Uniform boundedness of solutions to linear difference equations with periodic forcing functions

In this paper we give criteria on the uniform boundedness of the solutions to linear difference equations (LEs) with periodic forcing functions. First, we give a necessary and sufficient condition that the sequence $\{L^n\}$ of a square matrix $L$ is bounded, from which a criterion on the uniform boundedness of the solutions to LEs is obtained. Second, a criterion on the uniform boundedness of the solutions for LEs with periodic forcing functions is given by applying a certain representation of solutions. In connection with LEs with delay, we give the characteristic equation of a matrix under the commuting condition.