Initial problem for a nonhomogeneous linear differential-difference equation in a Banach space for a class of exponential type entire functions

Abstract

We prove the existence and uniqueness of a solution to a one-point initial problem for the nonhomogeneous linear differential-difference equation $$u'(z)=Au(z+h)+f(z)$$, $$z\in \mathbb {C}$$, in some classes of exponential type entire vector-valued functions. The obtained formula for the unique solution can be considered as a generalization of the classical Cauchy formula for a solution to the nonhomogeneous linear differential equation.