We propose a new successive approximation technique for the solvability analysis and approximate solution of general non-local boundary value  problems for non-linear systems of differential equations with locally Lipschitzian non-linearities. It will be studied the non-linear boundary value problem 
$\frac{dx(t)}{dt}=f(t,x(t)),\,t\in \left[ a,b\right] ,  \Phi (x)=d, $
where $\Phi :C(\left[ a,b\right] ,\mathbb{R}^{n})\rightarrow \mathbb{R}^{n}$ is a vector functional (possibly non-linear), $~f:\left[ a,b\right]
\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}~$is a function satisfying the Caratheodory conditions in a certain bounded set $D$, which will be concretized later, $d$ is a given vector and $f\in Lip(K,D),$ i.e. $f$ locally Lipschitzian 
$\left\vert f(t,u)-f(t,v)\right\vert \leq K\left\vert u-v\right\vert ,~\text{for all}\ \left\{ u,v\right\} \subset D~\text{ }a.e.t\in \left[ a,b\right] . $
By a solution of the problem, one means an absolutely continuous function satisfying the differential system almost everywhere on $\left[ a,b\right] .$ The analysis is constructive in the sense that it allows one to both study the solvability of the problem and approximately construct its solutions by operating with objects that are determined explicitly in finitely many steps of computation. The practical application of the technique is explained on a numerical example.