OF

We present a collection of techniques for demonstrating the existence of un-bounded solutions. We then use these techniques to determine the boundedness character of rational difference equations and systems of rational difference equa-tions. We study the rational difference equation xn = α + xn−1 Cxn−2 + xn−3, n ∈ N. Particularly, we show that for nonnegative α and C, whenever Cα = 0 and C+α> 0, unbounded solutions exist for some choice of nonnegative initial conditions. Moreover, we study the rational difference equation xn = α + βxn−1 + xn−2 xn−3, n ∈ N. Particularly, we show that whenever 0 < β < 1 3 and α ∈ [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 kth order rational difference equation xn = α + ∑k i=1 βixn−