International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 391265, 15 pages
doi:10.1155/2008/391265

On the Rational Recursive Sequence xn+1=(α−βxn)/(γ−δxn−xn−k)

Abstract

We study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation xn+1=(α−βxn)/(γ−δxn−xn−k),  n=0,1,2,…,  k∈{1,2,…}, in the two cases: (i) δ≥0,  α>0,  γ>β>0; (ii) δ≥0,  α=0,  γ,β>0, where the coefficients α,  β,  γ,   and   δ, and the initial conditions x−k,x−k+1,…,x−1,x0 are real numbers. We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients of this equation.