Global linear Cauchy problem

Abstract

This paper is devoted to the study of the Fuchsian Cauchy problem. We establish the global existence and uniqueness of a solution in spaces of functions of class $C^k$, $k\in N\cup \{\infty \}$, with respect to the Fuchsian variable and entire with respect to the other variables. The method of proof is based on the application of the fixed point theorem in some Banach spaces defined by suitable majorant functions.