Advances in Difference Equations
Volume 2009 (2009), Article ID 463169, 11 pages
doi:10.1155/2009/463169

On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn−1)/(1+xn) for q>1+p>1

Abstract

We study the boundedness of the difference equation xn+1=(pxn+qxn−1)/(1+xn), n=0,1,…, where q>1+p>1 and the initial values x−1,x0∈(0,+∞). We show that the solution {xn}n=−1∞ of this equation converges to x¯=q+p−1 if xn≥x¯ or xn≤x¯ for all n≥−1; otherwise {xn}n=−1∞ is unbounded. Besides, we obtain the set of all initial values (x−1,x0)∈(0,+∞)×(0,+∞) such that the positive solutions {xn}n=−1∞ of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenović and Ladas (2002).