Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations

We study linear systems of functional differential equations of mixed type, both autonomous and (asymptotically hyperbolic) nonautonomous. Such equations arise naturally in various contexts, for example, in lattice differential equations. We obtain a decomposition of the state space into stable and unstable subspaces with associated semigroups or evolutionary processes. In the autonomous case we additionally obtain representations of the semigroups in terms of retarded and advanced equations. We also obtain a factorization of the characteristic function, analogous to a Wiener-Hopf factorization, with which we define an integer invariant for the system. Finally, we study the boundary value problem on intervals of long but finite length in the spirit of the finite section method.