We present a collection of techniques for demonstrating the existence of unbounded solutions. We then use these techniques to determine the boundedness character of rational difference equations and systems of rational difference equations.

We study the rational difference equation [special characters omitted] Particularly, we show that for nonnegative &agr; and C, whenever C&agr; = 0 and C + &agr; > 0, unbounded solutions exist for some choice of nonnegative initial conditions. Moreover, we study the rational difference equation [special characters omitted] Particularly, we show that whenever 0 < β < ⅓ and &agr; ∈ [0,1], unbounded solutions exist for some choice of nonnegative initial conditions.

Following these two results, we then present some new results regarding the boundedness character of the k^th order rational difference equation [special characters omitted] When applied to the general fourth order rational difference equation, these results prove the existence of unbounded solutions for 49 special cases of the fourth order rational difference equation, where the boundedness character has not been established yet. This resolves 49 conjectures posed by E. Camouzis and G. Ladas.

Finally, we study k^th order systems of two rational difference equations [special characters omitted] In particular, we assume non-negative parameters and non-negative initial conditions. We develop several approaches, which allow us to prove that unbounded solutions exist for certain initial conditions in a range of the parameters.