Alaa E. Hamza and M. A. El-Sayed

Copyright © 1998 Alaa E. Hamza and M. A. El-Sayed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we investigate the asymptotic stability of the recursive sequence xn+1=α+βxn21+γxn−1,    n=0,1,… and the existence of certain monotonic solutions of the equation xn+1=xnpf(xn,xn−1,…,xn−k),     n=0,1,… which includes as a special case the rational recursive sequence xn+1=βxnp1+∑i=1kγixn−1p−r, where α≥0,β>0,γ>0,γi≥0, i=1,2,…,k,∑i=1kγi>0, p∈{2,3,…} and r∈{1,2,…,p−1}. The case when r=0 has been investigated by Camouzis et. al. [1], and for r=0 and p=2 by Camouzis et. al. [2].