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  • Research Article
  • Open Access

Exponential dichotomy of difference equations in l p -phase spaces on the half-line

Advances in Difference Equations20062006:058453

  • Received: 14 November 2005
  • Accepted: 17 May 2006
  • Published:


For a sequence of bounded linear operators on a Banach space X, we investigate the characterization of exponential dichotomy of the difference equations vn+1 = A n v n . We characterize the exponential dichotomy of difference equations in terms of the existence of solutions to the equations vn+1 = A n v n + f n in l p spaces (1 ≤ p < ∞). Then we apply the results to study the robustness of exponential dichotomy of difference equations.


  • Differential Equation
  • Banach Space
  • Phase Space
  • Partial Differential Equation
  • Ordinary Differential Equation


Authors’ Affiliations

Department of Applied Mathematics, Hanoi University of Technology, Khoa Toan Ung Dung Dai Hoc Bach Khoa Ha Noi, 1 Dai Co Viet Street, Hanoi, Vietnam


  1. Aulbach B, Minh NV: The concept of spectral dichotomy for linear difference equations. II. Journal of Difference Equations and Applications 1996,2(3):251–262. 10.1080/10236199608808060MathSciNetView ArticleMATHGoogle Scholar
  2. Bainov DD, Kostadinov SI, Zabreiko PP: Stability of the notion of dichotomy of linear impulsive differential equations in a Banach space. Italian Journal of Pure and Applied Mathematics 1997, 1: 43–50 (1998).MathSciNetMATHGoogle Scholar
  3. Baskakov AG: Semigroups of difference operators in the spectral analysis of linear differential operators. Functional Analysis and Its Applications 1996,30(3):149–157 (1997). 10.1007/BF02509501MathSciNetView ArticleMATHGoogle Scholar
  4. Chow S-N, Leiva H: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces. Journal of Differential Equations 1995,120(2):429–477. 10.1006/jdeq.1995.1117MathSciNetView ArticleMATHGoogle Scholar
  5. Coffman CV, Schäffer JJ: Dichotomies for linear difference equations. Mathematische Annalen 1967,172(2):139–166. 10.1007/BF01350095MathSciNetView ArticleMATHGoogle Scholar
  6. Dalec'kiĭ JuL, Kreĭn MG: Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs. Volume 43. American Mathematical Society, Rhode Island; 1974:vi+386.Google Scholar
  7. Henry D: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Volume 840. Springer, Berlin; 1981:iv+348.Google Scholar
  8. Huy NT: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. Journal of Functional Analysis 2006,235(1):330–354. 10.1016/j.jfa.2005.11.002MathSciNetView ArticleMATHGoogle Scholar
  9. Huy NT, Minh NV: Exponential dichotomy of difference equations and applications to evolution equations on the half-line. Computers & Mathematics with Applications 2001,42(3–5):301–311.MathSciNetView ArticleMATHGoogle Scholar
  10. Kato T: Perturbation Theory for Linear Operators. Springer, New York; 1980.MATHGoogle Scholar
  11. Latushkin Y, Tomilov Y: Fredholm differential operators with unbounded coefficients. Journal of Differential Equations 2005,208(2):388–429. 10.1016/j.jde.2003.10.018MathSciNetView ArticleMATHGoogle Scholar
  12. Li T: Die Stabilitatsfrage bei Differenzengleichungen. Acta Mathematica 1934, 63: 99–141. 10.1007/BF02547352MathSciNetView ArticleMATHGoogle Scholar
  13. Minh NV, Huy NT: Characterizations of dichotomies of evolution equations on the half-line. Journal of Mathematical Analysis and Applications 2001,261(1):28–44. 10.1006/jmaa.2001.7450MathSciNetView ArticleMATHGoogle Scholar
  14. Minh NV, Räbiger F, Schnaubelt R: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line. Integral Equations and Operator Theory 1998,32(3):332–353. 10.1007/BF01203774MathSciNetView ArticleMATHGoogle Scholar
  15. Ngoc PHA, Naito T: New characterizations of exponential dichotomy and exponential stability of linear difference equations. Journal of Difference Equations and Applications 2005,11(10):909–918. 10.1080/00423110500211947MathSciNetView ArticleMATHGoogle Scholar
  16. Peterson AC, Raffoul YN: Exponential stability of dynamic equations on time scales. Advances in Difference Equations 2005,2005(2):133–144. 10.1155/ADE.2005.133MathSciNetView ArticleMATHGoogle Scholar
  17. Rodkina A, Schurz H: Global asymptotic stability of solutions of cubic stochastic difference equations. Advances in Difference Equations 2004,2004(3):249–260. 10.1155/S1687183904309015MathSciNetView ArticleMATHGoogle Scholar
  18. Sell GR, You Y: Dynamics of Evolutionary Equations, Applied Mathematical Sciences. Volume 143. Springer, New York; 2002:xiv+670.View ArticleMATHGoogle Scholar
  19. Slyusarchuk VE: Exponential dichotomy of solutions of discrete systems. Ukrainskiı Matematicheskiĭ Zhurnal 1983,35(1):109–115, 137.MathSciNetMATHGoogle Scholar
  20. Stefanidou G, Papaschinopoulos G: Trichotomy, stability, and oscillation of a fuzzy difference equation. Advances in Difference Equations 2004,2004(4):337–357. 10.1155/S1687183904311015MathSciNetView ArticleMATHGoogle Scholar


© N.T. Huy and V.T.N. Ha 2006

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