(Szegedi Tudományegyetem, Bolyai Intézet)
Simasági problémák I.
Absztrakt: For differential equations with state-dependent delays a satisfactory theory is developed by H.-O. Walther on the so called solution manifold to guarantee $C^1$-smoothness for the solution operators. We present examples showing that better than $C^1$-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 difficulties 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 $C^1$ can be obtained on the solution manifold.