Ulam’s Type Stability of a Discrete Matrix Equation with Second-Order Difference and Single Delay

In this paper, we investigate the Ulam stability problem for a second-order discrete matrix difference equation with a single delay. The primary aim of this work is to establish rigorous Ulam-type stability criteria for this class of delayed matrix difference systems. Firstly, we formally define Hyers–Ulam stability (HUS) and Hyers–Ulam–Rassias stability (HURS) tailored to delayed discrete matrix equations, and derive an essential norm estimate for the delayed discrete matrix function (DDMF). Relying on this critical norm inequality, we further derive sufficient conditions to guarantee the Hyers–Ulam and Hyers–Ulam–Rassias stability of the addressed equation, which constitute the core theoretical contributions of this study. Compared with the existing literature, the obtained stability judgments are less conservative and applicable to second-order matrix delayed difference models rarely discussed in previous works. Finally, several numerical examples are presented to demonstrate the correctness and effectiveness of the derived theoretical conclusions.