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

Abstract

Abstract By using a comparison method and some difference inequalities we show that the following higher order difference equation x n + k = 1 f ( x n + k − 1 , … , x n ) , n ∈ N , $$ x_{n+k}=\frac{1}{f(x_{n+k-1},\ldots ,x_{n})},\quad n\in{\mathbb{N}},$$ where k ∈ N $k\in{\mathbb{N}}$ , f : [ 0 , + ∞ ) k → [ 0 , + ∞ ) $f:[0,+\infty )^{k}\to [0,+\infty )$ 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.

Keywords

• Difference equation
• Unbounded solutions
• Homogeneous function