Some discussions on the difference equation x<inf>n+1</inf> = α + (x^m<inf>n-1</inf>/x^k<inf>n</inf>)

Abstract

We give in this work the sufficient conditions on the positive solutions of the difference equation xn+1 = α + (xmn-1/xkn), n = 0,1,..., where α, k, and m ∈ (0, ∞) under positive initial conditions x-1, x0 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 cases m = k = 1 of Amleh et al. (1999) and m = 1 of Hamza and Morsy (2009). We offer improving conditions in the case of m = 1 of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.