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Monotone and oscillatory solutions of a rational difference equation containing quadratic terms

We show that the second order rational difference equation has several qualitatively different types of positive solutions. Depending on the non-negative parameter values A,B,C,α,β, all solutions may converge to 0, or they may all be unbounded. For some parameter values both cases can occur, or coexist depending on the initial values. We find converging solutions of both monotonic and oscillatory types, as well as periodic solutions with period two. A semiconjugate relation facilitates derivations of these results by providing a link to a rational first order equation.