The author gives a constructive approach to the solution of the linear nonhomogeneous boundary value problem \(z'= A(t)z+ f(z)\) in \([a,b]\), \(\langle z^*,z\rangle=0\), where \(A\in C([0,1], M_ n({\mathbb R}))\) and \(z^*\in C^ 1[a,b]^*\), and also to the weakly perturbed nonlinear problem \(z'= A(t)z+ f(t)+\varepsilon g(z,t,\varepsilon)\). The approach is based on perturbation theory and the use of generalized Green matrices. This book is an interesting contribution to the theory of nonlinear oscillations of weakly perturbed boundary value problems.