In this paper the existence and uniqueness of the solutions to boundary value problems for the first order non-linear system of the ordinary differential equations with three-point boundary conditions are investigated. For the first time the Green function is constructed and the considered problem is reduced to the equivalent integral equations that allow us to prove the existence and uniqueness theorems in differ from existing works, applying the Banach contraction mapping principle and Schaefer's fixed point theorem. An example is given to illustrate the obtained results