BOUNDEDNESS AND EXPONENTIAL STABILITY OF SOLUTIONS TO DYNAMIC EQUATIONS ON TIME SCALES

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1 Electronic Journal of Differential Equations, Vol , No. 12, pp ISSN: URL: or ftp ejde.math.txstate.edu login: ftp BOUNDEDNESS AND EXPONENTIAL STABILITY OF SOLUTIONS TO DYNAIC EQUATIONS ON TIE SCALES AI-LIAN LIU Abstract. aking use of the generalized time scales exponential function, we give a new definition for the exponential stability of solutions for dynamic equations on time scales. Employing Lyapunov-type functions on time scales, we investigate the boundedness and the exponential stability of solutions to first-order dynamic equations on time scales, and some sufficient conditions are obtained. Some examples are given at the end of this paper. 1. Introduction In this paper, we consider the boundedness and exponential stability of solutions to the first-order dynamic equations subject to the initial condition x = ft, x t, t T, 1.1 x = x 0 T, x 0 R n, 1.2 where f : T R n R n is a so called rd-continuous function and t is from a so called time scale T. If T = R, then x = x and 1.1, 1.2 is the following initial value-problem for ordinary differential equations, x = ft, x, 1.3 x = x If T = Z, then x = x the forward difference calculus, and 1.1, 1.2 corresponds to the initial value-problem for the O E xn + 1 xn = fn, xn, 1.5 xn 0 = x Recently Raffoul [6] used Lyapunov-type function to formulate some sufficient conditions that ensure all solutions of 1.3, 1.4 are uniformly bounded. In [7], by 2000 athematics Subject Classification. 34K11, 34K40, 39A12. Key words and phrases. Time scale; Lyapunov-type function on time scales; boundedness; exponential stability. c 2006 Texas State University - San arcos. Submitted January 16, Published January 8, Supported by grants and from NNSF of China, and by gran512 from Zhongshan University Advanced Research Centre. 1

2 2 A.-L. LIU EJDE-2006/12 a kind of suitable and easy-to-calculate Lyapunov type I function on time scales, Peterson and Tisdell formulated some appropriate inequalities on these functions that guarantee solutions to 1.1, 1.2 are uniformly bounded. Other results on boundedness can be found, for example, in [3] and [4]. The investigation of stability analysis of nonlinear systems has produced a vast body of important results. In [10], uhammad made use of non-negative definite Lyapunov functions to study the exponential stability of the zero solution of nonlinear discrete system 1.5, 1.6, and in [11], [12] they gave sufficient conditions for the exponential stability of a class of nonlinear time-varying differential equations. In this paper, we first define the boundedness and the exponential stability of solutions on time scales, then making use of the Lyapunov-type function on time scales, we get sufficient conditions that guarantee the boundedness and exponential stability of zero solution to 1.1, 1.2, which generalize the results in [10, 11, 12]. Some examples are also presented at the end of this paper. Throughout this paper, the following notation will be used: R n is the n-dimensional Euclidean vector space; R + is the set of all non-negative real numbers; x is the Euclidean norm of a vector x R n. 2. Preliminaries The theory of dynamic equations on time scales or more generally, measure chains was introduced in Stefan Hilger s PhD thesis in The theory presents a structure where, once a result is established for general time scales, special cases include a result for differential equations obtained by taking the time scale to be the real numbers and a result for difference equations obtained by taking the time scales to be the integers. A great deal of work has been done since 1988, unifying the theory of differential equations and the theory of difference equations by establishing the corresponding results in time scale setting. Definition 2.1. A time scale T is an arbitrary nonempty closed subset of the set R, with the topology and ordering inherited from R. We assume throughout the paper that T is unbounded above. Definition 2.2. For t T the forward jump operator σt : T T is defined by σt := inf{s T : s > t}. The jump operator σ gives the classifications of points on time scales. A point t T is called right dense if σt = t, and right scattered if σt > t. The graininess function µt : T R is defined by µt = σt t. Definition 2.3. Fix t T and let x : T R n, define the delta-derivative x t of x at t T to be the vector if it exists with the property that given any ɛ > 0, there is a neighborhood U T of t such that [x i σt x i s] x i t[σt s] ɛ σt s for all s U and i = 1, 2,..., n. At this time we say xt is delta differentiable. In the case of T = R, x t = x t. When T = Z, x t is the standard forward difference operator xn + 1 xn. Definition 2.4. For f : T R n and F : T R n, if F t = ft for all t T, then F is said to be an antiderivative of f. And define the cauchy integral by the

3 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 3 formula b a fτ τ = F b F a for a, b T. Definition 2.5. A function f : T R is called right-dense continuous provided it is continuous at right dense points of T and its left sided limit exists finite at left dense points of T. The set of all right-dense continuous function on T is denoted by C rd = C rd T = C rd T, R. Consequences of these definitions and properties can be found in [1, 2, 5]. Definition 2.6. A function p : T R is regressive provided that 1 + µtpt 0 for all t T. The set of all regressive and right-dense continuous function is denoted by R = RT = RT, R. The set R + of all positively regressive function is R + = R + T, R = {p R : 1 + µtpt > 0 for all t T}. Definition 2.7. For pt R, we define the generalized exponential function as e p t, = exp Log1 + µτpτ τ for, t T. µτ Remark 2.8. Consider the dynamic initial-value problem x = ptx, x = x 0, 2.1 where T and pt R. The exponential function xt = x 0 e p t, is the unique solution to system 2.1. Theorem 2.9. i If p R +, then e p t, > 0; ii e p σt, = 1 + µtpte p t, ; iii e p t, = 1 e pt,, where p = p 1 + pµt ; iv If p, q R, then e p t, e q t, = e p q t, ; v If p is a positive constant, then lim t e p t, =, lim t e p t, = 0. Other relevant theorems can be found in [1],[2]. In the following discussions, we assume conditions are imposed on system 1.1,1.2 such that the existence of solutions is guaranteed when t T + = {t T : t }. 3. Boundedness of Solutions Definition 3.1. We say a solution xt of system 1.1,1.2 is bounded if there exists a constant C, x 0 that may depend on and x 0 such that xt C, x 0 for t T +. We say that solutions of 1.1,1.2 are uniformly bounded if C is independent of.

4 4 A.-L. LIU EJDE-2006/12 Assume V : T + R n R + is delta differentiable in variable t and continuously differentiable in variable x, and xt is any solution of dynamic system 1.1, 1.2, then from [13, 14] we know the delta derivative along xt for V t, x is the following V t, x = V t, xt = V t t, xσt + = V t t, xσt V xt, xt + hµtx tdh x t V xt, xt + hµtft, xdh ft, x, where Vt is considered as the delta derivative in the first variable t and V x is taken as the normal derivative in variable x. Then we call V t, x a Lyapunov-type function on time scales. To calculate the derivative is not an easy work generally, but if V t, x is explicitly independent of t, i.e. V : R n R + and V x = V 1 x V n x n, this is the type I Lyapunov function introduced in [7] and V x t is easy to handle at this time. In this section we want to point out that the results in [7] are held to be true theoretically for general Lyapunov-type function on time scales. Theorem 3.2. Assume D is an open and convex set in R n. Suppose there exists a Lyapunov-type function V : T + D R + that satisfies λ 1 x p V t, x λ 2 x q, 3.1 V t, x λ 3 x r + L 1 + µt, 3.2 V t, x V r/q t, x γ, 3.3 where λ 1, λ 2, λ 3, p, q, r are positive constants, L and γ are nonnegative constants, and = λ 3 /λ r/q 2. Then all solutions of 1.1, 1.2 that stay in D are uniformly bounded. Proof. Note that = λ 3 /λ r/q 2, so R + and e t, is well defined and is positive. From the derivative formula of products and condition 3.2, V t, xe t, = V t, xe σt, + V t, xe t, λ 3 x r + L 1 + µt 1 + µte t, + V t, xe t, = λ 3 x r + L + V t, xe t,. From 3.1, we have x q V t, x/λ 2, consequently x r V t, x/λ 2 r/q. So by 3.3, V t, xe t, [ λ 3 /λ r/q 2 V r/q t, x + V t, x + L]e t, = [V t, x V r/q t, x + L]e t, γ + Le t,.

