The difference equation xn+1 = α + xn−k k−1 i=0 cixn−i has solutions converging to zero

The aim of this note is to show that the following difference equation: xn+1 = α + xn−k k−1 i=0 cixn−i , n= 0, 1, . . . , where k ∈ N, ci 0, i = 0, . . . , k − 1, k−1 i=0 ci = 1, and α < −1, has solutions which monotonically converge to zero. This result shows the existence of such solutions which was not shown in the recently accepted paper: A.E. Hamza, On the recursive sequence xn+1 = α + xn−1 xn , J. Math. Anal. Appl., in press. © 2006 Published by Elsevier Inc