Ukrainian Mathematical Journal

, Volume 48, Issue 10, pp 1600–1608 | Cite as

On the exponential dichotomy of linear difference equations

  • V. I. Tkachenko
Article

Abstract

We consider a system of linear difference equationsx n+1 =A (n)xn in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln εZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k εZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.

Keywords

Unstable Manifold Stable Manifold Finite Interval Fundamental Matrix Pointwise Convergence 

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References

  1. 1.
    S. U. Coffman and J. J. Schaffer, “Dichotomies for linear difference equations,”Math. Ann.,172, No. 2, 139–166 (1967).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    V. E. Slyusarchuk, “On the exponential dichotomy of solutions of discrete systems,”Ukr. Mat. Zh.,35, No. 1, 109–114 (1983).CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. M. Samoilenko and N. A. Perestyuk,Differential Equations with Pulse Influence [in Russian], Vyshcha Shkola, Kiev 1987.Google Scholar
  4. 4.
    A. Halanay and D. Wexler,Teoria Calitativa a Sistemelor cu Impulsuri, Editura Academici Republicii Socialiste Romania, Bucharest 1968.MATHGoogle Scholar
  5. 5.
    K. J. Palmer, “Exponential dichotomies for almost periodic equations,”Proc. Am. Math. Soc,101, No. 2, 293–298 (1987).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Papaschinopoulos, “Exponential dichotomy for almost periodic linear difference equations,”Ann. Soc. Sci. Bruxelles, Ser. 1,102, Nos. 1–2, 19–28(1988).MATHMathSciNetGoogle Scholar
  7. 7.
    W. A. Coppel, “Dichotomies in stability theory,”Lect. Notes Math.,629 (1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • V. I. Tkachenko

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