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  • Research Article
  • Open Access

Lyapunov functions for linear nonautonomous dynamical equations on time scales

Advances in Difference Equations20062006:069106

  • Received: 25 January 2006
  • Accepted: 13 April 2006
  • Published:


The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation


Authors’ Affiliations

Institut für Mathematik, Johann Wolfgang Goethe Universität, Frankfurt am Main 60054, Germany
Institut für Mathematik, Technische Universität Berlin, Berlin, 10623, Germany


  1. Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 155. Marcel Dekker, New York; 1992:xiv+777.Google Scholar
  2. Agarwal RP, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002,141(1–2):1–26. 10.1016/S0377-0427(01)00432-0MathSciNetView ArticleMATHGoogle Scholar
  3. 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.Google Scholar
  4. Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  5. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  6. Conway JB: Functions of One Complex Variable I, Graduate Texts in Mathematics. Volume 11. 2nd edition. Springer, New York; 1978:xiii+317.View ArticleGoogle Scholar
  7. Döffinger A: Theorie dynamischer Gleichungen—ein einheitlicher Zugang zur kontinuierlichen und diskreten Dynamik, Diplomarbeit. Universität Augsburg, Augsburg; 1995.Google Scholar
  8. Hilger S: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Dissertation. Universität Würzburg, Würzburg; 1988.MATHGoogle Scholar
  9. Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.MathSciNetView ArticleMATHGoogle Scholar
  10. Hilger S: Special functions, Laplace and Fourier transform on measure chains. Dynamic Systems and Applications 1999,8(3–4):471–488.MathSciNetMATHGoogle Scholar
  11. Keller S: Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Dissertation. Universität Augsburg, Augsburg; 1999.Google Scholar
  12. Kloeden PE, Khilger S: The effect of time granularity on the asymptotic stability of dynamical systems. Automation and Remote Control 1994,55(9, part 1):1293–1298 (1995).MathSciNetGoogle Scholar
  13. Pötzsche C, Siegmund S, Wirth F: A spectral characterisation of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical Systems 2002, 9: 255–265.Google Scholar
  14. Remmert R: Funktionentheorie. Springer, Berlin; 1995.MATHGoogle Scholar
  15. Yoshizawa T: Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, no. 9. The Mathematical Society of Japan, Tokyo; 1966:viii+223.Google Scholar
  16. Zmorzynska A: Lyapunovfunktionen auf Zeitskalen, Diplomarbeit. J. W. Goethe Universität, Frankfurt am Main; 2004.Google Scholar