5 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 5 Integrating the above inequality from to t t T +, we obtain Hence V t, xe t, V, x 0 + γ + L e t, e, V, x 0 + γ + L e t, λ 2 x 0 q + γ + L e t,. V t, x λ 2 x 0 q e t, + γ + L λ 2 x 0 q + γ + L. From 3.1, we have λ 1 x p V t, x, which implies xt 1 1/p λ2 x 0 q + γ + L 1/p for all t T + t λ 1 0. This completes the proof. Theorem 3.3. Assume D R n is open and convex, and there exists a Lyapunovtype function V : T + D R + that satisfies λ 1 t x p V t, x λ 2 t x q, 3.4 V t, x λ 3t x r + L, µt V t, x V r/q t, x γ, 3.6 for some positive constants p, q, r and positive functions λ 1 t, λ 2 t, λ 3 t, where λ 1 t is nondecreasing, L and γ are nonnegative constants, and = inf t T + λ 3 t/λ r/q 2 t > 0. Then all solutions of 1.1, 1.2 that stay in D are bounded. Proof. For = inf t T + λ 3 t/λ r/q 2 t > 0, by calculating V t, xe t, and then by the similar argument as in Theorem 3.2, we obtain V t, x λ 2 x 0 q e t, + γ + L λ 2 x 0 q + γ + L. Using condition 3.4, we arrive at x V t, x λ 1 t 1/p V t, x 1/p. λ 1 Combining the above two inequalities, we get xt 1 1/p λ2 x 0 q + γ + L 1/p for all t T + t λ 1 0, which concludes the proof.

6 6 A.-L. LIU EJDE-2006/12 4. Exponential Stability of Solutions For the dynamic equation 1.1, we assume ft, 0 = 0 for all t T, so xt = 0 is the trivial solution of 1.1. There are different definitions for the exponential stability of the zero solution according to different authors [8, 9]. Pötzsche [8] gave the definition by the regular exponential function e pt t0 constant p > 0; and Dacunha [9] defined the exponential stability in terms of e p t, constant p > 0, p R +. For constant p > 0, p R +, taking into consideration the following relationship e p t, e pt t0 e p t, t, T, t, here we introduce a new definition, which is more general than the present, by the use of the generalized time scales exponential function e p t,. Definition 4.1. The zero solution to system 1.1 is exponentially stable if any solution xt of 1.1, 1.2 satisfies xt β x 0, e α pt, t T +, where β : R + T R + is a nonnegative function, and α, p are positive constants. If β x 0, does not depend on, the zero solution is called uniformly exponentially stable. For the rest of this article, to shorten expressions, instead of saying the zero solution is stable, we say that the system 1.1,1.2 is stable. Lemma 4.2. For any time scale T, assume the graininess function µt is bounded above, i.e. there exists a constant B T that may depend on T such that µt B T, then for any positive constants, δ satisfying < δ, we have t e δ τ, τ is bounded above. Proof. From the properties of generalized exponential function Theorem 2.9, e δ τ, τ = e τ, e δ τ, τ = 1 e τ, e δ τ, τ. By the integration by parts formula [1, Theorem 1.77], we have e δ τ, τ = 1 e δ e δ t, 1 e στ, τ, t 0 e δ τ, e δ στ, τ = 1 e στ, e δ t, 1 + δ e δ στ, τ = µτe τ, e δ t, 1 + δ τ. e δ στ, Due to the assumption, we have e δ τ, τ 1 e δ t, 1 + δ1 + B T From the formula in [1, Theorem 2.38], e τ, e δ στ, τ = 1 e δ δτ, τ. e τ, e δ στ, τ.

