Variation of parameters and Lipschitz-type stability in unperturbed differential systems with initial time difference

Abstract

In this study, we explore the connection between unperturbed differential systems that exhibit variations in both their initial conditions and the starting time. Through the application of variation-of-parameters methods, we construct integral representations that reveal this relationship. To formulate Lipschitz-type stability criteria in the presence of an initial time shift in nonlinear differential systems, it is essential to employ the variational framework derived from the unperturbed model.