Abstract
We establish sufficient conditions for the existence of a periodic solution of a system of two linear firstorder ordinary differential equations with degenerate asymmetric matrix with derivatives in the case of an arbitrary periodic inhomogeneity.
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A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori [in Russian], Nauka, Moscow (1987).
Yu. Mozer, “Rapidly convergent iterative method and nonlinear differential controls,” Usp. Mat. Nauk, 23, No. 4, 179–238 (1968).
V. A. Eremenko, “On the reduction of a linear system of differential equations having a degenerate matrix with derivatives,” Ukr. Mat. Zh., 32, No. 2, 168–174 (1980).
V. A. Eremenko, Investigation of Oscillating Systems with Quasiperiodic Coefficients [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1981).
V. L. Kulik and V. A. Eremenko, “On quasiperiodic solutions of a linear system of differential equations having a degenerate matrix with derivatives,” Ukr. Mat. Zh., 32, No. 6, 746–753 (1980).
V. A. Eremenko, “Invariant tori of linear extensions of dynamical systems on a torus having a degenerate matrix with derivatives,” Ukr. Mat. Zh., 36, No. 6, 865–868 (1984).
V. Kh. Simokon’ and E. P. Trofimchuk, “On the regularity of linear systems having a degenerate matrix with derivatives,” Ukr. Mat. Zh., 45, No. 2, 279–286 (1993).
V. A. Eremenko, “On periodic solutions of linear degenerate second-order ordinary differential equations,” Ukr. Mat. Zh., 49, No. 8, 1137–1142 (1997).
Jk. V. Sibuya, “Some global properties of matrices of functions of one variable,” Math. Ann., 161, No. 1, 67–77 (1965).
V. Vazov, Asymptotic Expansions of Solutions of Ordinary Differential Equations [Russian translation], Mir, Moscow (1968).
A. M. Samoilenko, “Quasiperiodic solutions of a system of linear algebraic equations with quasiperiodic coefficients,” in: Analytic Methods for Investigation of Solutions of Nonlinear Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1975), pp. 5–26.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 350–356, March, 1998.
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Eremenko, V.A. Periodic solutions of systems of two linear first-order ordinary differential equations with degenerate asymmetric matrix with derivatives. Ukr Math J 50, 400–407 (1998). https://doi.org/10.1007/BF02528805
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DOI: https://doi.org/10.1007/BF02528805