Unboundedness Results for Rational Difference Equations

Abstract

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 [Math formulas unable to be displayed here, refer to PDF] Particularly, we show that for nonnegative α and C, whenever C + α = 0 and C + α > 0, unbounded solutions exist for some choice of nonnegative initial conditions. Moreover, we study the rational difference equation [Math formulas unable to be displayed here, refer to PDF] Particularly, we show that whenever 0 > B < 1/3 and α E [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 [Math formulas unable to be displayed here, refer to PDF] 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 kth order systems of two rational difference equations [Math formulas unable to be displayed here, refer to PDF] 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.