Global asymptotic behavior of yn+1 = (pyn + yn-1)/(r + qyn + yn-1)
Date of Original Version
We investigate the global stability character of the equilibrium points and the period-two solutions of yn+1 = (pyn + yn-1)/(r + qyn + yn-1), n = 0, 1,..., with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.
Advances in Difference Equations
Brett, A., and M. R. Kulenović. "Global asymptotic behavior of yn+1 = (pyn + yn-1)/(r + qyn + yn-1)." Advances in Difference Equations 2007, (2007). doi:10.1155/2007/41541.