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ASYMPTOTIC EQUIVALENCE BETWEEN TWO LINEAR DYNAMIC SYSTEMS ON TIME SCALES

  • Received : 2013.08.17
  • Published : 2014.07.31

Abstract

In this paper we investigate asymptotic properties about asymptotic equilibrium and asymptotic equivalence for linear dynamic systems on time scales by using the notion of $u_{\infty}$-similarity. Also, we give some examples to illustrate our results.

Keywords

asymptotic equivalence;asymptotic equilibrium;$u_{\infty}$-similarity;strong stability;linear dynamic systems;time scales

References

  1. V. Cormani, Liouville's formula on time scales, Dynam. Systems Appl. 12 (2003), no. 1-2, 79-86.
  2. I. Gohberg, M. A. Kaashoek, and J. Kos, Classification of linear time-varying difference equations under kinematic similarity, Integral Equations Operator Theory 25 (1996), no. 4, 445-480. https://doi.org/10.1007/BF01203027
  3. S. Hilger, Analysis on measure chains - a unified approach to continous and discrete calculus, Results Math. 18 (1990), no. 1-2, 18-56. https://doi.org/10.1007/BF03323153
  4. V. Lakshmikantham and S. Leela, Differential and Integral Inequalites with Theory and Applications, Academic Press, New York and London, 1969.
  5. L. Markus, Continuous matrices and the stability of differential systems, Math. Z. 62 (1955), 310-319. https://doi.org/10.1007/BF01180637
  6. W. F. Trench, On $t_{\infty}$ quasi-similarity of linear systems, Ann. Mat. Pura Appl. 142 (1985), 293-302. https://doi.org/10.1007/BF01766598
  7. W. F. Trench, Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems, Comput. Math. Appl. 36 (1998), no. 10-12, 261-267. https://doi.org/10.1016/S0898-1221(98)80027-8
  8. B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in "Qualitative Theory of Differential Equations"(Szeged, 1988), 37-56, Colloq. Math. Soc. Janos Bolyai, 53, North Holland, Amsterdam, 1990.
  9. B. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, in "Nonlinear Dynamics and Quantum Dynamical Systems"(Gaussig, 1990), 9-20, Math. Res. 59, Akademie-Verlag, Berlin, 1990.
  10. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
  11. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
  12. S. K. Choi, Y. H. Goo, and N. J. Koo, Variationally asymptotically stable difference systems, Adv. Difference Equ. 2007 (2007), Article ID 35378, 21 pp.
  13. S. K. Choi, Y. H. Goo, and N. Koo, h-Stability of dynamic equations on time scales with nonregressivity, Abstr. Appl. Anal. 2008 (2008), Article ID 632473, 13 pp.
  14. S. K. Choi and N. J. Koo, Variationally stable difference systems by $n_{\infty}$-similarity, J. Math. Anal. Appl. 249 (2000), no. 2, 553-568. https://doi.org/10.1006/jmaa.2000.6910
  15. S. K. Choi and N. Koo, On the stability of linear dynamic systems on time scales, J. Difference Equ. Appl. 15 (2009), no. 2, 167-183. https://doi.org/10.1080/10236190802008528
  16. S. K. Choi, N. J. Koo, and S. Dontha, Asymptotic property in variation for nonlinear differential systems, Appl. Math. Lett. 18 (2005), no. 1, 117-126. https://doi.org/10.1016/j.aml.2003.07.019
  17. S. K. Choi, N. J. Koo, and D. M. Im, Asymptotic equivalence between linear differential systems, Bull. Korean Math. Soc. 42 (2005), no. 4, 691-701. https://doi.org/10.4134/BKMS.2005.42.4.691
  18. S. K. Choi, N. J. Koo, and K. Lee, Asymptotic equivalence for linear differential systems, Commun. Korean Math. Soc. 26 (2011), no. 1, 37-49. https://doi.org/10.4134/CKMS.2011.26.1.037
  19. R. Conti, Sulla $t_{\infty}$-similitudine tra matrici e la stabilita dei sistemi differenziali lineari, Atti. Acc. Naz. Lincei, Rend. Cl. Fis. Mat. Nat. 49 (1955), 247-250.