Geodesic
On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2022), pp. 37-48.

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

In this paper is investigated the question of an a priori estimate and the existence of periodic solutions for one class of systems of ordinary differential equations, in which the main nonlinear part is gradient of a positively homogeneous function. Found the necessary and sufficient conditions that provide an a priori estimate for periodic solutions. It is proved that under the conditions of an a priori estimate, periodic solutions exist if and only if not equal to zero degree mapping of the gradient of a positively homogeneous function on the unit sphere. The novelty of the work is that, firstly, previously obtained results of the authors are generalized for multidimensional systems, and secondly, is proved the formula for calculating degree mapping of the gradient of a positively homogeneous function on the unit sphere.
Keywords: periodic problem, positive homogeneous function, method of guiding functions, a priori estimate, solvability of a periodic problem, degree mapping of a vector field. 
@article{IVM_2022_4_a3,
     author = {E. Mukhamadiev and A. N. Naimov},
     title = {On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {37--48},
     publisher = {mathdoc},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_4_a3/}
}
TY  - JOUR
AU  - E. Mukhamadiev
AU  - A. N. Naimov
TI  - On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2022
SP  - 37
EP  - 48
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2022_4_a3/
LA  - ru
ID  - IVM_2022_4_a3
ER  - 
%0 Journal Article
%A E. Mukhamadiev
%A A. N. Naimov
%T On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2022
%P 37-48
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2022_4_a3/
%G ru
%F IVM_2022_4_a3
E. Mukhamadiev; A. N. Naimov. On a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2022), pp. 37-48. http://geodesic.mathdoc.fr/item/IVM_2022_4_a3/

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

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

[3] Krasnoselskii M. A., Operator sdviga po traektoriyam differentsialnykh uravnenii, Nauka, M., 1966 | MR

[4] Mukhamadiev E., Naimov A. N., “Kriterii suschestvovaniya periodicheskikh i ogranichennykh reshenii dlya trekhmernykh sistem differentsialnykh uravnenii”, Tr. IMM UrO RAN, 27, no. 1, 2021, 157–172

[5] Zvyagin V. G., Kornev S. V., “Metod napravlyayuschikh funktsii v zadache o suschestvovanii periodicheskikh reshenii differentsialnykh uravnenii”, Sovremen. matem. Fundament. napravleniya, 58, 2015, 59–81

[6] Perov A. I., Kaverina V. K., “Ob odnoi zadache Vladimira Ivanovicha Zubova”, Differents. uravneniya, 55:2 (2019), 269–272 | MR | Zbl

[7] Obukhovskii V., Zecca P., Van Loi N., Kornev S., Method of guiding functions in problems of nonlinear analysis, Lect. Notes Math., 2076, Springer-Verlag, Berlin–Heidelberg, 2013 | DOI | MR | Zbl

[8] Mukhamadiev E., “Ogranichennye resheniya i gomotopicheskie invarianty sistem nelineinykh differentsialnykh uravnenii”, Dokl. RAN, 351:5 (1996), 596–598 | MR | Zbl

[9] Borisovich Yu. G., Bliznyakov N. M., Izrailevich Ya. A., Fomenko T. N., Vvedenie v topologiyu, URSS, LENAND, M., 2014 | MR