Advances in Difference Equations
Volume 2009 (2009), Article ID 985161, 7 pages
doi:10.1155/2009/985161

Convergence results on a second-order rational difference equation with quadratic terms

Abstract

We investigate the global behavior of the second-order difference equation xn+1=xn−1((αxn+βxn−1)/(Axn+Bxn−1)), where initial conditions and all coefficients are positive. We find conditions on A,B,α,β under which the even and odd subsequences of a positive solution converge, one to zero and the other to a nonnegative number; as well as conditions where one of the subsequences diverges to infinity and the other either converges to a positive number or diverges to infinity. We also find initial conditions where the solution monotonically converges to zero and where it diverges to infinity.