The solution manifold and C1-smoothness for differential equations with state-dependent delay

Let h>0, open, and continuously differentiable. If f satisfies two mild additional smoothness conditions then the set is a C1-submanifold of codimension n in , the maximal solutions xφ of the initial value problems define a continuous semiflow F on X, and all operators F(t,•) are continuously differentiable. Their derivatives D2F(t,φ) are given in the usual way by solutions v to the variational equation along xφ, with segments vt in the tangent spaces TF(t,φ)X. The additional conditions on f are motivated by properties of differential equations with state-dependent delay, and are verified for an example.