Samusenko P. Asymptotic integration of singularly perturbed systems of differential-functional equations with degenerations.

The Dissertation deals with the research into the properties of the solutions of some types of the singular systems of differential-functional equations. The paper presents the methodology of the research into the singular systems of differential-functional equations by means of which it is possible to describe the asymptotic properties of the solution of a corresponding initial value problem for the case when Kronecker matrix pencil structure generated by a linear part of these systems is not stable. The sufficient conditions of the existence and uniqueness of the solution of the Cauchy problem are presented for the singular system of differential equations. The asymptotic expansion of this solution is constructed. The theorems on the accuracy with which constructed asymptotic solution satisfies the system are proved. Similar results are obtained for the Cauchy problem for the singular system of singularly perturbed differential equations and for an initial value problem for the singular system of singularly perturbed differential equations with small delay.