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Advances in Difference Equations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Stability criteria for linear Hamiltonian dynamic systems on time scales Advances in Difference Equations 2011, 2011:63 doi:10.1186/1687-1847-2011-63 Xiaofei He (hexiaofei525@sina.com) Xianhua Tang (tangxh@mail.csu.edu.cn) Qi-Ming Zhang (zhqm20082008@sina.com) ISSN 1687-1847 Article type Research Submission date 5 August 2011 Acceptance date 20 December 2011 Publication date 20 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/63 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 He et al. ; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Stability criteria for linear Hamiltonian dynamic systems on time scales Xiaofei He1,2 , Xianhua Tang ∗1 and Qi-Ming Zhang1 1 School of Mathematical Sciences and Computing Technology, Central South University, Changsha 410083, Hunan, P.R. China 2 College of Mathematics and Computer Science, Jishou University, Jishou 416000, Hunan, P.R.China ∗ Corresponding author: tangxh@mail.csu.edu.cn Email address: XH: hexiaofei525@sina.com Q-MZ: zhqm20082008@sina.com Abstract In this article, we establish some stability criteria for the polar linear Hamiltonian dynamic system on time scales x (t) = α(t)x(σ (t))+ β (t)y (t), y (t) = −γ (t)x(σ (t)) − α(t)y (t), t∈T by using Floquet theory and Lyapunov-type inequalities. 2000 Mathematics Subject Classiﬁcation: 39A10. 1
Keywords: Hamiltonian dynamic system; Lyapunov-type inequality; Floquet theory; stability; time scales. 1 Introduction A time scale is an arbitrary nonempty closed subset of the real numbers R. We assume that T is a time scale. For t ∈ T, the forward jump operator σ : T → T is deﬁned by σ (t) = inf {s ∈ T : s > t}, the backward jump operator ρ : T → T is deﬁned by ρ(t) = sup{s ∈ T : s < t}, and the graininess function µ : T → [0, ∞) is deﬁned by µ(t) = σ (t) − t. For other related basic concepts of time scales, we refer the reader to the original studies by Hilger [1–3], and for further details, we refer the reader to the books of Bohner and Peterson [4, 5] and Kaymakcalan et al. [6]. Deﬁnition 1.1. If there exists a positive number ω ∈ R such that t + nω ∈ T for all t ∈ T and n ∈ Z, then we call T a periodic time scale with period ω . Suppose T is a ω -periodic time scale and 0 ∈ T. Consider the polar linear Hamiltonian dynamic system on time scale T x (t) = α(t)x(σ (t)) + β (t)y (t), y (t) = −γ (t)x(σ (t)) − α(t)y (t), t ∈ T, (1.1) where α(t), β (t) and γ (t) are real-valued rd-continuous functions deﬁned on T. Throughout this article, we always assume that 1 − µ(t)α(t) > 0, ∀t∈T (1.2) 2
