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Strongly Invariant Subspaces of Nonautonomous Linear Periodic Systems and Solutions Whose Period Is Incommensurable with the Period of the System Itself

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Abstract

We introduce the notions of quasi-invariant and strongly invariant subspaces of a one-parameter family of linear operators acting on a finite-dimensional vector space. The geometric meaning of these notions is that the restrictions of all operators of the family to a quasiinvariant subspace coincide and that the restrictions to a strongly invariant subspace are, in addition, an endomorphism of that subspace. These notions are used to reduce the well-known problem on Ω-periodic solutions of an ω-periodic linear differential system with incommensurable Ω and ω to the algebraic problem on the eigenvalues and eigenvectors of some matrix constructed from the right-hand side of the system.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 220072, Belarus

    V. T. Borukhov

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  1. V. T. Borukhov
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Correspondence to V. T. Borukhov.

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Original Russian Text © V.T. Borukhov, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 5, pp. 585–591.

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Borukhov, V.T. Strongly Invariant Subspaces of Nonautonomous Linear Periodic Systems and Solutions Whose Period Is Incommensurable with the Period of the System Itself. Diff Equat 54, 578–585 (2018). https://doi.org/10.1134/S0012266118050026

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  • DOI: https://doi.org/10.1134/S0012266118050026

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