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On the Solvability of a Periodic Problem for Nonlinear Ordinary Differential Equation of the Second Order

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Abstract

The article deals with a periodic problem for a nonlinear ordinary differential equation of the second order with the main positively homogeneous part selected. The paper uses the research scheme previously implemented by the first author in the study of the third two-point boundary value problem for a nonlinear ordinary differential equation of the second order. According to the research scheme, first in terms of the properties of the main positively homogeneous part, the conditions for a priori estimate of periodic solutions are found. Then, in conditions of a priori estimate, the theorems on the solvability of the periodic problem are formulated and proved using methods for calculating the rotation of vector fields. The obtained results can subsequently be generalized for systems of nonlinear ordinary differential equations of the second order.

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ACKNOWLEDGMENTS

The authors express their sincere gratitude to Professor E. Mukhamadiev for his attention to this work.

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Authors and Affiliations

  1. Vologda State University, 15 Lenin str., 160000, Vologda, Russia

    A. N. Naimov

  2. Russian-Tajik (Slavonic) University, 30 M. Tursun-zade str., 734000, Dushanbe, Republic of Tajikistan

    M. M. Kobilzoda

Authors
  1. A. N. Naimov
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  2. M. M. Kobilzoda
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Corresponding authors

Correspondence to A. N. Naimov or M. M. Kobilzoda.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 8, pp. 56–65.

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Naimov, A.N., Kobilzoda, M.M. On the Solvability of a Periodic Problem for Nonlinear Ordinary Differential Equation of the Second Order. Russ Math. 65, 49–57 (2021). https://doi.org/10.3103/S1066369X21080065

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  • DOI: https://doi.org/10.3103/S1066369X21080065

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