Some Discussions on the Difference Equation xn+1 = alpha+ (xn-1mx(n)(k))

摘要

We give in this work the sufficient conditions on the positive solutions of the difference equation x(n+1) = alpha + (x(n-1)(m)/x(n)(k)), n = 0, 1,..., where alpha, k, and m is an element of (0, infinity) under positive initial conditions x(-1), x(0) to be bounded, alpha-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 Gumus and Ocalan (2012) and explain our results by some numerical examples with figures.