On entire solutions of explicit and implicit linear differential equations in a Banach space

Let E be a Banach space and A be a closed linear operator on E with domain of definition D(A) that may be not dense in E. We suppose that A has a bounded inverse operator. The proof of well-posedeness of the differential equation w'=Aw+f(z) in special spaces of entire functions is considered in the talk . The proof is based on study of properties of entire solutions of implicit differential equation Tw'+g(z)=w(z). The general statements are ilustrated with meaningful examples