Existence of periodic solutions in a linear higher order system of difference equations

In this paper we investigate the existence of nontrivial periodic solutions of a higher order system of difference equations. Our framework is based on an earlier periodicity condition of us. We also use the theory of circulant matrices combined by a theorem of Silvester on the computation the determinant of block matrices. An illustrative application is given to show the effectiveness of our framework and to point out the connection between our periodicity results and some known stability conditions.