ON THE REDUCTION OF THE LINEAR SYSTEM OF THE DIFFERENTIAL EQUATIONS WITH COEFFICIENTS OF OSCILLATING TYPE TO THE TRIANGULAR KIND IN THE RESONANT CASE

For the linear homogeneous differential system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, the conditions of the existence of the transformation which leads it to triangular kind, are obtained in the resonant cases. MSC 2010: 34A30, 34C25.


Introduction
In the theory of linear systems of differential equations is well known problem of the construction for the linear homogeneous system of the differential equations where ( ) = ( ( )) , =1, , ( ) ≡ 0 ( < ) [1][2][3][4]. In this paper, we assume, that the system (1) already reduced to a kind, close to triangular: where -small parameter, and the matrix ( ) has a some special kind. And we study the problen on bringing the system (2) to a purely triangular form where ( ) = ( ( )) , =1, , ≡ 0 ( < ). This paper continues the research, begun in the paper [5]. The basic notation and definitions of the paper [5] are retained. As in the paper [5], we will study this problem for a third-order system ( = 3) so as not to clutter up the presentation with secondary technical difficulties associated with the dimension of the system. All fundamental difficulties take place in this case too.
Proof. We substitute the expression (8) into system (7), and require that the transformed system has the kind (9). We obtain the next chain of matrix differential equations for detemining matrices Ψ 1 , ..., Ψ : Then the matrix at sufficiently small values is determined from the equation: We consider the equation (11). In the component it looks like this: Define 1 , 1 , 1 by the following expression: All the elements of matrix 1 belongs to the class ( ; 0 ). All the elements of matrix Ψ 1 belongs to the class ( ; 0 ; ). All the elements of matrix 1 belongs to the class ( − 1; 0 ; ).
All the equations (12) are considered similarly to equations (11), and so the matrices Ψ , , ( = 1, ) are determined. And also all the elements of matrix Ψ belongs to the class ( ; 0 ; ), all the elements of matrix belongs to the class ( ; 0 ), all the elements of matrix belongs to the class ( − 1; 0 ; ) ( = 1, ). Matrix are determined from the equations (13).
Lemma are proved.
Proof is complete analogous to the proof of the Theorem 3 from [5]. Now for different relationships between we get for the system (17) more specific conditions of existence of the transformation (24). We will check only condition 1) of the theorem, assuming conditions 2) and 3) to be satisfied.
(28) We assume, that the system (28) has a solution ( , ), such that Then, in accordance with the small parameter method, all systems (23) will have a solution, belongs to the class ( ; 0 ; ). Consequently, the functions * ( , , , ) in (20) will also belongs to the class ( ; 0 ; ).
Based on the lemma, using the transformation of kind )︃ (32) we will lead the system (31) to the kind (25).
We assume, that the system (34) has a solution 12 ( , ), 23 ( , ) such that Further reasonings are the same as in case 1.
We assume, that the system (37) has a solution 12 ( , ), 23 ( , ) such that Further reasonings are the same as in case 1.

Conclusion
Thus, for the system (2) the conditions of the existence of the transformation with coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, which leads it to triangular kind, are obtained in the resonant cases.