On a study of the representation of solutions of a $\mathrm{\Psi }$-Caputo fractional differential equations with a single delay

We give a representation of solutions to linear nonhomogeneous $\mathrm{\Psi }$-fractional delayed differential equations with noncommutative matrices. We newly define $\mathrm{\Psi }$-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $\mathrm{\Psi }$-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.