On the recursive sequence xn+1 = α+βxn-k+1 over A+Bxn-k+1+Cxn-2k+1

In this paper, we study the global asymptotic stability, invariant intervals, the character of semicycles, and the boundedness of all positive solutions of the higher order nonlinear difference equation Xn+1 = alpha + beta x(n-k+1)/A + Bx(n-k+1) + Cx(n-2k+1), n = 0, 1,..., where A, B, C and alpha, beta are positive, k is an element of {1,2,3,...}, and the initial conditions x(-2k+1),...,x(-1),x(0) are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable. Finally, we present some informative examples. (c) 2005 Elsevier Inc. All rights reserved.