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Global Behavior of the Difference Equation xn+1 = xn−1g(xn)

Author

Listed:
  • Hongjian Xi
  • Taixiang Sun
  • Bin Qin
  • Hui Wu

Abstract

We consider the following difference equation xn+1 = xn−1g(xn), n = 0,1, …, where initial values x−1, x0 ∈ [0, +∞) and g : [0, +∞)→(0,1] is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to a, 0, a, 0, …, or 0, a, 0, a, … for some a ∈ [0, +∞). (2) Assume a ∈ (0, +∞). Then the set of initial conditions (x−1, x0)∈(0, +∞)×(0, +∞) such that the positive solutions of this equation converge to a, 0, a, 0, …, or 0, a, 0, a, … is a unique strictly increasing continuous function or an empty set.

Suggested Citation

  • Hongjian Xi & Taixiang Sun & Bin Qin & Hui Wu, 2014. "Global Behavior of the Difference Equation xn+1 = xn−1g(xn)," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:705893
    DOI: 10.1155/2014/705893
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    References listed on IDEAS

    as
    1. Qiuli He & Taixiang Sun & Hongjian Xi, 2013. "Dynamics of a Family of Nonlinear Delay Difference Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-4, May.
    2. Qiuli He & Taixiang Sun & Hongjian Xi, 2013. "Dynamics of a Family of Nonlinear Delay Difference Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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