Kocherga O. The asymptotic solution of Cauchy problem for the degenerated singularly perturbed systems of the differential equations

The thesis work is devoted to constructing the asymptotic solution of Cauchy problem for the degenerated singularly perturbed systems of differential equations. The sufficient conditions of existence and uniqueness of Cauchy problem solution in case of the system regularity have been found out. Proceeding from the known results of the asymptotic analysis of the given sys-tem's general solution, the method of direct construction of the initial problem's formal solution has been worked out. But it is necessary to take into account the found conditions of its existence and uniqueness. The criterion for determining the degrees of small parameter has been fixed. From this power the corresponding formal expansions in different cases of spectrum behaviour of the limit bundle of matrixes began. The algorithm for determining the coefficients of the corresponding formal expansions in case of simple and multiple spectrum of the limit bundle of the matrices has been worked out. The peculiarities of constructing solutions in the so-called non-critical and critical cases have been studied. The conditions under which the constructed formal solutions are the asymptotic expansions of the corresponding exact solutions have been found out.