The behaviour of the positive solutions of the difference equation x n = A + (xn-2/xn-1)p

Abstract

This paper studies the boundedness, global asymptotic stability and periodicity for solutions of the equation xn = A + (x n-2/xn-1)p, n = 0, 1,..., with p, A ∈ (0, ∞), p ≠ 1 and x-2, x-1 ∈ (0, ∞). It is shown that: (a) all solutions converge to the unique equilibrium, x̄ = A + 1, whenever p ≤ min{1, (A + 1)/2}; (b) all solutions converge to period two solutions whenever (A + 1)/2 < p < 1; and (c) there exist unbounded solutions whenever p > 1. These results complement those for the case p = 1 in A.M. Amleh et al., On the recursive sequence yn+1 = α + (yn-1/yn, Journal of Mathematical Analysis and Applications 233 (1999), 790-798.