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

G. G. Braichev

Matematičeskie zametki, Tome 19 (1976) no. 2, pp. 225-236

Résumé

In this paper it is shown that under conditions of applicability of the operator $\mathfrak Ly=\sum_{k\ge0}{a_kD^ky(x)}$ to the class $[\rho,\sigma]$, $\rho=(1,\rho_2$, $\rho_2<1$, $\sigma=(\sigma_1, \sigma_2)$, $\sigma_1,\sigma_2<\infty$ the equation $\mathfrak Ly=f$ has a particular solution of this class $\forall\,f\in[\rho,\sigma]$. The general form of a solution of the homogeneous equation $\mathfrak Ly=0$ 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 $E(T)=\bigcup\limits_{\sigma\in T}[\rho,\sigma]$, where $T$ is a certain set in $R_+^2$ depending on the operator $\mathfrak L$.