We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonautonomous nonlinear periodic boundary-value problem for a Rayleightype equation unsolved with respect to the derivative. The urgency of investigation of nonautonomous boundary-value problems unsolved with respect to the derivative is explained by the fact that the analysis of traditional problems solved with respect to the derivative is sometimes significantly complicated, e.g., in the presence of nonlinearities that are not integrable in elementary functions. We consider the critical case in which the equation for generating amplitudes of a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation does not turn into the identity. The least-squares method is used to establish constructive conditions for the solvability and propose convergent iterative schemes for the construction of approximate solutions to a nonautonomous nonlinear boundary-value problem unsolved with respect to the derivative. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems unsolved with respect to the derivative in the case of periodic problem for the equation that describes the motion of a satellite on the elliptic orbit. We obtain an estimate for the range of values of a small parameter in which the iterative procedure used for the construction of solutions to a weakly nonlinear periodic boundary-value problem for a Rayleigh-type equation unsolved with respect to the derivative is convergent. To check the accuracy of the proposed approximations, we estimate the discrepancies appearing in the equation used to simulate the motion of satellites along the elliptic orbits.
Similar content being viewed by others
References
V. F. Zaitsev and A. D. Polyanin, A Handbook on Nonlinear Ordinary Differential Equations [in Russian], Faktorial, Moscow (1997).
Yu. D. Shlapak, “Periodic solutions of nonlinear second-order equations which are not solved for the highest derivative,” Ukr. Mat. Zh., 26, No. 6, 850–854 (1974); English translation: Ukr. Math. J., 26, No. 6, 702–706 (1974).
A. M. Samoilenko, S. M. Chuiko, and O. V. Nesmelova, “Nonlinear boundary-value problems unsolved with respect to the derivative,” Ukr. Mat. Zh., 72, No. 8, 1106–1118 (2020); English translation: Ukr. Math. J., 72, No. 8, 1280–1293 (2021).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, de Gruyter, Berlin (2016).
A. P. Torzhevskii, “Periodic solutions of the equation of plane oscillations of a satellite on the elliptic orbit,” Kosm. Issled., 2, No. 5, 667–678 (1964).
S. M. Chuiko and O. V. Nesmelova (Starkova), “Autonomous Noether boundary-value problems not solved with respect to the derivative,” J. Math. Sci., 230, No. 5, 799–803 (2018).
D. Nucez and P. J. Torres, “Stable odd solutions of some periodic equations modeling satellite motion,” J. Math. Anal., 279, No. 2, 700–709 (2003).
A. Cabada and A. J. A. Cid, “On a class of singular Sturm–Liouville periodic boundary value problems,” Nonlin. Anal. Real World Appl., 12, No. 4, 2378–2384 (2011).
L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).
S. M. Chuiko, “To the generalization of the Newton–Kantorovich theorem,” Visn. Karazin Kharkiv Nats. Univ., Ser. Mat., Prykl. Mat. Mekh., 85, No. 1, 62–68 (2017).
N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).
S. M. Chuiko, “On approximate solution of boundary-value problems by the least squares method,” Nelin. Kolyv., 11, No. 4, 554–573 (2008); English translation: Nonlin. Oscilat., 11, No. 4, 585–604 (2008).
I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).
G. T. Gilbert, “Positive definite matrices and Sylvester’s criterion,” Amer. Math. Monthly, 98, No. 1, 44–46 (1991).
S. M. Chuiko, “Domain of convergence of an iterative procedure for an autonomous boundary-value problem,” Nelin. Kolyv., 9, No. 3, 416–432 (2006); English translation: Nonlinear Oscillat., 9, No. 3, 405–422 (2006).
S. Chuiko, “Weakly nonlinear boundary value problem for a matrix differential equation,” Miskolc Math. Notes, 17, No. 1, 139–150 (2016).
S. M. Chuiko and A. S. Chuiko, “On approximate solution of periodic boundary-value problems with delay by the least-squares method in the critical case,” Nelin. Kolyv., 14, No. 3, 419–432 (2011); English translation: Nonlin. Oscillat., 14, No. 3, 445–460 (2012).
G. V. Demidenko and S. V. Uspensky, Equations and Systems Unsolved with Respect to the Derivative [in Russian], Nauch. Kniga, Novosibirsk (1998).
A. F. Filippov, “Uniqueness of the solution of a system of differential equations unsolved for the derivatives,” Different. Equat., 41, No. 1, 90–95 (2005).
A. V. Arutyunov, E. S. Zhukovskii, and S. E. Zhukovskii, “On the well-posedness of differential equations unsolved for the derivative,” Different. Equat., 47, No. 11, 1541–1555 (2011).
S. M. Chuiko, O. V. Starkova, and O. E. Pirus, “Nonlinear Noetherian boundary-value problems unsolved with respect to the derivative,” Dinam. Sist., 2(30), No. 1-2, 169–186 (2012).
A. A. Boichuk and A. A. Pokutnyi, “Perturbation theory of operator equations in the Fréchet and Hilbert spaces,” Ukr. Mat. Zh., 67, No. 9, 1181–1188 (2015); English translation: Ukr. Math. J., 67, No. 9, 1327–1335 (2016).
O. A. Boichuk and O. A. Pokutnyi, Normally Solvable Boundary-Value Problems for Operator-Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2022).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 10, pp. 1429–1440, October, 2023. Ukrainian https://doi.org/10.37863/umzh.v75i10.7362.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chuiko, S., Nesmelova, O. Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative. Ukr Math J 75, 1621–1633 (2024). https://doi.org/10.1007/s11253-024-02282-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-024-02282-2