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Asymptotic stability for dynamic equations on time scales

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Abstract

We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales. To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.

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References

  1. 1.

    Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. In Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Math. Res.. Volume 59. Akademie, Berlin; 1990:9–20.

  2. 2.

    Birkhoff JD: Quantum mechanics and asymptotic series. Bulletin of the American Mathematical Society 1933, 32: 681–700.

  3. 3.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.

  4. 4.

    Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2002.

  5. 5.

    Gard T, Hoffacker J: Asymptotic behavior of natural growth on time scales. Dynamic Systems and Applications 2003,12(1–2):131–147.

  6. 6.

    Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.

  7. 7.

    Hoffacker J, Tisdell CC: Stability and instability for dynamic equations on time scales. Computers & Mathematics with Applications 2005,49(9–10):1327–1334. 10.1016/j.camwa.2005.01.016

  8. 8.

    Hovhannisyan G: Asymptotic stability for second-order differential equations with complex coefficients. Electronic Journal of Differential Equations 2004,2004(85):1–20.

  9. 9.

    Hovhannisyan G: Asymptotic stability and asymptotic solutions of second-order differential equations. to appear in Journal of Mathematical Analysis and Applications

  10. 10.

    Levinson N: The asymptotic nature of solutions of linear systems of differential equations. Duke Mathematical Journal 1948,15(1):111–126. 10.1215/S0012-7094-48-01514-2

  11. 11.

    Peterson AC, Raffoul YN: Exponential stability of dynamic equations on time scales. Advances in Difference Equations 2005,2005(2):133–144. 10.1155/ADE.2005.133

  12. 12.

    Pötzsche C, Siegmund S, Wirth F: A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical Systems 2003,9(5):1223–1241.

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Correspondence to Gro Hovhannisyan.

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Keywords

  • Differential Equation
  • Error Estimate
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis