Browse > Article
http://dx.doi.org/10.14403/jcms.2011.24.4.24

LYAPUNOV FUNCTIONS FOR NONLINEAR DIFFERENCE EQUATIONS  

Choi, Sung Kyu (Department of Mathematics Chungnam National University)
Cui, Yinhua (Department of Applied Mathematics Paichai University)
Koo, Namjip (Department of Mathematics Chungnam National University)
Publication Information
Journal of the Chungcheong Mathematical Society / v.24, no.4, 2011 , pp. 883-893 More about this Journal
Abstract
In this paper we study h-stability of the solutions of nonlinear difference system via the notion of $n_{\infty}$-summable similarity between its variational systems. Also, we show that two concepts of h-stability and h-stability in variation for nonlinear difference systems are equivalent. Furthermore, we characterize h-stability for nonlinear difference systems by using Lyapunov functions.
Keywords
$n_{\infty}$-summable similarity; nonlinear difference systems; h-stability; variational systems; Lyapunov functions;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 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), 371-383.
2 S. K. Choi and N. J. Koo, Variationally stable difference systems by $n_{\infty}$- similarity, J. Math. Anal. Appl. 249 (2000), 553-568.   DOI   ScienceOn
3 S. K. Choi, N. J. Koo, and Y. H. Goo, Variationally stable difference systems, J. Math. Anal. Appl. 256 (2001), 587-605.   DOI   ScienceOn
4 S. K. Choi, N. J. Koo, and S. M. Song, h-stability for nonlinear perturbed difference systems, Bull. Korean Math. Soc. 41 (2004), 435-450.
5 S. K. Choi, Y. H. Goo, and N. J. Koo, Variationally asymptotically stable difference systems, Adv. Difference Equ. 2007, Article ID 35378, 21 pages.
6 S. K. Choi, W. Kang, N. Koo, and H. M. Lee, On h-stability of linear difference systems via $n_{\infty}$-quasisimilarity, J. Chungcheong Math. Soc. 21 (2008), no. 2, 189-196.
7 S. K. Choi, W. Kang, and N. Koo, On h-stability of linear dynamic equations on time scales via $u_{\infty}$-similarity, J. Chungcheong Math. Soc. 21 (2008), no. 3, 395-401.
8 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.   DOI   ScienceOn
9 R. Conti, Sulla $u_{\infty}$-similitudine tra matrici e la stabilitµa dei sistemi differenziali lineari, Atti. Acc. Naz. Lincei, Rend. Cl. Fis. Mat. Nat. 49 (1955), 247-250.
10 V. Lakshmikantham and D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications, 2nd ed., Marcel Dekker, New York, 2002.
11 R. Medina, Asymptotic behavior of nonlinear difference systems, J. Math. Anal. Appl. 219 (1998), 294-311.   DOI   ScienceOn
12 R. Medina, Stability results for nonlinear difference equations, Nonlinear Stud. 6 (1999), 73-83.
13 R. P. Agarwal, Difference Equations and Inequalities, 2nd ed., Marcel Dekker, New York, 2000.
14 M. Pinto, Perturbations of asymptotically stable differential systems, Analysis 4 (1984), 161-175.
15 R. Medina, Stability of nonlinear difference systems, Dynam. Systems Appl. 9 (2000), 1-14.
16 R. Medina and M. Pinto, Stability of nonlinear difference equations, Proc. Dynamic Systems and Appl. 2 (1996), 397-404.
17 R. Medina and M. Pinto, Variationally stable difference equations, Nonlinear Anal. 30 (1997), no. 2, 1141-1152.   DOI   ScienceOn
18 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.   DOI   ScienceOn