ON THE RECURSIVE SEQUENCE $x_{n+1} = \dfrac{\alpha + \beta x_{n-k}}{f(x_n,...,x_{n-k+1})}$

Abstract

The boundedness, the oscillatory behavior and the global stability of the nonnegative solutions of the difference equation $$ x_{n+1} = \frac{\alpha + \beta x_{n-k}}{f(x_n,...,x_{n-k+1})}$$ is investigated, where $k \in \mathbf{N},$ the parameters $\alpha$ and $\beta$ are nonnegative real numbers and $f: \mathbf{R}^k_+ \to \mathbf{R}_+$ is a continuous function nondecreasing in each variable such that $f(0,...,0) \gt 0$.