A note on positive non-oscillatory solutions of the difference equation xn+1 = α + xn-kp/xn p

Abstract

The aim of this note is to show that the following difference equationwhere α > - 1, p > 0 and k is fixed, has positive non-oscillatory solutions which converge to the positive equilibrium This result solves Open Problem 1 in Stević, 2005, On the recursive sequence 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 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 is variable.