Sharpness results for state dependent delay differential equations: An overview

Abstract

This manuscript presents many detailed studies of the numerical solution (by the restart method) of a general class of functional differential equations with state dependent lags; that is, where the delays depend upon the (unknown) solution. It emphasizes the sharpness of earlier published results, including those that relate the multiplicity of various shifted zeros of the lag function to the rates of convergence of methods for locating these zeros and to the rates of convergence of the global solution of the delay equation. The studies were selected to clarify the unusual aspects of numerical methods for delay equations, particularly those with state dependent lags. Although the results given here are mainly analytical in nature, they are ones that every numerical analyst and software designer interested in delay equations should master since they form the technological basis for modern software for state dependent delay differential equations. Indeed, almost all modern delay equation software, in one form or another, is based on these results.