Solvability of partial differential equations of infinite order in certain classes of entire functions

Abstract

In this paper it is shown that under conditions of applicability of the operator$\mathfrak{L}y = \sum\nolimits_{k \geqslant 0} {a_k D^k y(x)}$ to the class [ρ,σ] ρ=(I,ρs), ρ₂ <i, σ=(σ₁, σ₂), σ₁, σ₂<∞ the equation$\mathfrak{L}$ y=f has a particular solution of this class vfε[ρ, σ]. The general form of a solution of the homogeneous equation$\mathfrak{L}$ y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class$E(T) = \mathop \cup \limits_{\sigma \in T} [\rho ,\sigma ]$, where T is a certain set in R ²₊ depending on the operator$\mathfrak{L}$.