USA

The method of variation of parameters has been very useful in the qualitative theory of differential equations, since it is a practical tool in the investigation of the properties of solutions of differential equations, especially to study nonlinear problems. It has been applied to investigate the boundedness properties of perturbed systems, the relationship of unperturbed and perturbed systems in nonlinear systems, and again to find a formula for solutions of perturbed integro-differential equations. Recently, the variation of parameter formulas were used to investigate the relationship between: (1) unperturbed systems with different initial conditions and (2) unperturbed and perturbed systems with different initial conditions. Previously, the investigation of initial value problems in differential equations has been restricted to a perturbation in the space variable only with the initial time unchanged. But, as in all measurements, errors occur in measurements of starting time and the solutions of an unperturbed differential system may start at an initial time that is different than the starting time of the perturbed differential system. In real situations it may be impossible to have only a change in the space variable and not also in he initial time. Thus, there is a need to investigate the stability properties of these differential systems.

In this paper, we utilize the variational system associated with the unperturbed differential system to establish stability criteria for initial time difference stability in variation, initial time difference exponential asymptotic stability in variation, and initial time difference slowly growing motion in variation. This is not done with respect to the null solution, but with respect to the unperturbed differential system where the unperturbed differential system and the perturbed differential system have a change in initial position and an initial time difference. We also show variation of parameters formulas that relate an unperturbed differential system and a perturbed differential system with different initial times and initial positions.