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

By using a comparison method and some difference inequalities we show that the following higher order difference equation Xn+k = 1/f(Xn+k-1, ..., X-n), n is an element of N, where k is an element of N , f : [0, +infinity)(k) -> [0,+infinity) 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.

Automated summary

By using a comparison method and difference inequalities, we prove that a certain class of higher order difference equations with specific conditions has unbounded solutions, serving as counterexamples for the boundedness character of solutions to difference equations.