Some Discussions on the Difference Equation xn+1=α+(xn-1mxnk)‎

Abstract EN

We give in this work the sufficient conditions on the positive solutions of the difference equation xn+1=α+(xn-1m/xnk), 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.