Representation of solutions and finite‐time stability for fractional delay oscillation difference equations

Abstract

In this article, an explicit solution of the homogeneous fractional delay oscillation difference equation of order is given by constructing discrete sine‐ and cosine‐type delayed Mittag‐Leffler functions. Then, the discrete Laplace transform technique as an effective tool for solving the nonhomogeneous term, which is utilized to explore the solution of corresponding nonhomogeneous equation. Next, finite‐time stability of the homogeneous equation is studied based on the representation of the solution. Furthermore, we show a numerical example to elaborate the correctness of stability theory. Finally, an exact solution for the nonhomogeneous fractional difference equation with is further presented via using the discrete two‐parameter delayed Mittag‐Leffler function.