and β (t) ≥ 0, ∀ t ∈ T. (1.3) For the second-order linear dynamic equation [p(t)x (t)] + q (t)x(σ (t)) = 0, t ∈ T, (1.4) if let y (t) = p(t)x (t), then we can rewrite (1.4) as an equivalent polar linear Hamiltonian dynamic system of type (1.1): 1 x (t) = y (t), y (t) = −q (t)x(σ (t)), t ∈ T, (1.5) p(t) where p(t) and q (t) are real-valued rd-continuous functions deﬁned on T with p(t) > 0, and 1 α(t) = 0, β (t) = , γ (t) = q (t). p(t) Recently, Agarwal et al. [7], Jiang and Zhou [8], Wong et al. [9] and He et al. [10] established some Lyapunov-type inequalities for dynamic equations on time scales, which generalize the corresponding results on diﬀerential and diﬀer- ence equations. Lyapunov-type inequalities are very useful in oscillation theory, stability, disconjugacy, eigenvalue problems and numerous other applications in the theory of diﬀerential and diﬀerence equations. In particular, the stability criteria for the polar continuous and discrete Hamiltonian systems can be ob- tained by Lyapunov-type inequalities and Floquet theory, see [11–16]. In 2000, Atici et al. [17] established the following stablity criterion for the second-order linear dynamic equation (1.4): 3
Theorem 1.2 [17]. Assume p(t) > 0 for t ∈ T, and that p(t + ω ) = p(t), q (t + ω ) = q (t), ∀ t ∈ T. (1.6) If ω q (t) t ≥ 0, q (t) ≡ 0 (1.7) 0 and ω ω 1 q + (t) t ≤ 4, p0 + t (1.8) p(t) 0 0 then equation (1.4) is stable, where σ (t) − t q + (t) = max{q (t), 0}, p0 = max , (1.9) p(t) t∈[0,ρ(ω )] where and in the sequel, system (1.1) or Equation (1.4) is said to be unstable if all nontrivial solutions are unbounded on T; conditionally stable if there exist a nontrivial solution which is bounded on T; and stable if all solutions are bounded on T. In this article, we will use the Floquet theory in [18, 19] and the Lyapunov- type inequalities in [10] to establish two stability criteria for system (1.1) and equation (1.4). Our main results are the following two theorems. Theorem 1.3. Suppose (1.2) and (1.3) hold and α(t + ω ) = α(t), β (t + ω ) = β (t), γ (t + ω ) = γ (t), ∀ t ∈ T. (1.10) Assume that there exists a non-negative rd-continuous function θ(t) deﬁned on T such that |α(t)| ≤ θ(t)β (t), ∀ t ∈ T[0, ω ] = [0, ω ] ∩ T, (1.11) 4
ω γ (t) − θ2 (t)β (t) t > 0, (1.12) 0 and 1 /2 ω ω ω γ + (t) t |α(t)| t + β (t) t < 2. (1.13) 0 0 0 Then system (1.1) is stable. Theorem 1.4. Assume that (1.6) and (1.7) hold, and that ω ω 1 q + (t) t ≤ 4. t (1.14) p(t) 0 0 Then equation (1.4) is stable. Remark 1.5. Clearly, condition (1.14) improves (1.8) by removing term p0 . We dwell on the three special cases as follows: 1. If T = R, system (1.1) takes the form: x (t) = α(t)x(t) + β (t)y (t), y (t) = −γ (t)x(t) − α(t)y (t), t ∈ R. (1.15) In this case, the conditions (1.12) and (1.13) of Theorem 1.3 can be transformed into ω γ (t) − θ2 (t)β (t) dt > 0, (1.16) 0 and 1 /2 ω ω ω γ + (t)dt |α(t)|dt + β (t)dt < 2. (1.17) 0 0 0 Condition (1.17) is the same as (3.10) in [12], but (1.11) and (1.16) are better than (3.9) in [12] by taking θ(t) = |α(t)|/β (t). A better condition than (1.17) can be found in [14, 15]. 5
