Solutions of finitely smooth nonlinear singular differential equations and problems of diagonalization and triangularization

摘要

It is known that existence of a formal power series solution (y) over cap(x) to a system of nonlinear ordinary differential equations (ODEs) with analytic or infinitely smooth coefficients at an irregular singular point implies the existence of an actual solution y(x), which possesses the asymptotic expansion (y) over cap(x). In the present paper we extend this result for systems with finitely smooth coefficients. In this case one cannot speak about a formal power series solution (y) over cap(x); it has therefore to be replaced by the requirement of existence of an "approximate" solution y(o)(x). The existence of a corresponding actual solution is a subject of certain conditions that link the smoothness of the system, the "accuracy" of the approximation y(o)(x), and the "degeneracy" of the system, linearized with respect to y(o)(x). As applications, problems of reduction of linear time dependent systems of ODEs into diagonal and triangular forms, as well as some other problems, are considered. In particular, the well-known theorem on integration of linear systems with irregular singularities is extended from analytical to finitely smooth systems. In one of the simplest cases, our result is simultaneously a consequence of the classical Levinson theorem. [References: 21]