Variation of Parameters Representations and Lipschitz-Type Stability for Unperturbed Matrix Differential Systems with Initial Time Difference

In this study, we explore the connection between unperturbed matrix 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 under the presence of an initial time shift in nonlinear differential systems, it is essential to employ the variational framework derived from the unperturbed matrix model.