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We determine the global behaviours of all solutions of the following rational difference equationsThese equations are related to each other via semiconjugate relations that also let us reduce them to first-order equations. Using this approach, we determine the forbidden sets of each equation explicitly and show that for initial values outside the forbidden sets, their solutions may converge to 0, or to a positive fixed point, or they may be periodic of period 2 or unbounded. In some cases, different types of solutions coexist depending on the initial values.

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Dedicated to Gerry Ladas on the occasion of his 70th birthdayWe show that the second order rational difference equationax n þ bx n21 has several qualitatively different types of positive solutions. Depending on the non-negative parameter values A,B,C,a,b, 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.