Vira M. The asymptotic solution of the boundary-value problem for the linear singularly perturbed systems of differential equations with the degenerations

The dissertation research is devoted to construction of the asymptotic solutions of the boundary-value problem for the linear singularly perturbed system of differential equations with degenerated matrix at derivative.The method of construction of asymptotic solutions of the given boundary-value problem is suggested. This method is based on the idea of reducing of the given boundary-value problem to Fredholm's or Noether's one by neglecting of exponentially small items in corresponding boundary conditions. The algorithm of construction of the asymptotics in the cases of simple and multiple spectrum of the boundary bundle of matrixes is worked out. The existence and uniqueness conditions for the solution of this boundary-value problem in each case have been found. The existence conditions for the multi-parametric family of solutions of the given boundary-value problem have been found as well. The asymptotic estimates for the constructed solutions are obtained; the algorithm of finding of the coefficients of the corresponding series in explicit form is developed. The particularities of construction of the solutions in stable and semi-stable cases are investigated.