Stability with respect to initial time difference for generalized delay differential equations

Stability with initial data difference for nonlinear delay differential equations is introduced. This type of stability generalizes the known concept of stability in the literature. It gives us the opportunity to compare the behavior of two nonzero solutions when both initial values and initial intervals are different. Several sufficient conditions for stability and for asymptotic stability with initial time difference are obtained. Lyapunov functions as well as com-parison results for scalar ordinary differential equations are employed. Several examples are given to illustrate the theory.