{R}}^{n}$ since V is globally Lipschitz continuous in x for fixed $t\in \mathbb{T}$.
Suppose that $t\in \mathbb{T}$ is right dense and let $\stackrel{\u02c6}{t}=t+\nu $ for $\nu \in {A}_{t}$ with $\nu \ge 0$. Then we have
Since $V(t,x)$ is Lipschitzian in x and $x(t+\nu ,t,x)$ is continuous in ν, the first two terms (2.22) and (2.23) on the righthand side of the preceding inequality are small when $\stackrel{\u02c6}{x}x$ and ν are small. That is, we have
for all $\stackrel{\u02c6}{x}\in {B}_{{\delta}_{2}}(x)$ when ${\delta}_{2}<\frac{\epsilon}{3c}$ and
since it follows from ${lim}_{\nu \to 0,\nu \in {A}_{t}}x(t+\nu ,t,x)=x$ that there exists a ${\stackrel{\u02c6}{\delta}}_{1}>0$ such that
for all $\nu \in {A}_{t}$ with $0\le \nu <{\stackrel{\u02c6}{\delta}}_{1}$.
Let us consider the third term in (2.24). We note that
Thus we have
where $\alpha (\nu )={sup}_{\tau \in \{\tau \in [\nu ,\mathrm{\infty}):t+\tau \in \mathbb{T}\}}x(t+\tau ,t,x(t))h{(t+\tau )}^{1}h(t)$ for $\nu \in {A}_{t}$.
We have $\alpha (\nu )\le \alpha (0)$ for all $\nu \in {A}_{t}$ with $\nu \ge 0$. Furthermore, $\alpha (\nu )$ is a nonincreasing function in ν with
Hence, there exists a ${\stackrel{\u02c6}{\delta}}_{3}>0$ such that
Now, choose ${\delta}_{1}=min\{{\stackrel{\u02c6}{\delta}}_{1},{\stackrel{\u02c6}{\delta}}_{3}\}$. For $\stackrel{\u02c6}{t}=t+\nu $ with $\nu \in {A}_{t}$ where $0\le \nu <{\delta}_{1}$ and $\stackrel{\u02c6}{x}\in {B}_{{\delta}_{2}}(x)$, combining all of the above estimates of the terms in (2.22)(2.24) gives
which proves the continuity of $V(t,x)$.
Sufficiency: Assume that $V(t,x)$ satisfies the properties (i)(iv). Let $x(t,{t}_{0},{x}_{0})$ be any solution of system (2.1). Then it follows from condition (iii) of $V(t,x)$ that
From Gronwall’s inequality on time scale [5] and Lemma 2.15 [14], we obtain
where $p(t)=\frac{{h}^{\mathrm{\Delta}}(t)}{h(t)}$. From (2.25) and property (i) of $V(t,x)$, we have
for each $t\ge {t}_{0}$ and ${x}_{0}\in {\mathbb{R}}^{n}$. Hence the zero solution $x=0$ of (2.1) is Gh S. This completes the proof of the theorem. □
Remark 2.10 Assume that condition (H) is satisfied for $\mathbb{T}=\mathbb{R}$. Furthermore, suppose that ${h}^{\prime}(t)$ exists and is continuous on ${\mathbb{R}}_{+}$. Then we can obtain Theorems 2.4 and 2.6 in [7] as a continuous version of Theorem 2.12.
Also, we can obtain the following result as a discrete version of Theorem 2.12.
Corollary 2.13 [11], Theorem 3.7]
Assume that ${f}_{x}(n,0)$ is ${n}_{\mathrm{\infty}}$summably similar to ${f}_{x}(n,x(n,{n}_{0},{x}_{0}))$ for $n\ge {n}_{0}\ge 0$ and every ${x}_{0}\in {\mathbb{R}}^{m}$ with $\frac{h(n)}{h(n+1)}F(n)\in {l}_{1}(\mathbb{N}({n}_{0}))$. Then system (2.1) is GhS if and only if there exists a function $V(n,z)$ defined on $\mathbb{N}({n}_{0})\times {\mathbb{R}}^{m}$ such that the following properties hold:

