Cherevko I. Integral manifolds and approximate methods of investigating differential-functional equations

The dissertation is devoted to the development of research methodology of investigation of singularly perturbed functional differential equations with the help of integral manifolds method, construction and establishment of approximation algorithms for differential equations with delay. New theorems of existence of integral manifolds of fast and slow variables of nonlinear singularly perturbed functional differential equations have been proved. The change of variables is constructed, which decomposes nonlinear singularly perturbed system to the triangle-block form in the neighbourhood of manifolds of slow variables, and splitted linear singularly perturbed system into the two independent fast and slow subsystems. The theorems of existence of stable, center-stable, center, center-unstable integral manifolds for the system of singularly perturbed functional differential equations with time lag on fast and slow variables are proved. Properties of these manifolds are investigated. The approximation schemes of initial and boundary value problems for differential-difference equations are studied, the algorithm of investigation of asymptotiс stability (unstability) of the trivial solution of equations with delay is constructed. The iteration methods for obtaining the solutions of boundary value problems for differential-difference equations with delay have been established.