Asymptotic expansions for regularized state-dependent neutral delay equations

Singularly perturbed delay differential equations arising from the regularization of state dependent neutral delay equations are considered. Asymptotic expansions of their solutions are constructed and their limit for $eps o 0^+$ is studied. Due to discontinuities in the derivative of the solution of the neutral delay equation and the presence of different time scales when crossing breaking points, new difficulties have to be managed. A two-dimensional dynamical system is presented which characterizes whether classical or weak solutions are approximated by the regularized problem. A new type of expansion (in powers of $sqrteps$) turns out to be necessary for the study of the transition from weak to classical solutions. The techniques of this article can also be applied to the study of general singularly perturbed delay equations