DYNAMICAL PROPERTIES OF NONAUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

A type of nonautonomous n-dimensional state-dependent delay differential equation (SDDE) is studied. The evolution law is supposed to satisfy standard conditions ensuring that it can be imbedded, via the Bebutov hull construction, in a new map which determines a family of SDDEs when it is evaluated along the orbits of a flow on a compact metric space. Additional conditions on the initial equation, inherited by those of the family, ensure the existence and uniqueness of the maximal solution of each initial value problem. The solutions give rise to a skew-product semiflow which may be noncontinuous, but which satisfies strong continuity properties. In addition, the solutions of the variational equation associated to the SDDE determine the Frechet differential with respect to the initial state of the orbits of the semiflow at the compatibility points. These results are key points to start using topological tools in the analysis of the long-term behavior of the solution of this type of nonautonomous SDDEs.