Floquet representations and asymptotic behavior of solutions to periodic linear difference equations

Abstract

We give new representations of solutions for the periodic linear difference equation of the type $x(n+1)=B(n)x(n)+b(n)$, where complex nonsingular matrices $B(n)$ and vectors $b(n)$ are $\rho$-periodic. These are based on the Floquet multipliers and the Floquet exponents, respectively. By using these representations, asymptotic behavior of solutions is characterized by initial values. In particular, we can characterize necessary and sufficient conditions that the equation has a bounded solution(or a $\rho$-periodic solution), and the Massera type theorem by initial values.