Some Discussions on the Difference Equation xn+1=α+(xn-1m/xnk). (10th May 2015) - Bakery, Awad A.
Abstract : We give in this work the sufficient conditions on the positive solutions of the difference equationx n + 1 = α + ( x n - 1 m / x n k ), n = 0, 1, …, where α, k, andm ∈ ( 0, ∞ ) under positive initial conditionsx - 1, x 0 to be bounded, α -convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm = k = 1 of Amleh et al. (1999) andm = 1 of Hamza and Morsy (2009). We offer improving conditions in the case ofm = 1 of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.