Closed-Form Solutions and Stability of Delayed Discrete Matrix Equations with Second-Order Differences and Multiple Delays

This paper presents a new technique for finding explicit solutions to multi-delay linear discrete matrix equations. The method suggested here does not need the coefficient matrices to commute with each other or the coefficient matrices and the nonhomogeneous term. The fundamental solutions are derived using generalized multinomial expansions and multi-delayed discrete matrix functions of the sine and cosine types in an environment without commutativity. The solutions presented are of the closed forms and they have been investigated for their effect on the system in the sense of the stability of the system with specific attention paid to the Hyers–Ulam stability of the same. From the analytical study of the problem, we come to know the influence of the commutation and configuration of delays on both the behavior of the solution and the stability of the system. Finally, a numerical example has been investigated to explain the practical relevance of the proposed method where noncommuting coefficients and delays have profound impacts.