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On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

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Abstract

We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is 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.

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REFERENCES

  1. Mukhamadiev, E., On the theory of periodic solutions of systems of ordinary differential equations, Dokl. Akad. Nauk SSSR, 1970, vol. 194, no. 3, pp. 510–513.

    MathSciNet  MATH  Google Scholar 

  2. Krasnosel’skii, M.A. and Zabreiko, P.P., Geometricheskie metody nelineinogo analiza (Geometrical Methods of Nonlinear Analysis), Moscow: Nauka, 1975.

    Google Scholar 

  3. Mukhamadiev, E. and ˙ Naimov, A.N., Criteria for the existence of periodic and bounded solutions of three-dimensional systems of differential equations, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2021, vol. 27, no. 1, pp. 157–172.

    MathSciNet  Google Scholar 

  4. Mukhamadiev, E. and ˙ Naimov, A.N., On an a priori estimate and the existence of periodic solutions for a class of systems of nonlinear ordinary differential equations, Izv. VUZov. Mat., 2022, no. 4, pp. 37–48.

  5. Zvyagin, V.G. and Kornev, S.V., Method of guiding functions for existence problems for periodic solutions of differential equations, J. Math. Sci., 2018, vol. 233, no. 4, pp. 578–601.

    Article  MathSciNet  MATH  Google Scholar 

  6. Perov, A.I. and Kaverina, V.K., On a problem posed by Vladimir Ivanovich Zubov, Differ. Equations, 2019, vol. 55, no. 2, pp. 274–278.

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation, project no. 23-21-00032.

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Authors and Affiliations

  1. Vologda State University, Vologda, 160000, Russia

    E. Mukhamadiev & A. N. Naimov

Authors
  1. E. Mukhamadiev
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  2. A. N. Naimov
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Correspondence to E. Mukhamadiev or A. N. Naimov.

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Translated by V. Potapchouck

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Mukhamadiev, E., Naimov, A.N. On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity. Diff Equat 59, 289–291 (2023). https://doi.org/10.1134/S0012266123020131

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  • DOI: https://doi.org/10.1134/S0012266123020131

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