Some Discussions on the Difference Equationxn+1=α+(xn-1m

Abstract

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk), n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1, x0to 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=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.