7 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 7 Hence e δ τ, τ δ1 + B T δ = 1 δ1 + BT δ That completes the proof. δ1 + B T δ = Constant. e δ t, 1 δ1 + B T δ e δt, The results obtained in this section are under the assumption that Lemma 4.2 holds to be true, i.e., throughout this section we assume sup t T + µt < with its bound dependent on the time scale T. Theorem 4.3. Assume D R n is an open and convex set containing the origin and there exists a Lyapunov-type function V : T + D R + which satisfies λ 1 x p V t, x λ 2 x q, 4.1 V t, x λ 3 x r + Ke δ t,, µt where λ 1, λ 2, λ 3, K, p, q, r, δ are positive numbers and the following two conditions hold for all t, x T + D and there exists a γ 0 such that δ > λ 3 /λ 2 r/q =, 4.3 V t, x V r/q t, x γe δ t,. 4.4 Then system 1.1, 1.2 is uniformly exponentially stable. Proof. Let xt be a solution of 1.1, 1.2 and let Then Qt, x = V t, xe t,. Q t, x = V t, xe σt, + V t, xe t,. Taking 4.2 into account, for all t T +, x D, we have Q t, x λ 3 x r + Ke δ t, 1 + µte t, + V t, xe t,. 1 + µt By 4.1, we have x q V t, x/λ 2, or equivalently x r V t, x/λ 2 r/q. Therefore, Q t, x V r/q t, xλ 3 /λ r/q 2 + Ke δ t, + V t, x e t,. Since λ 3 /λ r/q 2 =, we have Using 4.4, we obtain Q t, x V t, x V r/q t, xe t, + Ke δ t,. Q t, x γ + Ke δ t,.

8 8 A.-L. LIU EJDE-2006/12 Integrating both sides of the above inequality from to t, we obtain Qt, x Q, x 0 γ + Ke δ τ, τ = γ + K From Lemma 4.2, we have γ + K Qt, x Q, x 0 Since Q, x 0 = V, x 0 λ 2 x 0 q, we have δ1 + B T δ. e δ τ, τ. Qt, x λ 2 x 0 q γ + K δ1 + B T + δ =: β x 0. We have Qt, x β x 0. On the other hand, from 4.1 it follows that V t, x 1/p x. λ 1 Substituting V t, x = Qt, xe t, in the last inequality, we obtain Qt, xe t, 1/p β x0 1/p 1/p xt e λ 1 λ t,. 1 This inequality shows that system 1.1, 1.2 is uniformly exponentially stable. Therefore the proof is complete. Theorem 4.4. Assume D R n is an open and convex set containing the origin and there exists a Lyapunov-type function V : T + D R + which satisfies λ 1 t x p V t, x λ 2 t x q, 4.5 V t, x λ 3t x r + Ke δ t,, µt δ > inf t T + λ 3 t/λ 2 t r/q = > 0, 4.7 γ 0, such that V t, x V r/q t, x γe δ t,, 4.8 where λ 1 t, λ 2 t, λ 3 t are positive functions, λ 1 t is nondecreasing for all t T +, and K, p, q, r, δ are positive constants. Then system 1.1, 1.2 is exponentially stable. Proof. We consider the function Qt, x = V t, xe t,. By a similar argument used in the proof of Theorem 4.3, we arrive at Q t, x λ 3 t x r + Ke δ t, e t, + V t, xe t,. Taking condition 4.5 into account and by the assumption λ 2 t > 0 for all t T +, we have x q V t, x/λ 2 t, so equivalently x r V t, x/λ 2 t r/q. Therefore, Q t, x V r/q t, xλ 3 t/λ 2 t r/q + Ke δ t, + V t, x e t,. Since λ 3 t/λ 2 t r/q, by condition 4.8, we obtain Q t, x V t, x V r/q t, xe t, + Ke δ t, γ + Ke δ t,.

