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Nonlinear Boundary-Value Problems Unsolved with Respect to the Derivative

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Ukrainian Mathematical Journal Aims and scope

We establish constructive necessary and sufficient conditions of solvability and a scheme of construction of the solutions for a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems for a Rayleigh-type equation unsolved with respect to the derivative, in particular, in the case of periodic problem for the equation used to describe the motion of satellites on elliptic orbits.

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References

  1. A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, de Gruyter, Berlin (2016).

    Book  Google Scholar 

  2. 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).

  3. S. M. Chuiko and O. V. Starkova, “Autonomous Noether boundary-value problems not solved with respect to the derivative,” J. Math. Sci., 232, No. 5, 783–799 (2018).

    MathSciNet  Google Scholar 

  4. A. P. Torzhevskii, “Periodic solutions of the equation of plane oscillations of a satellite on an elliptic orbit,” Kosmich. Issled., 2, No. 5, 667–678 (1964).

    Google Scholar 

  5. 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).

  6. S. M. Chuiko, A. S. Chuiko, and O. V. Starkova, “Periodic problem for the Liénard equation unsolved with respect to the derivative in the critical case,” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 29 (2015), pp. 157–171.

  7. I. G. Malkin, Some Problems of the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).

    MATH  Google Scholar 

  8. S. M. Chuiko, “Weakly nonlinear boundary-value problem in a special critical case,” Ukr. Mat. Zh., 61, No. 4, 548–562 (2009); English translation: Ukr. Math. J., 61, No. 4, 657–673 (2009).

  9. A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Normally Solvable Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (2019).

    Google Scholar 

  10. A. S. Chuiko, “Domain of convergence of an iteration procedure for a weakly nonlinear boundary-value problem,” Nelin. Kolyv., 8, No. 2, 278–288 (2005); English translation: Nonlin. Oscillat., 8, No. 2, 277–287 (2005).

  11. D. K. Lika and Yu. A. Ryabov, Iterative Methods and Majorizing Lyapunov Equations in the Theory of Nonlinear Oscillations [in Russian], Shtiintsa, Kishinev (1974).

    Google Scholar 

  12. L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  13. E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods for the Analysis of Nonlinear Systems [in Russian], Nauka, Moscow (1979).

    MATH  Google Scholar 

  14. O. B. Lykova and A. A. Boichuk, “Construction of periodic solutions of nonlinear systems in critical cases,” Ukr. Mat. Zh., 40, No. 1, 62–69 (1988); English translation: Ukr. Math. J., 40, No. 1, 51–58 (1988).

  15. 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).

    MATH  Google Scholar 

  16. S. M. Chuiko, “On the generalization of the Newton–Kantorovich theorem in Banach spaces,” Dop. Nats. Akad. Nauk Ukr., No. 6, 22–31 (2018).

    Article  MathSciNet  Google Scholar 

  17. V. F. Zaitsev and A. D. Polyanin, A Handbook of Nonlinear Ordinary Differential Equations [in Russian], Faktorial, Moscow (1997).

    Google Scholar 

  18. S. L. Campbell, Singular Systems of Differential Equations, Pitman Advanced Publishing Program, San Francisco (1980).

    MATH  Google Scholar 

  19. A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degenerations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).

    Google Scholar 

  20. O. A. Boichuk and L. M. Shehda, “Degenerate nonlinear boundary-value problems,” Ukr. Mat. Zh., 61, No. 9, 1174–1188 (2009); English translation: Ukr. Math. J., 61, No. 9, 1387–1403 (2009).

  21. S. M. Chuiko, “On a reduction of the order in a differential-algebraic system,” J. Math. Sci., 235, No. 1, 2–18 (2018).

    Article  MathSciNet  Google Scholar 

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Correspondence to O. V. Nesmelova.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 8, pp. 1106–1118, August, 2020. Ukrainian DOI: 10.37863/umzh.v72i8.5986.

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Samoilenko, A.M., Chuiko, S.M. & Nesmelova, O.V. Nonlinear Boundary-Value Problems Unsolved with Respect to the Derivative. Ukr Math J 72, 1280–1293 (2021). https://doi.org/10.1007/s11253-020-01852-4

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  • DOI: https://doi.org/10.1007/s11253-020-01852-4

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