Operator Bruwier Series and Initial Problem for a Linear Differential–Difference Equation in a Banach Space

We prove the existence and uniqueness of a solution to the one-point initial problem for the linear differential–difference equation u′(z) = Au(z + h), h ϵ ℂ, in some classes of exponential type entire functions. We obtain a representation of a unique solution to this problem by using the operator Bruwier series.