Existence and Ulam Type Stability for Impulsive Fractional Differential Systems with Pure Delay

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs.