517.919

degenerate systems of linear differential equations, periodic solutions.

The effective sufficient conditions for a positive definite symmetrization of a differential operator based on a system of two linear first order ordinary differential equations with an asymmetric variable rank matrix in the derivatives were established. According to these conditions, the existence of a periodic solution for the arbitrary periodic inhomogeneity and the Galerkin iterative method of its approximate construction was confirmed. The approach for the research of  numbers of the equations, where , was described.

1. Samoilenko A. M., Shkil M. I., Yakovets V. P. Liniini systemy dyferentsialnykh rivnian z vyrodzhenniamy. Kyiv: Vyshcha shkola, 2000. 294 p. [Іn Ukrainian].
2. Mozer Yu. Bystrosxodyashhijsya metod iteracij i nelinejnye uravneniya. Uspexi mat. Nauk. Vol. 23.
   No. 4. 1968. Р. 179–238. [Іn Russian].
3. Kulik V. L., Eremenko V. A. Quasiperiodic solutions of a linear system of differential equations with a singular matrix in the derivatives.Ukr. Mat. Zh. Vol. 32. No. 6. 1980. Р. 502–508.
4. Simokon' V. Kh., Trokhimchuk E. P. On regularity of linear systems with a degenerate matrix by the derivative. Ukr. Mat. Zh. Vol. 45. No. 3. 1993. Р. 299–308.
5. Yeromenko V. O. Periodychni rozviazky funktsionalno-synhuliarno zburenykh liniinykh zvychainykh dyferentsialnykh rivnian vyshchykh poriadkiv, Matematychne ta kompiuterne modeliuvannia. Seriia: fizyko-matematychni nauky: zb. nauk. prats. Vol. 6. 2012. Р. 97–113. [Іn Ukrainian].
6. Yeromenko V. O., Aliluiko A. M. Kvaziperiodychni rozviazky funktsionalno-synhuliarno zburenykh liniinykh zvychainykh dyferentsialnykh rivnian vyshchykh poriadkiv. Neliniini kolyvannia. Vol. 21.
   No. 4. 2018. Р. 457–469. [Іn Ukrainian].
7. Eremenko V. A. Periodic solutions of systems of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives. Ukr. Mat. Zh. Vol. 50. No. 3. 1998. Р. 400–407.
8. Samojlenko A. M. E'lementy matematicheskoj teorii mnogochastotnyx kolebanij. Invariantnye tory, Moscva: Nauka, 1987. 304 p. [Іn Russian].
9. Er’omenko V. O., Aliluiko A. M. Periodic solutions of linear degenerate systems of ordinary differential equations of the second order. Nonlinear Oscill. Vol. 13. No. 3. 2011. Р. 361–371.

1. Samoilenko A. M., Shkil M. I., Yakovets V. P. Liniini systemy dyferentsialnykh rivnian z vyrodzhenniamy. Kyiv: Vyshcha shkola, 2000. 294 p. [Іn Ukrainian].
2. Mozer Yu. Bystrosxodyashhijsya metod iteracij i nelinejnye uravneniya. Uspexi mat. Nauk. Vol. 23.
   No. 4. 1968. Р. 179–238. [Іn Russian].
3. Kulik V. L., Eremenko V. A. Quasiperiodic solutions of a linear system of differential equations with a singular matrix in the derivatives.Ukr. Mat. Zh. Vol. 32. No. 6. 1980. Р. 502–508.
4. Simokon' V. Kh., Trokhimchuk E. P. On regularity of linear systems with a degenerate matrix by the derivative. Ukr. Mat. Zh. Vol. 45. No. 3. 1993. Р. 299–308.
5. Yeromenko V. O. Periodychni rozviazky funktsionalno-synhuliarno zburenykh liniinykh zvychainykh dyferentsialnykh rivnian vyshchykh poriadkiv, Matematychne ta kompiuterne modeliuvannia. Seriia: fizyko-matematychni nauky: zb. nauk. prats. Vol. 6. 2012. Р. 97–113. [Іn Ukrainian].
6. Yeromenko V. O., Aliluiko A. M. Kvaziperiodychni rozviazky funktsionalno-synhuliarno zburenykh liniinykh zvychainykh dyferentsialnykh rivnian vyshchykh poriadkiv. Neliniini kolyvannia. Vol. 21.
   No. 4. 2018. Р. 457–469. [Іn Ukrainian].
7. Eremenko V. A. Periodic solutions of systems of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives. Ukr. Mat. Zh. Vol. 50. No. 3. 1998. Р. 400–407.
8. Samojlenko A. M. E'lementy matematicheskoj teorii mnogochastotnyx kolebanij. Invariantnye tory, Moscva: Nauka, 1987. 304 p. [Іn Russian].
9. Er’omenko V. O., Aliluiko A. M. Periodic solutions of linear degenerate systems of ordinary differential equations of the second order. Nonlinear Oscill. Vol. 13. No. 3. 2011. Р. 361–371.