We consider linear partial differential equations with several Fuchsian
variables in the sense of M.S. Baouendi and C. Goulaouic [1].
For a holomorphic Fuchsian operator with holomorphic Fuchsian principal
part,
we prove existence and uniqueness of a holomorphic local solution.
Our theorem generalizes the results of ([3, 1, 11]),
precises the one of [4] and reduces the proof
of their theorems to the proof of the fixed-point theorem.
For a holomorphic Fuchsian operator
with constant Fuchsian principal part,
we establish the existence and uniqueness
of a holomorphic global solution. Our aim is to simplify its proof.
The methods of proof are based on the application
of the fixed-point theorem in some Banach spaces defined
by majorant functions that are suitable to this kind of equations.