Geodesic
On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 37-45.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we examine the solvability of a periodic problem for a system of ordinary differential equations whose principal nonlinear part is a quasi-homogeneous mapping. We prove that if an unperturbed system with quasi-homogeneous nonlinearity has no nonzero bounded solutions, then the periodic problem admits an a priori estimate. The results obtained are of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.
Keywords: periodic problem, quasi-homogeneous nonlinearity, a priori estimate, vector field, rotation of a vector field, homotopic vector fields 
@article{INTO_2024_233_a3,
     author = {A. N. Naimov and M. V. Bystretskii},
     title = {On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {37--45},
     publisher = {mathdoc},
     volume = {233},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2024_233_a3/}
}
TY  - JOUR
AU  - A. N. Naimov
AU  - M. V. Bystretskii
TI  - On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2024
SP  - 37
EP  - 45
VL  - 233
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2024_233_a3/
LA  - ru
ID  - INTO_2024_233_a3
ER  - 
%0 Journal Article
%A A. N. Naimov
%A M. V. Bystretskii
%T On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2024
%P 37-45
%V 233
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2024_233_a3/
%G ru
%F INTO_2024_233_a3
A. N. Naimov; M. V. Bystretskii. On the solvability of a periodic problem for a system of ordinary differential equations with quasi-homogeneous nonlinearity. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 37-45. http://geodesic.mathdoc.fr/item/INTO_2024_233_a3/

[1] Bobylev N. A., “O postroenii pravilnykh napravlyayuschikh funktsii”, Dokl. AN SSSR., 183:2 (1968), 265–266 | Zbl

[2] Zvyagin V. G., Kornev S. V., “Metod napravlyayuschikh funktsii v zadache o suschestvovanii periodicheskikh reshenii differentsialnykh uravnenii”, Sovr. mat. Fundam. napr., 58:1 (2015), 59–81 | MR

[3] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975

[4] Lions Zh. L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972

[5] Mukhamadiev E., “K teorii periodicheskikh reshenii sistem obyknovennykh differentsialnykh uravnenii”, Dokl. AN SSSR., 194:3 (1970), 510–513 | Zbl

[6] Mukhamadiev E., Naimov A. N., “O razreshimosti periodicheskoi zadachi dlya sistemy obyknovennykh differentsialnykh uravnenii s glavnoi polozhitelno odnorodnoi nelineinostyu”, Differ. uravn., 59:2 (2023), 280–282 | DOI | MR | Zbl

[7] Mukhamadiev E., Naimov A. N., “Ob apriornoi otsenke i suschestvovanii periodicheskikh reshenii dlya odnogo klassa sistem nelineinykh obyknovennykh differentsialnykh uravnenii”, Izv. vuzov. Mat., 2022, no. 4, 37–48 | Zbl

[8] Perov A. I., Kaverina V. K., “Ob odnoi zadache Vladimira Ivanovicha Zubova”, Differ. uravn., 55:2 (2019), 269–272 | DOI | Zbl