Exponential dichotomies for linear non-autonomous functional differential equations of mixed type

Functional differential equations with forward and backward delays arise naturally, for instance, in the study of travelling waves in lattice equations and as semi-discretizations of partial differential equations (PDEs) on unbounded domains. Linear functional differential equations of mixed type are typically ill-posed, i.e., there exists, in general, no solution to a given initial condition. We prove that Fredholm properties of these equations imply the existence of exponential dichotomies. Exponential dichotomies can be used in discretized PDEs and in lattice differential equations to construct multi-pulses, to perform Evans-function type calculations, and to justify numerical computations using artificial boundary conditions.