On the rational (K + 1,K + 1)-type difference equation

Abstract

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form

$$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$

where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb ₀+∑ ^k _j=1 b _j x _n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6).