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The Cauchy problem in spaces of entire functions. (Problème de Cauchy dans des espaces de fonctions entières.) (French) Zbl 0858.35001
In the first part we are interested in the case of differential operators with constant coefficients. We show directly that the solution of the Cauchy problem is an entire function if the data are so. In the second part we study the case of a differential operator, the principal part of which has constant coefficients, first in the space of entire functions and then in the space of entire functions of finite order. The third part concerns the global Cauchy-Kowalevski problem. Our studies are restricted to Cauchy problems, the same approach permits analogous theorems for the Goursat problem. We reduce the proof of the theorems to the proof of the fixed-point theorem in Banach spaces that we define by means of majorant functions.

35A10 Cauchy-Kovalevskaya theorems
30D20 Entire functions of one complex variable, general theory
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients