The behaviour of the positive solutions of the difference equation

This paper studies the boundedness, global asymptotic stability and periodicity for solutions of the equation with p, A ∊ (0, ∞), p ≠ 1 and x − 2, x − 1 ∊ (0, ∞). It is shown that: (a) all solutions converge to the unique equilibrium, , 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 Journal of Mathematical Analysis and Applications 233 (1999), 790–798.