Existence and Ulam-type stability for impulsive oscillating systems with pure delay

In this paper, we first introduce the new concept of an impulsive delayed vector function, which helps us to build the representation of an exact solution for the linear impulsive oscillating differential delay systems (IODDSs). Second, we derive some sufficient conditions to guarantee the existence and uniqueness of solution by Schauder's and Banach's fixed point theorems for a new class of nonlinear IODDSs. Finally, we obtain the Ulam-Hyers stability (UHs) and Ulam-Hyers-Rassias stability (UHRs) for nonlinear IODDSs, and two examples are presented to illustrate the main results of our study.