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Variationally Asymptotically Stable Difference Systems

Advances in Difference Equations volume 2007, Article number: 035378 (2007) Cite this article

Abstract

We characterize the h-stability in variation and asymptotic equilibrium in variation for nonlinear difference systems via n∞-summable similarity and comparison principle. Furthermore we study the asymptotic equivalence between nonlinear difference systems and their variational difference systems by means of asymptotic equilibria of two systems.

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References

  1. Conti R: Sulla t -similitudine tra matrici e la stabilità dei sistemi differenziali lineari. Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1955, 19: 247–250.

    MATH  MathSciNet  Google Scholar 

  2. Choi SK, Koo N, Dontha S: Asymptotic property in variation for nonlinear differential systems. Applied Mathematics Letters 2005,18(1):117–126. 10.1016/j.aml.2003.07.019

    Article  MATH  MathSciNet  Google Scholar 

  3. Trench WF: On t ∞ quasisimilarity of linear systems. Annali di Matematica Pura ed Applicata 1985, 142: 293–302. 10.1007/BF01766598

    Article  MATH  MathSciNet  Google Scholar 

  4. Trench WF: Linear asymptotic equilibrium and uniform, exponential, and strict stability of linear difference systems. Computers & Mathematics with Applications 1998,36(10–12):261–267.

    Article  MATH  MathSciNet  Google Scholar 

  5. Choi SK, Koo N: Variationally stable difference systems by n ∞ -similarity. Journal of Mathematical Analysis and Applications 2000,249(2):553–568. 10.1006/jmaa.2000.6910

    Article  MATH  MathSciNet  Google Scholar 

  6. Choi SK, Koo N, Goo YH: Variationally stable difference systems. Journal of Mathematical Analysis and Applications 2001,256(2):587–605. 10.1006/jmaa.2000.7330

    Article  MATH  MathSciNet  Google Scholar 

  7. Choi SK, Koo N, Ryu HS: Asymptotic equivalence between two difference systems. Computers & Mathematics with Applications 2003,45(6–9):1327–1337.

    Article  MATH  MathSciNet  Google Scholar 

  8. Galescu G, Talpalaru P: Stability results for perturbed difference equations of Volterra type. Libertas Mathematica 2003, 23: 95–109.

    MATH  MathSciNet  Google Scholar 

  9. Choi SK, Koo N: Asymptotic property of linear Volterra difference systems. Journal of Mathematical Analysis and Applications 2006,321(1):260–272. 10.1016/j.jmaa.2005.08.028

    Article  MATH  MathSciNet  Google Scholar 

  10. Lakshmikantham V, Trigiante D: Theory of Difference Equations: Numerical Methods and Applications, Mathematics in Science and Engineering. Volume 181. Academic Press, Boston, Mass, USA; 1988:x+242.

    Google Scholar 

  11. Medina R: Asymptotic behavior of nonlinear difference systems. Journal of Mathematical Analysis and Applications 1998,219(2):294–311. 10.1006/jmaa.1997.5798

    Article  MATH  MathSciNet  Google Scholar 

  12. Medina R: Stability of nonlinear difference systems. Dynamic Systems & Applications 2000,9(1):1–13.

    MATH  MathSciNet  Google Scholar 

  13. Medina R, Pinto M: Stability of nonlinear difference equations. In Proceedings of Dynamic Systems and Applications, Vol. 2 (Atlanta, GA, 1995), 1996, Atlanta, Ga, USA. Dynamic; 397–404.

  14. Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.

    Google Scholar 

  15. Sheng Q, Agarwal RP: On nonlinear variation of parameter methods for summary difference equations. Dynamic Systems & Applications 1993,2(2):227–242.

    MATH  MathSciNet  Google Scholar 

  16. Pinto M: Discrete dichotomies. Computers & Mathematics with Applications 1994,28(1–3):259–270.

    Article  MATH  MathSciNet  Google Scholar 

  17. Medina R, Pinto M: Asymptotic equivalence and asymptotic behavior of difference systems. Communications in Applied Analysis 1997,1(4):511–523.

    MathSciNet  Google Scholar 

  18. Medina R: Asymptotic properties of solutions of nonlinear difference equations. Journal of Computational and Applied Mathematics 1996,70(1):57–66. 10.1016/0377-0427(95)00139-5

    Article  MATH  MathSciNet  Google Scholar 

  19. Morchało J: Asymptotic equivalence of difference equations. Mathematica Slovaca 1998,48(1):57–68.

    MATH  MathSciNet  Google Scholar 

  20. Zafer A: On asymptotic equivalence of linear and quasilinear difference equations. Applicable Analysis 2005,84(9):899–908. 10.1080/00036810500048350

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Chungnam National University, Daejeon, 305-764, Korea

    Sung Kyu Choi & Namjip Koo

  2. Department of Mathematics, Hanseo University, Seosan, Chungnam, 352-820, Korea

    Yoon Hoe Goo

Authors
  1. Sung Kyu Choi
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  2. Yoon Hoe Goo
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  3. Namjip Koo
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Correspondence to Sung Kyu Choi.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Choi, S.K., Goo, Y.H. & Koo, N. Variationally Asymptotically Stable Difference Systems. Adv Differ Equ 2007, 035378 (2007). https://doi.org/10.1155/2007/35378

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  • DOI: https://doi.org/10.1155/2007/35378

Keywords

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