Solvability problems for a linear homogeneous functional-differential equation of the pointwise type

The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities of these equations and is based on the description of spectral properties of an operator that is induced by the right-hand side of the equation and acts in the scale of spaces of infinite sequences. © 2017, Pleiades Publishing, Ltd.