A note on unbounded solutions of a class of second order rational difference equations
Date of Original Version
We investigate the unbounded solutions of the second order difference equation xn+1 = α + βxn + γx n-1/A + Bxn + Cxn-1, n = 0,1,... where all parameters α, β, γ, A, B, and C and initial conditions x -1,x0 are nonnegative and such that A + Bxn + Cxn-1 > 0 for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to ∞ and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC). © 2006 Taylor & Francis.
Journal of Difference Equations and Applications
Kulenović, M. R.S., and O. Merino. "A note on unbounded solutions of a class of second order rational difference equations." Journal of Difference Equations and Applications 12, 7 (2006): 777-781. doi:10.1080/10236190600734184.