Higher order difference equations with homogeneous governing functions nonincreasing in each variable with unbounded solutions

Abstract

By using a comparison method and some difference inequalities we show that the following higher order difference equation $x_{n+k}=\frac{1}{f(x_{n+k-1},\dots ,x_n)},n\in \mathbb{N},$ where $k\in \mathbb{N}$, $f:[0,+\mathrm{\infty }{)}^k\to [0,+\mathrm{\infty })$ is a homogeneous function of order strictly bigger than one, which is nondecreasing in each variable and satisfies some additional conditions, has unbounded solutions, presenting a large class of such equations. The class can be used as a useful counterexample in dealing with the boundedness character of solutions to some difference equations. Some analyses related to such equations and a global convergence result are also given.