2. If T = Z, system (1.1) takes the form: x(n) = α(n)x(n+1)+β (n)y (n), y (n) = −γ (n)x(n+1)−α(n)y (n), n ∈ Z. (1.18) In this case, the conditions (1.11), (1.12), and (1.13) of Theorem 1.3 can be transformed into |α(n)| ≤ θ(n)β (n), ∀ n ∈ {0, 1, . . . , ω − 1}, (1.19) ω −1 γ (n) − θ2 (n)β (n) > 0, (1.20) n=0 and 1 /2 ω −1 ω −1 ω −1 + |α(n)| + β (n) γ (n) < 2. (1.21) n=0 n=0 n=0 Conditions (1.19), (1.20), and (1.21) are the same as (1.17), (1.18) and (1.19) in [16], i.e., Theorem 1.3 coincides with Theorem 3.4 in [16]. However, when p(n) and q (n) are ω -periodic functions deﬁned on Z, the stability conditions ω −1 ω −1 4 q + (n) ≤ 0≤ q (n) ≤ , q (n) ≡ 0, ∀ n ∈ {0, 1, . . . , ω − 1} ω −1 1 n=0 p(n) n=0 n=0 (1.22) in Theorem 1.4 are better than the one ω −1 ω −1 4 q + (n) < 0< q ( n) ≤ (1.23) ω −1 1 n=0 p(n) n=0 n=0 in [16, Corollary 3.4]. More related results on stability for discrete linear Hamil- tonian systems can be found in [20–24]. 3. Let δ > 0 and N ∈ {2, 3, 4, . . .}. Set ω = δ + N , deﬁne the time scale T as follows: T= [kω, kω + δ ] ∪ {kω + δ + n : n = 1, 2, . . . , N − 1}. (1.24) k ∈Z 6
Then system (1.1) takes the form: x (t) = α(t)x(t) + β (t)y (t), y (t) = −γ (t)x(t) − α(t)y (t), t∈ [kω, kω + δ ), k ∈Z (1.25) and x(t) = α(t)x(t + 1) + β (t)y (t), y (t) = −γ (t)x(t + 1) − α(t)y (t), t∈ {kω + δ + n : n = 0, 1, . . . , N − 2}. (1.26) k ∈Z In this case, the conditions (1.11), (1.12), and (1.13) of Theorem 1.3 can be transformed into |α(t)| ≤ θ(t)β (t), ∀ t ∈ [0, δ ] ∪ {δ + 1, δ + 2, . . . , δ + N − 1}, (1.27) N −1 δ γ (t) − θ2 (t)β (t) dt + γ (δ + n) − θ2 (δ + n)β (δ + n) > 0, (1.28) 0 n=0 and N −1 δ |α(t)|dt + |α(δ + n)| 0 n=0 1/ 2 N −1 N −1 δ δ + + + β (t)dt + |β (δ + n)| γ (t)dt + |γ (δ + n)| < 2. 0 0 n=0 n=0 (1.29) 2 Proofs of theorems Let u(t) = (x(t), y (t)) , uσ (t) = (x(σ (t)), y (t)) and    α(t) β (t)  A(t) =  .   −γ (t) −α(t) 7
Then, we can rewrite (1.1) as a standard linear Hamiltonian dynamic system u (t) = A(t)uσ (t), t ∈ T. (2.1) Let u1 (t) = (x10 (t), y10 (t)) and u2 (t) = (x20 (t), y20 (t)) be two solutions of system (1.1) with (u1 (0), u2 (0)) = I2 . Denote by Φ(t) = (u1 (t), u2 (t)). Then Φ(t) is a fundamental matrix solution for (1.1) and satisﬁes Φ(0) = I2 . Sup- pose that α(t), β (t) and γ (t) are ω -periodic functions deﬁned on T (i.e. (1.10) holds), then Φ(t + ω ) is also a fundamental matrix solution for (1.1) ( see [18]). Therefore, it follows from the uniqueness of solutions of system (1.1) with initial condition ( see [9, 18, 19]) that Φ(t + ω ) = Φ(t)Φ(ω ), ∀ t ∈ T. (2.2) From (1.1), we have x10 (t) x20 (t) x10 (σ (t)) x20 (σ (t)) (det Φ(t)) = + = 0, ∀ t ∈ T. y10 (t) y20 (t) y10 (t) y20 (t) (2.3) It follows that det Φ(t) = det Φ(0) = 1 for all t ∈ T. Let λ1 and λ2 be the roots (real or complex) of the characteristic equation of Φ(ω ) det(λI2 − Φ(ω )) = 0, which is equivalent to λ2 − Hλ + 1 = 0, (2.4) where H = x10 (ω ) + y20 (ω ). 8
