Sylvester‐Type Delay Differential Systems Driven by Tempered Fractional Operators With Arbitrary Kernel

This paper investigates a class of tempered fractional Sylvester‐type delay matrix systems defined with arbitrary kernel. A tempered delay perturbation matrix function is introduced and employed to derive explicit solution representations for the corresponding linear problem. The obtained formulation provides a systematic framework for the analysis of matrix‐valued delay systems involving generalized tempered memory effects. Furthermore, the stability in the sense of Ulam‐Hyers of the linear system is established. The analysis is then extended to a semi‐linear setting. By applying the Banach fixed‐point theorem, sufficient conditions ensuring the existence and uniqueness of solutions are obtained. In addition, Ulam‐Hyers stability results for the semi‐linear system are derived. Finally, numerical examples are presented to illustrate the applicability of the theoretical findings.