On entire solutions of exponential type of some implicit linear differential-difference equation in a Banach space

Let $A$ be a closed linear operator on a Banach space with a possibly domain. Entire solutions of exponential type of the linear differential-difference equation $w'(z)=Aw(z-h)+f(z)$ are studied nondense. Assuming that operator $A$ has a bounded inverse, the well-posedness of this equation in a special space of entire $E$-valued function is proved.