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 Mat. Zametki, 2011, Volume 89, Issue 4, Pages 558–576 (Mi mz6337)

Completeness Theorem for Singular Differential Pencils

D. V. Poplavsky

Saratov State University named after N. G. Chernyshevsky

Abstract: A theorem completeness theorem of special vector functions induced by the products of the so-called Weyl solutions of a fourth-order differential equation and by their derivatives on the semiaxis is presented. We prove that such nonlinear combinations of Weyl solutions and their derivatives constitute a linear subspace of decreasing (at infinity) solutions of a linear singular differential system of Kamke type. We construct and study the Green function of the corresponding singular boundary-value problems on the semiaxis for operator pencils defining differential systems of Kamke type. The required completeness theorem is proved by using the analytic and asymptotic properties of the Green function, operator spectral theory methods, and analytic function theory.

Keywords: singular differential pencil, fourth-order differential equation, Weyl solution, Green function, boundary-value problem, operator spectral theory

DOI: https://doi.org/10.4213/mzm6337

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English version:
Mathematical Notes, 2011, 89:4, 528–544

Bibliographic databases:

UDC: 517.925
Revised: 14.06.2010

Citation: D. V. Poplavsky, “Completeness Theorem for Singular Differential Pencils”, Mat. Zametki, 89:4 (2011), 558–576; Math. Notes, 89:4 (2011), 528–544

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz6337
• https://doi.org/10.4213/mzm6337
• http://mi.mathnet.ru/eng/mz/v89/i4/p558

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. A. Yurko, “Inverse Problems for First-Order Integro-Differential Operators”, Math. Notes, 100:6 (2016), 876–882
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