We establish sufficient conditions for the existence of quasiperiodic solutions of a system of ordinary second-order differential equations with degenerate symmetric matrix of the second derivatives for an arbitrary quasiperiodic inhomogeneity.
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A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Degenerate Linear Systems of Differential Equations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).
V. O. Er’omenko, “On quasiperiodic solutions of degenerate linear ordinary second-order differential equations,” in: Nonlinear Problems of Analysis, Abstr. of the Fourth All-Ukrainian Scientific Conference [in Ukrainian], Plai, Ivano-Frankivs’k (2008), p. 33.
P. Lankaster, Theory of Matrices [Russian translation], Nauka, Moscow (1978).
A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori [in Russian], Moscow, Nauka (1987).
J. Moser, “A rapidly convergent iteration method and nonlinear differential equations. I,” Annali della Scuola Norm Supper. de Pisa Ser. III, 20, No. 2, 265–315 (1966); “A rapidly convergent iteration method and nonlinear differential equations. II,” Annali della Scuola Norm Supper. de Pisa Ser. III, 20, No. 3, 499–535 (1966).
A. M. Samoilenko, V. O. Er’omenko, and A. A. Davydenko, “Smoothness of quasiperiodic solutions of linear systems of ordinary differential equations with degenerate symmetric matrix by the derivatives,” Dop. Nats. Akad. Nauk Ukr., No. 4, 21–27 (2001).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 6, pp. 773–783, June, 2010.
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Er’omenko, V.O., Aliluiko, A.M. Quasiperiodic solutions of degenerate linear systems of second-order ordinary differential equations. Ukr Math J 62, 894–906 (2010). https://doi.org/10.1007/s11253-010-0398-3
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DOI: https://doi.org/10.1007/s11253-010-0398-3