  Classically, solution theories for state-dependent delay equations are
developed in spaces of continuous or continuously differentiable functions. The
former can be technically challenging to apply in as much as suitably Lipschitz
continuous extensions of mappings onto the space of continuous functions are
required; whereas the latter approach leads to restrictions on the class of
initial pre-histories. Here, we establish a solution theory for state-dependent
delay equations for arbitrary Lipschitz continuous pre-histories and suitably
Lipschitz continuous right-hand sides on the Sobolev space $H^1$. The provided
solution theory is independent of previous ones and is based on the contraction
mapping principle on exponentially weighted spaces. In particular, initial
pre-histories are not required to belong to solution manifolds and the
generality of the approach permits the consideration of a large class of
functional differential equations even for which the continuity of the
right-hand side has constraints on the derivative.
