A note on positive non-oscillatory solutions of the difference equation xn+1 = α + xpn-k xpn

The aim of this note is to show that the following difference equation x(n+1) = alpha + x(n-k)(p)/x(n)(p), n = 0,1,..., where alpha > -1, p > 0 and k is an element of N is fixed, has positive non-oscillatory solutions which converge to the positive equilibrium (x) over bar alpha + 1: This result solves Open Problem 1 in Stevic, 2005, On the recursive sequence x(n+1) = alpha + (x(n-1)(p)/xn(n)p), Journal of Applied Mathematics and Computing 18(1-2), 229-234, as well as, Open Problem 1 in DeVault, Kent and Kosmala, 2003, On the recursive sequence x(n+1) = p + (x(n-k)/x(n)), Journal of Difference Equations and Application, 9(8), 721-730. It is interesting that the method described here can, in some cases, be applied also when the parameter a is variable.