Linear functional-differential equations on the line Original Research Article

Multidimensional equation j=1l(Djy)(Fjx) = γ(x) is considered. Here xε ℝ1 and Dj: C∞(ℝ1, ℂm) → C∞(ℂn) are given differential linear operators. We build a theory of this equation via “common dynamics” of maps Fj. In particular, it is proved that the operator T0: C∞(ℝ1, ℂm) → C∞(ℂn) with affine transformations Fjx = αjx + βj and differential linear operators Dj with constant coefficients is semi-Fredholm.