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  • Research Article
  • Open Access

Stability of a delay difference system

Advances in Difference Equations20062006:031409

  • Received: 28 January 2006
  • Accepted: 1 June 2006
  • Published:


We consider the stability problem for the difference system x n = Axn-1 + Bxn-k, where A, B are real matrixes and the delay k is a positive integer. In the case A = -I, the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If ||A|| + ||B|| < 1, then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A I and A -I.


  • Differential Equation
  • Positive Integer
  • Partial Differential Equation
  • Stability Condition
  • Ordinary Differential Equation


Authors’ Affiliations

Department of Mathematics, Chelyabinsk State Pedagogical University, 69 Lenin Avenue, Chelyabinsk, 454080, Russia
Department of Mathematics, Southern Ural State University, 76 Lenin Avenue, Chelyabinsk, 454080, Russia


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© M. Kipnis and D. Komissarova. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.