(i)
$V(n,z)$ is continuous on $\mathbb{N}({n}_{0})\times {\mathbb{R}}^{m}$;

(ii)
$xy\le V(n,xy)\le cxy$ for $(n,x,y)\in \mathbb{N}({n}_{0})\times {\mathbb{R}}^{m}\times {\mathbb{R}}^{m}$;

(iii)
$V(n,{z}_{1})V(n,{z}_{2})\le c{z}_{1}{z}_{2}$ for $n\in \mathbb{N}({n}_{0})$, ${z}_{1},{z}_{2}\in {\mathbb{R}}^{m}$;

(iv)
$\frac{\mathrm{\Delta}V(n,xy)}{V(n,xy)}\le \frac{\mathrm{\Delta}h(n)}{h(n)}$ for $(n,x,y)\in \mathbb{N}({n}_{0})\times {\mathbb{R}}^{m}\times {\mathbb{R}}^{m}$ with $x\ne y$.
Remark 2.11 Choi et al. [8, 9] introduced the notion of ${n}_{\mathrm{\infty}}$similarity which is slightly different from ${n}_{\mathrm{\infty}}$summable similarity and studied a general variational stability for a nonlinear difference system via ${n}_{\mathrm{\infty}}$similarity and Lyapunov functions. We can obtain the discrete analogues [8], Theorem 5, Corollary 8] and [9], Theorem 2.1] as a discrete version of Theorem 2.12.
Remark 2.12 Choi et al. [14], Theorem 2.16] studied hstability for linear dynamic equations on time scales by using the unified time scale quadratic Lyapunov functions. Also, Mukdasai and Niamsup [31], Theorem 3.13] derived a sufficient condition for hstability for a linear timevarying system with nonlinear perturbation on time scales by constructing appropriate Lyapunov functions.
We can obtain the following Massera type converse theorem for the uniform exponential asymptotic stability of linear dynamic equations on time scales as a special case of Theorem 2.12.
Corollary 2.14 [25], Theorem 3.1]
Assume that $f(t,x)=A(t)x$ is linear, where $A\in {C}_{\mathrm{rd}}\mathcal{R}(\mathbb{T},{\mathbb{R}}^{n}\times {\mathbb{R}}^{n})$. If system (2.1) is hS with $h(t)={e}^{\lambda t}$ on time scales $\mathbb{T}$ for a nonnegative constant λ, then there exists a function $V:\mathbb{T}\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ such that

(i)
$x\le V(t,x)\le Kx$ for all $t\in \mathbb{T},x\in {\mathbb{R}}^{n}$.

(ii)
$V(t,{x}_{1})V(t,{x}_{2})\le K{x}_{1}{x}_{2}$ for any fixed $t\in \mathbb{T}$ and all ${x}_{1},{x}_{2}\in {\mathbb{R}}^{n}$.