Hence λ1 + λ2 = H, λ 1 λ2 = 1 . (2.5) Let v1 = (c11 , c21 ) and v2 = (c12 , c22 ) be the characteristic vectors associated with the characteristic roots λ1 and λ2 of Φ(ω ), respectively, i.e. Φ(ω )vj = λj vj , j = 1, 2. (2.6) Let vj (t) = Φ(t)vj , j = 1, 2. Then it follows from (2.2) and (2.6) that vj (t + ω ) = λj vj (t), ∀ t ∈ T, j = 1, 2. (2.7) On the other hand, it follows from (2.1) that vj (t) = Φ (t)vj = u1 (t), u2 (t) vj A(t) (uσ (t), uσ (t)) vj = 1 2 σ = A(t)vj (t), j = 1, 2. (2.8) This shows that v1 (t) and v2 (t) are two solutions of system (1.1) which satisfy (2.7). Hence, we obtain the following lemma. Lemma 2.1. Let Φ(t) be a fundamental matrix solution for (1.1) with Φ(0) = I2 , and let λ1 and λ2 be the roots (real or complex) of the characteristic equation (2.4) of Φ(ω ). Then system (1.1) has two solutions v1 (t) and v2 (t) which satisfy (2.7). Similar to the continuous case, we have the following lemma. 9
Lemma 2.2. System (1.1) is unstable if |H | > 2, and stable if |H | < 2. Instead of the usual zero, we adopt the following concept of generalized zero on time scales. Deﬁnition 2.3. A function f : T → R is said to have a generalized zero at t0 ∈ T provided either f (t0 ) = 0 or f (t0 )f (σ (t0 )) < 0. Lemma 2.4. [4] Assume f, g : T → R are diﬀerential at t ∈ Tk . If f (t) exists, then f (σ (t)) = f (t) + µ(t)f (t). Lemma 2.5. [4] (Cauchy-Schwarz inequality). Let a, b ∈ T. For f, g ∈ Crd we have 1 b b b 2 2 2 |f (t)g (t)| t≤ f (t) t · g (t) t . a a a The above inequality can be equality only if there exists a constant c such that f (t) = cg (t) for t ∈ T[a, b]. Lemma 2.6. Let v1 (t) = (x1 (t), y1 (t)) and v2 (t) = (x2 (t), y2 (t)) be two solutions of system (1.1) which satisfy (2.7). Assume that (1.2), (1.3) and (1.10) hold, and that exists a non-negative function θ(t) such that (1.11) and (1.12) hold. If H 2 ≥ 4, then both x1 (t) and x2 (t) have generalized zeros in T[0, ω ]. Proof. Since |H | ≥ 2, then λ1 and λ2 are real numbers, and v1 (t) and v2 (t) are also real functions. We only prove that x1 (t) must have at least one generalized zero in T[0, ω ]. Otherwise, we assume that x1 (t) > 0 for t ∈ T[0, ω ] and so (2.7) 10
implies that x1 (t) > 0 for t ∈ T. Deﬁne z (t) := y1 (t)/x1 (t). Due to (2.7), one sees that z (t) is ω -periodic, i.e. z (t + ω ) = z (t), ∀ t ∈ T. From (1.1), we have x1 (t)y1 (t) − x1 (t)y1 (t) z (t) = x1 (t)x1 (σ (t)) 2 −γ (t)x1 (t)x1 (σ (t)) − α(t)[x1 (t) + x1 (σ (t))]y1 (t) − β (t)y1 (t) = x1 (t)x1 (σ (t)) y1 (t) y1 (t) y1 (t) y1 (t) = −γ (t) − α(t) + − β (t) x1 (t) x1 (σ (t)) x1 (t) x1 (σ (t)) y1 (t) y1 (t) = −γ (t) − α(t) z (t) + − β (t)z (t) . (2.9) x1 (σ (t)) x1 (σ (t)) From the ﬁrst equation of (1.1), and using Lemma 2.4, we have [1 − µ(t)α(t)]x1 (σ (t)) = x1 (t) + µ(t)β (t)y1 (t), t ∈ T. (2.10) Since x1 (t) > 0 for all t ∈ T, it follows from (1.2) and (2.10) that y1 (t) x1 (σ (t)) 1 + µ(t)β (t)z (t) = 1 + µ(t)β (t) = [1 − µ(t)α(t)] > 0, (2.11) x1 (t) x1 (t) which yields y1 (t) [1 − µ(t)α(t)]z (t) = . (2.12) x1 (σ (t)) 1 + µ(t)β (t)z (t) Substituting (2.12) into (2.9), we obtain [−2α(t) + µ(t)α2 (t)]z (t) − β (t)z 2 (t) z (t) = −γ (t) + . (2.13) 1 + µ(t)β (t)z (t) If β (t) > 0, together with (1.11), it is easy to verify that [−2α(t) + µ(t)α2 (t)]z (t) − β (t)z 2 (t) α2 (t) ≤ θ2 (t)β (t); ≤ (2.14) 1 + µ(t)β (t)z (t) β (t) If β (t) = 0, it follows from (1.11) that α(t) = 0, hence [−2α(t) + µ(t)α2 (t)]z (t) − β (t)z 2 (t) = 0 = θ2 (t)β (t). (2.15) 1 + µ(t)β (t)z (t) 11
