# ASYMPTOTIC EQUIVALENCE FOR LINEAR DIFFERENTIAL SYSTEMS

• Choi, Sung-Kyu (DEPARTMENT OF MATHEMATICS CHUNGNAM NATIONAL UNIVERSITY) ;
• Koo, Nam-Jip (DEPARTMENT OF MATHEMATICS CHUNGNAM NATIONAL UNIVERSITY) ;
• Lee, Keon-Hee (DEPARTMENT OF MATHEMATICS CHUNGNAM NATIONAL UNIVERSITY)
• Published : 2011.01.31

#### Abstract

We investigate the asymptotic equivalence for linear differential systems by means of the notions of $t_{\infty}$-similarity and strong stability.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. G. Ascoli, Osservazioni sopra alcune questioni di stabilita. I, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 9 (1950), 129-134.
2. L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Gottingen-Heidelberg, 1963.
3. 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
4. S. K. Choi and N. J. Koo, Asymptotic equivalence between two linear Volterra difference systems, Comput. Math. Appl. 47 (2004), no. 2-3, 461-471. https://doi.org/10.1016/S0898-1221(04)90038-7
5. 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
6. S. K. Choi, N. J. Koo, and Y. H. Goo, Asymptotic property of nonlinear Volterra difference systems, Nonlinear Anal. 51 (2002), no. 2, Ser. A: Theory Methods, 321-337. https://doi.org/10.1016/S0362-546X(01)00833-1
7. 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
8. S. K. Choi, N. J. Koo, and H. S. Ryu, h-stability of differential systems via $t_{\infty}-similarity$, Bull. Korean Math. Soc. 34 (1997), no. 3, 371-383.
9. R. Conti, Sulla t-similitudine tra matrici e la stabilita dei sistemi differenziali lineari, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 19 (1955), 247-250.
10. R. Conti, Linear differential equations asymptotically equivalent to x = 0, Riv. Mat. Univ. Parma (4) 5 (1979), part 2, 847-853.
11. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, 1965.
12. G. A. Hewer, Stability properties of the equation of first variation by $t_{\infty}-similarity$, J. Math. Anal. Appl. 41 (1973), 336-344. https://doi.org/10.1016/0022-247X(73)90209-6
13. L. A. Lusternik and W. S. Sobolev, Functional Analysis, Science, Moscov, 1965.
14. G. Sansone and R. Conti, Non-linear Differential Equatioins, The Macmillan Company, New York, 1964.
15. W. F. Trench, On $t_{\infty}$ quasisimilarity of linear systems, Ann. Mat. Pura Appl. (4) 142 (1985), 293-302. https://doi.org/10.1007/BF01766598
16. W. F. Trench, Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems, Advances in difference equations, II. Comput. Math. Appl. 36 (1998), no. 10-12, 261-267.

#### Cited by

1. ASYMPTOTIC EQUIVALENCE BETWEEN TWO LINEAR DYNAMIC SYSTEMS ON TIME SCALES vol.51, pp.4, 2014, https://doi.org/10.4134/BKMS.2014.51.4.1075