Delayed analogue of pseudo-Mittag-Leffler functions and their applications to pseudo-Hilfer fractional time retarded differential equations

In this write-up, we take account of pseudo-Hilfer-type fractional order delayed differential equations that provide a bounded definite integral initial condition on [0 ,T ]. After proving certain lemmas at the beginning of the article, we found the solution of the homogeneous pseudo-Hilfer-type fractional order retarded differential equation, which satisfies the appropriate initial condition using classical methods. Then, we found the explicit formula of solutions to linear inhomogeneous pseudo-Hilfer-type fractional time retarded differential equations with constant coefficients by imposing classical conceptions. Furthermore, the existence and uniqueness of the solution of the pseudo-Hilfer-type fractional order delayed differential equation was investigated in the article, and the stability of the given differential equation was demonstrated in the Ulam-Hyers sense on [0 ,T ].