Combining (2.14) with (2.15), we have [−2α(t) + µ(t)α2 (t)]z (t) − β (t)z 2 (t) ≤ θ2 (t)β (t). (2.16) 1 + µ(t)β (t)z (t) Substituting (2.16) into (2.13), we obtain z (t) ≤ −γ (t) + θ2 (t)β (t). (2.17) Integrating equation (2.17) from 0 to ω , and noticing that z (t) is ω -periodic, we obtain ω γ (t) − θ2 (t)β (t) 0≤− t, 0 which contradicts condition (1.12). Lemma 2.7. Let v1 (t) = (x1 (t), y1 (t)) and v2 (t) = (x2 (t), y2 (t)) be two solutions of system (1.1) which satisfy (2.7). Assume that α(t) = 0, β (t) > 0, γ (t) ≡ 0, ∀ t ∈ T, (2.18) β (t + ω ) = β (t), γ (t + ω ) = γ (t), ∀ t ∈ T, (2.19) and ω γ (t) t ≥ 0. (2.20) 0 If H 2 ≥ 4, then both x1 (t) and x2 (t) have generalized zeros in T[0, ω ]. Proof. Except (1.12), (2.18), and (2.19) imply all assumptions in Lemma 2.6 hold. In view of the proof of Lemma 2.6, it is suﬃcient to derive an inequality which contradicts (2.20) instead of (1.12). From (2.11), (2.13), and (2.18), we have y1 (t) x1 (σ (t)) 1 + µ(t)β (t)z (t) = 1 + µ(t)β (t) = >0 (2.21) x1 (t) x1 (t) 12
and β (t)z 2 (t) z (t) = −γ (t) − . (2.22) 1 + µ(t)β (t)z (t) Since z (t) is ω -periodic and γ (t) ≡ 0, it follows from (2.22) that z 2 (t) ≡ 0 on T[0, ω ]. Integrating equation (2.22) from 0 to ω , we obtain ω ω β (t)z 2 (t) 0=− γ (t) + t 4. β (t) t (2.24) a a Proof. In view of the proof of [10, Theorem 3.5] (see (3.8), (3.29)–(3.34) in [10]), we have x(a) = −ξµ(a)β (a)y (a), (2.25) 13
τ x(τ ) = (1 − ξ )µ(a)β (a)y (a) + β (t)y (t) t, σ (a) ≤ τ ≤ b, (2.26) σ (a) b b ϑ1 µ(a)β (a)y 2 (a) + β (t)y 2 (t) t = γ (t)x2 (σ (t)) t, (2.27) σ (a) a and b 2|x(τ )| ≤ ϑ2 µ(a)β (a)|y (a)| + β (t)|y (t)| t, σ (a) ≤ τ ≤ b, (2.28) σ (a) where ξ ∈ [0, 1), and ϑ1 = 1 − ξ + κ2 ξ, ϑ2 = 1 − ξ + |κ|ξ. (2.29) Let |x(τ ∗ )| = maxσ(a)≤τ ≤b |x(τ )|. There are three possible cases: (1) y (t) ≡ y (a) = 0, ∀ t ∈ T[a, b]; (2) y (t) ≡ y (a), |y (t)| ≡ |y (a)|, ∀ t ∈ T[a, b]; (3) |y (t)| ≡ |y (a)|, ∀ t ∈ T[a, b]. Case (1). In this case, κ = 1. It follows from (2.25) and (2.26) that b x(b) = (1 − ξ )µ(a)β (a)y (a) + β (t)y (t) t σ (a) b = y (a) (1 − ξ )µ(a)β (a) + β (t) t σ (a) b = x(a) + y (a) β (t) t a = x(a), which contradicts the assumption that x(b) = κx(a) = x(a). Case (2). In this case, we have b 2|x(τ )| < ϑ2 µ(a)β (a)|y (a)| + β (t)|y (t)| t, σ (a) ≤ τ ≤ b (2.30) σ (a) 14