9 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 9 Thus integrating both sides of the above inequality from to t and applying Lemma 4.2, Qt, x Q, x 0 γ + Ke δ τ, τ γ + K δ1 + B T = δ. Since Q, x 0 = V, x 0 λ 2 x 0 q, we have Qt, x λ 2 x 0 q γ + K δ1 + B T + δ =: β x 0,. Furthermore, from 4.5 it follows that V t, x 1/p. x λ 1 t Since λ 1 t is non-decreasing, hence λ 1 t λ 1. So V t, x 1/p. x λ 1 Substituting V t, x = Qt, xe t, into the last inequality, we obtain Qt, xe t, 1/p β x0, 1/pe 1/p xt λ 1 λ 1 t,. This inequality shows that system 1.1, 1.2 is exponential stable. Theorem 4.5. Assume D R n is an open and convex set containing the origin, and there exists a Lyapunov-type function V : T + D R + such that λ 1 t x p V t, x Φ x, V t, x Ψ x + Le δt,, 1 + µt ΨΦ 1 V t, x + V t, x γe δ t,, where Φ : [0, [0,, Ψ : [0,, 0], λ 1 t : T + 0, +, Ψ is nonincreasing, λ 1 t, Φ is nondecreasing, and Φ 1 exists, L and γ are nonnegative constants, δ > 1. Then system 1.1, 1.2 is uniformly exponentially stable. Proof. Let xt be a solution of system 1.1, 1.2, then [V t, xe 1 t, ] = V t, xe 1 σt, + V t, xe 1 t, Ψ x + Le δt, 1 + µte 1 t, + V t, xe 1 t, 1 + µt ΨΦ 1 V t, x + Le δ t, e 1 t, + V t, xe 1 t, γ + Le 1 δ t,. Integrating both sides from to t, we obtain V t, xte 1 t, V, x 0 + γ + L e 1 δ τ, τ V, x 0 + γ + L δ1 + B T δ 1 =: β x 0,,

10 10 A.-L. LIU EJDE-2006/12 so that V t, xt β x 0, e 1 t,. From the assumption, we have 1 1/p 1 1/pe x λ 1 t β x 0, e 1 t, λ 1 β x 1/p 0, 1 t,, which completes the proof. Remark 4.6. In Theorems 4.4 and 4.5, we can replace the nondecreasing assumption of λ 1 t by the following assumption: There exists a > 0 such that a < and λ 1 t e a t,, for all t T +, where = inf t T + λ 3 t/λ 2 t r/q. Take = 1 in Theorem 4.5. The following theorem does not require an upper bound on the Lyapunov function V t, x. Theorem 4.7. Assume D R n is an open and convex set containing the origin. Let V : T + D R + be a given Lyapunov-type function satisfying λ 1 x p V t, x, 4.9 V t, x λ 2V t, x + Ke δ t,, µtε for some positive constants λ 1, λ 2, p, K, δ, ε with ε λ 2, ε < δ. Then system 1.1, 1.2 is exponentially stable. Proof. Let Qt, x = V t, xe ε t,. By an argument similar to the one used in the proof of the two theorems above, and taking into consideration conditions 4.9, 4.10, we arrive at Q t, x = V t, xe ε σt, + V t, xe ε t, So by Lemma 4.2, λ 2V t, x + Ke δ t, 1 + µtεe ε t, + εv t, xe ε t, 1 + µtε = λ 2 V t, x + Ke δ t, e ε t, + εv t, xe ε t, = λ 2 + ε V t, xe ε t, + Ke ε δ t, Ke ε δ t,. V t, xe ε t, V, x 0 + K e ε δ τ, τ V, x 0 + K δ1 + B Tε εδ ε =: β x 0,. Hence V t, x β x 0, e ε t,. From 4.9, we get V t, x 1/p β x0, 1/p 1/p x e ε t,. λ 1 λ 1 This completes the proof. Now we will present some examples to illustrate the theory developed above.

