Explicit solutions and asymptotic behaviors of Caputo discrete fractional-order equations with variable coefficients

A new discrete fractional-order Peano-Baker series is established in this paper. Based on this series, the explicit solutions of Caputo linear discrete fractional-order equations (DFOEs) with variable matrix coefficients involving the homogeneous and inhomogeneous cases are obtained. On the basis of the comparison theorem and the explicit solutions of Caputo linear DFOEs with variable coefficients, the asymptotic behaviors of the solutions of the homogeneous and inhomogeneous Caputo linear DFOEs with variable coefficients are presented. Finally, the applicability and effectiveness of the proposed results are illustrated by four examples.