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

In this write-up, we focus on pseudo-Hilfer-type fractional order delayed differential equations with bounded definite integral initial conditions on the time interval [0, T]. We begin by establishing relevant lemmas. Then, we derive the solution to the homogeneous Hilfer-type pseudo-fractional order retarded differential equation that satisfies the appropriate initial condition using classical methods. Next, we obtain explicit formulas for solutions to linear inhomogeneous Hilfer-type pseudo-fractional time retarded differential equations with constant coefficients, employing classical ideas. Furthermore, we investigate the existence and uniqueness of the solution of the Hilfer-type pseudo-fractional order delayed differential equation and demonstrate the stability of the given differential equation in the Ulam-Hyers sense on the time interval [0, T].