THE ANALOGUE OF THE FLOQUET-LYAPUNOV THEOREM FOR THE LINEAR DIFFERENTIAL SYSTEMS OF THE SPECIAL KIND

The analogue of the well known in the theory of the linear differential systems Floquet’s– Lyapunov’s theorem are constructed by the certain condidtions for the linear differential system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency. MSC: 34A30, 34C25.

Introduction. In the theory of linear systems of differential equations is well known the Floquet-Lyapunov theorem [1]. The fundamental matrix ( ) of the linear homogeneous system where ( ) -is a continuous -periodic matrix, has a kind: where ( ) -is a -periodic matrix, and -is a constant matrix. There exists many analogues of this theorem for the linear systems of different types, for example, for the systems with quasiperiodic coefficients [2], for the countable systems of differential equations [3], for the differential equations in the Banach spaces [4] and other.
The purpose of this paper is to obtain of analogue of Floquet-Lyapunov theorem for the linear systems of differential equations whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency. Here we make substantial use of the results of our paper [5].
Under the slowly varying function we mean a function of class ( ; 0 ).
Main Results.
Theorem. Let for the system (3) the condition (6) By virtue Lemma 1 we concluding, that Conclusion. Thus, the analogue of the Floquet-Lyapunov theorem, well known in the theory of linear homogeneous systems of the differential equations, are obtained for the linear homogeneous systems, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency.