The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.
B. A. Biletskyi. A. A. Boichuk. A. A. Pokutnyi. "Periodic Problems of Difference Equations and Ergodic Theory." Abstr. Appl. Anal. 2011 (SI1) 1 - 12, 2011. https://doi.org/10.1155/2011/928587