Báo cáo sinh học: Stability criteria for linear Hamiltonian dynamic systems on time scales

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Báo cáo sinh học: " Stability criteria for linear Hamiltonian dynamic systems on time scales"Advances in DifferenceEquations 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 Zhang11 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 IntroductionA time scale is an arbitrary nonempty closed subset of the real numbers R. Weassume that T is a time scale. For t ∈ T, the forward jump operator σ : T → Tis defined by σ (t) = inf {s ∈ T : s > t}, the backward jump operator ρ : T → T isdefined 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, werefer 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 Kaymakcalanet al. [6].Definition 1.1. If there exists a positive number ω ∈ R such that t + nω ∈ Tfor 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 linearHamiltonian 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) 2and 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, ...