The Behaviour of the Positive Solutions of the Difference Equation x n = A + ( x n − 2 x n − 1 ) p Kenneth
نویسندگان
چکیده
This paper studies the boundedness, global asymptotic stability and periodicity for solutions of the equation xn = A + ( xn−2 xn−1 )p , n = 0, 1, . . . , with p, A ∈ (0,∞), p 6= 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 , J. Math. Anal. Appl. 233 (1999), 790-798.
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