Unboundedness results for systems

Abstract

We study k ^th order systems of two rational difference equations

$$ x_n = \frac{{\alpha + \sum\nolimits_{i = 1}^k {\beta _i x_{n - i} + } \sum\nolimits_{i = 1}^k {\gamma _i y_{n - i} } }} {{A + \sum\nolimits_{j = 1}^k {B_j x_{n - j} + } \sum\nolimits_{j = 1}^k {C_j y_{n - j} } }},n \in \mathbb{N}, $$

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.