Smoothness issues in differential equations with state-dependent delay

Abstract

For differential equations with state-dependent delays a satisfactory theory is developed by the second author [6] on the solution manifold to guarantee C1 -smoothness for the solution operators. We present examples showing that better than C1 -smoothness cannot be expected in general for the solution manifold and for local stable manifolds at stationary points on the solution manifold. Then we propose a new approach to overcome the diffi culties caused by the lack of smoothness. The mollification technique is used to approximate the nonsmooth evaluation map with smooth maps. Several examples show that the mollified systems can have nicer smoothness properties than the original equation. Examples are also given where better smoothness than C1 can be obtained on the solution manifold.