11 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 11 Example 4.8. Consider the dynamic equation x t = a x + R x 1/3 e 1/3 δ t,, 4.11 where a < 0, a R +, R > 0 and δ > 0. If there exist positive constants λ 3, K such that δ > λ 3, 1 + µtλ 3 2a + µta R µtar µtr2 λ 3, µtλ R µtar µtr2 K, then system 4.11 is uniformly exponentially stable. To see this, let V t, x = x , we obtain By calculating V t, x along the solutions of V t, x = 2xft, x + µtft, x 2 = 2x ax + Rx 1/3 e 1/3 δ t, = 2a + µta 2 x µtr 2 e 2/3 δ t, x 2/3. + µt ax + Rx 1/3 e 1/3 δ t, 2Re 1/3 δ t, + 2µtaRe 1/3 δ t, 2 x 4/3 Using the Young s inequality wz < we e + zf f with 1 e + 1 f = 1, we have x 4/3 e 1/3 x 4/3 δ t, t 3/2 0 + e1/3 δ t, 3 = 2 3/2 3 3 x e δt,, x 2/3 e 2/3 x 2/3 δ t, t e2/3 δ t, 3/2 = 1 3 3/2 3 x e δt,. Thus V t, x 2a + µta 2 2 x 2 + 2R + 2µtaR 3 x e δt, + µtr x e δt, 2a + µta R µtar µtr2 x R µtar µtr2 e δ t, 1 { = 1 + µtλ 3 2a + µta µtλ 3 3 R µtar µtr2 x µtλ 3 3 R µtar + 2 } 3 µtr2 e δ t,. Under the above assumptions, one can check that conditions of Theorem 4.3 are satisfied. Hence system 4.11 is uniformly exponentially stable.

12 12 A.-L. LIU EJDE-2006/12 In fact, if there exist constants λ 3 > 0, K > 0, such that δ > λ 3, 1 + µ λ 3 2a + µ a R µ ar µ R2 λ 3, µ λ R µ ar µ R2 K, here µ = sup t T µt, µ = inf t T µt, then 4.12 will hold evidently. Case 1: If T = R, then µt = µ = µ = 0 and the conditions in 4.13 reduce to that positive constants λ 3, K need to exist such that δ > λ 3 2a R, 2 3 R K, then system 4.11 is uniformly exponentially stable. Case 2: If T = hz, then µt = µ = µ = h. The conditions in 4.13 reduce to that there exist λ 3 > 0, K > 0 such that δ > λ 3, 2a + ha R har hr2 λ 3 /1 + hλ 3, 2 3 R har hr2 K/1 + hλ 3, then system 4.11 is uniformly exponentially stable. Case 3: When T = k=0 [kl + h, kl + h + l], here l, h are positive constants, this kind of time scales could exactly describe many phenomena which are common in nature, such as the life span of certain species and the change of electric circuit with time progressing etc. [1, 15]. At this time, µ = 0, µ = h. If there exist constants λ 3 > 0, K > 0, such that λ 3 < δ, 2a + ha R R2 h λ 3, 1 + hλ R R2 h K, then 4.13 will hold. So system 4.11 is uniformly exponentially stable. Example 4.9. Consider the system x 1 = ax 1 + ax 2, 4.14 x 2 = ax 1 ax 2, 4.15 x 1, x 2 = c, d, 4.16 for certain constants a > 0, a R +, and c, d are any constants. If there is a constant λ 2 > 0 such that for all t T + then system is exponentially stable. λ 2 /1 + λ 2 µt 2a1 aµt, 4.17