(iii)
The upper right Dini derivative of ${V}_{\ast}$ exists and the estimates ${V}_{\ast}^{\mathrm{\Delta}}(t)\le {\overline{\xi}}_{\lambda}(t){V}_{\ast}(t)<0$ hold for all $t\ge {t}_{0}$ for a positive constants K and λ. Here the function ${\overline{\xi}}_{\lambda}:\mathbb{T}\to \mathbb{R}$ is given by
$${\overline{\xi}}_{\lambda}(t)=\{\begin{array}{cc}\frac{exp(\mu (t)\lambda )1}{\mu (t)},\hfill & t<\sigma (t),\hfill \\ \lambda ,\hfill & t=\sigma (t).\hfill \end{array}$$ 
(iv)
V is continuous from the right in $(t,x)\in \mathbb{T}\times {\mathbb{R}}^{n}$, that is,
$$\underset{(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})\to (t,x),\stackrel{\u02c6}{t}\ge t}{lim}V(\stackrel{\u02c6}{t},\stackrel{\u02c6}{x})V(t,x)=0.$$
3 Examples
In this section we give some examples which illustrate some results from the previous section.
To illustrate the notion of ${u}_{\mathrm{\infty}}$similarity, we will give an example for scalar functions defined on time scales.
Example 3.1 Let $a,b:\mathbb{T}\to \mathbb{R}$ be scalar functions given by
where α is a negative regressive constant and ${lim}_{t\to \mathrm{\infty}}{e}_{\alpha}(t,{t}_{0})=0$ for fixed ${t}_{0}\in \mathbb{T}$. If we put $s(t)=1$ for each $t\in \mathbb{T}$, then $s(t)$ and ${s}^{1}(t)$ are bounded and nonzero functions. Moreover, we have
where $f(t)=\frac{1}{2}{e}_{\alpha}(t,{t}_{0})$.
Thus we have
This implies that a and b are ${u}_{\mathrm{\infty}}$similar on $\mathbb{T}$.
For the examples of nonscalar functions about ${t}_{\mathrm{\infty}}$similarity on $\mathbb{T}=\mathbb{R}$ and ${n}_{\mathrm{\infty}}$summable similarity on $\mathbb{T}=\mathbb{Z}$, see [7], Example] and [11], Example 2.6] respectively.
Example 3.2 To illustrate Lemma 2.5, we consider the linear dynamic system
where $A(t)=\left(\begin{array}{cc}\frac{{e}^{t}}{2+{e}^{t}}& 0\\ 0& 0\end{array}\right)$. If $\mu (t)<2{e}^{t}+1$ for $t\in \mathbb{T}$, then (3.1) is hstable.
Proof A matrix exponential function ${\mathrm{\Phi}}_{A}(t,{t}_{0})$ of (3.1) is given by
where $p(t)=\frac{{e}^{t}}{2+{e}^{t}}$ and ${e}_{p}(t,{t}_{0})=exp{\int}_{{t}_{0}}^{t}{\xi}_{\mu (\tau )}(p(\tau ))\mathrm{\Delta}\tau $. Here the cylinder transformation ${\xi}_{\mu}(z)$ is given by
It follows that
for each $t,{t}_{0}\in \mathbb{T}$ with $t\ge {t}_{0}$, where M is a positive constant. Thus we obtain
where $h(t)={e}_{p}(t,{t}_{1})$ is a positive bounded rdcontinuous function for a fixed point ${t}_{1}\in \mathbb{T}$, and $c:=M+1$ is a positive constant. Hence system (3.1) is hstable by Lemma 2.5. □
To illustrate that the converse of Theorem 2.8 does not hold in general, we give the following example.
Example 3.3 [11], Example 5.2]
Let $\mathbb{T}$ be the unbounded above time scales with $\mu (t)<2$ for each $t\in \mathbb{T}$. We consider the nonlinear dynamic equation
and its variational dynamic equation
where ${f}_{x}(t,x)=\frac{1}{2}+2x$. Then $v=0$ of (3.3) is hstable, but $x=0$ of (3.2) is not hstable.
Proof Since the fundamental solution is $\varphi (t)={e}_{\frac{1}{2}}(t,{t}_{0}){v}_{0}$ for each $t\ge {t}_{0}$, Eq. (3.3) is hstable with a positive bounded function $h(t)={e}_{\frac{1}{2}}(t,{\stackrel{\u02c6}{t}}_{0})$ for a fixed point ${\stackrel{\u02c6}{t}}_{0}\in \mathbb{T}$. But (3.2) is not hstable because there exists an unbounded solution $x(t,{\stackrel{\u02c6}{t}}_{0},1)$ of (3.2) such that
□
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Acknowledgement
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF20100008835). The authors are thankful to the anonymous referees for their valuable comments and corrections to improve this paper.
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Keywords
 dynamic system
 variational system
 ${u}_{\mathrm{\infty}}$similarity
 Lyapunov function
 hstability
 hstability in variation
 time scales