This paper is the translation by Giampiero Esposito of a paper originally published in French in Acta Mathematica 88, 141-225 (1952) under the title: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. The first three chapters are devoted to the solution of the Cauchy problem, in the nonanalytic case, for a system of nonlinear second-order hyperbolic partial differential equations with n unknown functions and four independent variables. This task is accomplished in chapter III by using the system of integral equations fulfilled by the solutions of partial differential equations that approximate the original nonlinear system. In chapter IV, such results are applied to the vacuum Einstein equations. The resulting Ricci-flatness condition is expressed, in isothermal coordinates, through nonlinear equations of the kind studied here. It is hence proved that the solution of the Cauchy problem, pertaining to such nonlinear equations, satisfies over the whole of its existence domain the isothermal conditions if the same is true for the initial data. One therefore obtains a solution of the vacuum Einstein equations which is unique up to a coordinate change.