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On sharp conditions for the global stability of a difference equation satisfying the Yorke condition

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Abstract

Continuing our previous investigations, we give simple sufficient conditions for the global stability of the zero solution of the difference equation x n+1 = qx n + ƒn(x n, …, x n−k), n ∈ ℤ, where the nonlinear functions ƒn satisfy the Yorke condition. For every positive integer k, we represent the interval (0, 1] as the union of [(2k + 2)/3] disjoint subintervals, and, for q from each subinterval, we present a global-stability condition in explicit form. The conditions obtained are sharp for the class of equations satisfying the Yorke condition.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 73–80, January, 2008.

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Nenya, O.I., Tkachenko, V.I. & Trofymchuk, S.I. On sharp conditions for the global stability of a difference equation satisfying the Yorke condition. Ukr Math J 60, 78–90 (2008). https://doi.org/10.1007/s11253-008-0043-6

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  • DOI: https://doi.org/10.1007/s11253-008-0043-6

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