The difference equation xn+1=α+xn-k/Σi=0k-1 cixn-i solutions converging to zero

The aim of this note is to show that the following difference equation: x(n+1) = alpha + x(n-k)/Sigma(i=0) (k-1)c(i)x(n-i), n=0, 1,..., where k is an element of N, c(i) >= 0, i =0,...,k-1, Sigma(k-1)(i=0)c(i) = 1, and alpha < -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. Harnza, On the recursive sequence x(n+1) = alpha + x(n-1)/x(n), J. Math. Anal. Appl., in press. (c) 2006 Published by Elsevier Inc.