instead of (2.28). Applying Lemma 2.5 and using (2.27) and (2.30), we have 2|x(τ ∗ )| b < ϑ2 µ(a)β (a)|y (a)| + β (t)|y (t)| t σ (a) 1/ 2 b b ϑ2 2 ϑ1 µ(a)β (a)y 2 (a) + β (t)y 2 (t) t ≤ µ(a)β (a) + β (t) t ϑ1 σ (a) σ ( a) 1 /2 b b ϑ2 2 γ (t)x2 (σ (t)) t = µ(a)β (a) + β (t) t ϑ1 σ (a) a 1/ 2 b b ϑ2 2 ∗ + ≤ |x(τ )| µ(a)β (a) + β (t) t γ (t) t . (2.31) ϑ1 σ (a) a Dividing the latter inequality of (2.31) by |x(τ ∗ )|, we obtain 1 /2 b b ϑ2 2 γ + (t) t µ(a)β (a) + β (t) t > 2. (2.32) ϑ1 σ ( a) a Case (3). In this case, applying Lemma 2.5 and using (2.27) and (2.28), we have 2|x(τ ∗ )| b ≤ ϑ2 µ(a)β (a)|y (a)| + β (t)|y (t)| t σ (a) 1/ 2 b b ϑ2 2 2 2 < µ(a)β (a) + β (t) t ϑ1 µ(a)β (a)y (a) + β (t)y (t) t ϑ1 σ (a) σ ( a) 1 /2 b b ϑ2 2 2 = µ(a)β (a) + β (t) t γ (t)x (σ (t)) t ϑ1 σ (a) a 1/ 2 b b ϑ2 2 ∗ + ≤ |x(τ )| µ(a)β (a) + β (t) t γ (t) t . (2.33) ϑ1 σ (a) a Dividing the latter inequality of (2.33) by |x(τ ∗ )|, we also obtain (2.32). It is easy to verify that ϑ2 [1 − ξ + |κ|ξ ]2 2 = ≤ 1. 1 − ξ + κ2 ξ ϑ1 15
Substituting this into (2.32), we obtain (2.24). Proof of Theorem 1.3. If |H | ≥ 2, then λ1 and λ2 are real numbers and λ1 λ2 = 1, it follows that 0 < min{λ2 , λ2 } ≤ 1. Suppose λ2 ≤ 1. By Lemma 1 2 1 2.6, system (1.1) has a non-zero solution v1 (t) = (x1 (t), y1 (t)) such that (2.7) holds and x1 (t) has a generalized zero in T[0, ω ], say t1 . It follows from (2.7) that (x1 (t1 + ω ), y1 (t1 + ω )) = λ1 (x1 (t1 ), y1 (t1 )). Applying Lemma 2.8 to the solution (x1 (t), y1 (t)) with a = t1 , b = t1 + ω and κ1 = κ2 = λ1 , we get 1 /2 t1 +ω t1 +ω t1 +ω γ + (t) t |α(t)| t + β (t) t ≥ 2. (2.34) t1 t1 t1 Next, noticing that for any ω -periodic function f (t) on T, the equality t0 + ω ω f (t) t = f (t) t t0 0 holds for all t0 ∈ T. It follows from (3.1) that 1 /2 ω ω ω γ + (t) t |α(t)| t + β (t) t ≥ 2. (2.35) 0 0 0 which contradicts condition (1.13). Thus |H | < 2 and hence system (1.1) is stable. Proof of Theorem 1.4. By using Lemmas 2.7 and 2.9 instead of Lemmas 2.6 and 2.8, respectively, we can prove Theorem 1.4 in a similar fashion as the proof of Theorem 1.3. So, we omit the proof here. Competing interests The authors declare that they have no competing interests. 16
Authors’ contributions XH carried out the theoretical proof and drafted the manuscript. Both XT and QZ participated in the design and coordination. All authors read and approved the ﬁnal manuscript. Acknowledgments The authors thank the referees for valuable comments and suggestions. This project is supported by Scientiﬁc Research Fund of Hunan Provincial Educa- tion Department (No. 11A095) and partially supported by the NNSF (No: 11171351) of China. References [1] S Hilger, Einßmakettenkalk¨l mit Anwendung auf Zentrumsmannig- u faltigkeiten. Ph.D. Thesis, Universit¨t W¨rzburg, 1988 (in German) a u [2] S Hilger, Analysis on measure chain—A uniﬁed approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990) [3] S Hilger, Diﬀerential and diﬀerence calculus-uniﬁed. Nonlinear Anal. 30, 2683–2694 (1997) [4] M Bohner, A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkh¨user, Boston, 2001) a 17
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