On the linearization problem for neutral equations with state-dependent delays

Neutral functional-differential equations of the form
$$x'(t)=g(\partial x_t,x_t)$$
define continuous semiflows $G$ on closed subsets in manifolds of $C^2$-functions under hypotheses designed for the application to equations with state-dependent delay. The differentiability of the solution operators $G(t,\,\cdot\,)$ in the usual sense is not available, but for a certain variational equation along flowlines, the initial value is well-posed. Using this variational equation, we prove a principle of linearized stability, which covers the prototype
$$x'(t)=A(x'(t+d(x(t))))+f(x(t+r(x(t))))$$
with nonlinear real functions $A$, $d<0$, $f$, and $r\le 0$. Special cases of the latter describe the interaction of two kinds of behaviour, namely, following a trend versus negative feedback with respect to a stationary state.
The author gratefully acknowledges support by FONDECYT project 7090086.