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

Abstract

Let h>0, U⊂C 1([−h,0], R n) open, and f:U→ R n continuously differentiable. If f satisfies two mild additional smoothness conditions then the set X={φ∈U : φ ̇ (0)=f(φ)} is a C 1-submanifold of codimension n in C 1([−h,0], R n) , the maximal solutions x φ of the initial value problems x ̇ (t)=f(x t), x 0=φ∈X define a continuous semiflow F on X, and all operators F( t,·) are continuously differentiable. Their derivatives D 2 F( t, φ) are given in the usual way by solutions v to the variational equation along x φ , with segments v t in the tangent spaces T F( 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.