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 xn+k=1f(xn+k−1,…,xn),n∈N, where k∈ N, f: [0 , + ∞) k→ [0 , + ∞) 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.