Asymptotic analysis of certain classes of singularly perturbed problems on the semiaxis

We propose a new method for asymptotic integration of certain classes of singularly perturbed Cauchy problems on the semiaxis for nonhomogeneous systems of linear ordinary differential equations; this method is a further development of the ideas of the regularization method. This method enables us to prove the existence of a unique bounded (as $\varepsilon\to+0$) solution of such problems and leads to a simpler and more constructive algorithm for obtaining the asymptotic expansion of the solution and singling out all of its singularities in closed analytic form (including the critical case in which the spectral points of the limit operator may touch the imaginary axis). The proposed method supplements and sharpens earlier results.