13 EJDE-2006/12 BOUNDEDNESS AND EXPONENTIAL STABILITY 13 Proof. We will show that, under the above assumptions, the conditions of Theorem 4.7 are satisfied. Choose D = R 2, V t, x = x 2 = x x 2 2, so 4.9 holds. From [7], we have V t, x = 2xft, x + µt ft, x 2 = 2a1 aµt x 2 λ 2V t, x 1 + λ 2 µt λ 2V t, x + Ke δ t, 1 + λ 2 µt = λ 2V t, x + Ke δ t,. 1 + εµt Here we let ε = λ 2, δ > ε > 0, and K be arbitrary positive constant, so 4.10 holds. Therefore, system is exponentially stable. In fact, if there exists a constant λ 2 > 0, such that 2a1 a µ 1 + λ 2 µ λ 2, 4.18 then condition 4.17 will hold. Case 1: If T = R, then µt = µ = µ = 0 and 4.18 will hold to be true if there exists a constant λ 2 satisfying 0 < λ 2 2a. So system is exponentially stable. Case 2: If T = hz, then µt = µ = µ = h. If we can find a constant λ 2 > 0 such that λ 2 2a1 ah, 1 + hλ 2 then condition 4.18 would hold. So system is exponentially stable. Case 3: If T = k=0 [kl + h, kl + h + l] as in the above example, then µ = 0, µ = h. The condition 4.18 reduces to that a constant λ 2 exists such that 0 < λ 2 2a1 ah, so that system is exponentially stable. References [1]. Bohner, A. C. Peterson. Dynamic Equations on Time Scales. Birkhäuser, Boston, [2]. Bohner, A. C. Peterson. Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, [3] P. Hartman. Ordinary Differential Equations, Classics in Applied athematics, 38. Society for Industrial and Applied athematics SIA, Philadelphia, PA, [4] J. Hoffacker, C. C. Tisdell. Stability and instability for dynamic equations on time scales. Comput. ath. Appl., 2005, , [5] S. Hilger. Analysis on easure Chains - A Unified Approach to Continuous and Discrete Calculus. Res. ath., 1990, 18, [6] Y. N. Raffoul. Boundedness in Nonlinear Differential Equations, Nonlinear Studies, 2003, 10, [7] A. C. Peterson, C. C. Tisdell. Boundedness and Uniqueness of Solutions to Dynamic Equations on Time Scales. Journal of Difference Equations and Applications, ,

14 14 A.-L. LIU EJDE-2006/12 [8] C. Pötzsche, S. Siegmund, F. Wirth. A Spectral Characterization of Exponential Stability for Linear Time-invariant Systems on Time Scales, Discrete and Continuous Dynamical System, 2003, 95, [9] J. J. Dacunha. Stability for Time-varying Linear Dynamic Systems on Time Scales. Journal of Computational and Applied athematics, 2005, 1762, [10]. N. Islam, Y. N. Raffoul. Exponential Stability in Nonlinear Difference Equations. Difference Equations and Its Applications, special session in Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA, ay 24-27, [11] N.. Linh, V.N. Phat. Exponential Stability of Nonlinear Time-varying Differential Equations and Applications, Electronic Journal of Differential Equations, 2001, , [12] Y.J. Sun,C.H. Lien, J.G. Hsieh. Global Exponential Stabilization for a Class of Uncertain Nonlinear Systems with Control Constraints. IEEE Trans. Aut. Contr., 1998, 43, [13] B.. Garay. Embeddability of Time Scale Dynamics in ODE Dynamics. Nonlinear Analysis, 2001, 47, [14] C. Pötzsche. Chain Rule and Invariance Principle on easure Chains. Journal of Computational and Applied athematics, 2002, 141, [15] R. Agarwal,. Bohner, D. O Regan, A. C. Peterson. Dynamic equations on Time Scales: A Survey. Journal of Computational and Applied athematics, 2002, 141, Ai-Lian Liu Department of Statistics and athematics, Shandong Economics University, Jinan , China address: ailianliu2002@163.com