On the existence of unbounded solutions for some rational equations

Abstract

We resolve several conjectures regarding the boundedness character of the rational difference equation

$$x_n=\frac{\alpha +\delta x_{n-3}}{A+B{x}_{n-1}+C{x}_{n-2}+E{x}_{n-4}},n\in \mathbb{N}.$$

We show that whenever parameters are nonnegative, $A<\delta$, and $C,E>0$, unbounded solutions exist for some choice of nonnegative initial conditions. We also partly resolve a conjecture regarding the boundedness character of the rational difference equation

$$x_n=\frac{x_{n-3}}{B{x}_{n-1}+x_{n-4}},n\in \mathbb{N}.$$

We show that whenever $B>2^5$, unbounded solutions exist for some choice of nonnegative initial conditions.