Skip to main content
SpringerLink
Log in

Completeness theorem for singular differential pencils

  • Published:
Mathematical Notes Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Borg, “Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte,” Acta Math. 78(1), 1–96 (1946).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. G. Gasymov and B. M. Levitan, “On the expansion in products of special solutions of two Sturm-Liouville equations,” Dokl. Akad. Nauk SSSR 310(4), 781–784 (1990) [Soviet Math. Dokl. 41 (1), 113–117 (1990)].

    MathSciNet  Google Scholar 

  3. E. Kh. Khristov, “Expansion in products of the solutions of two Sturm-Liouville problems on the semiaxis,” Differ. Uravn. 16(11), 2023–2029 (1980).

    MathSciNet  MATH  Google Scholar 

  4. E. Kh. Khristov, “On the Λ-operators associated with two Sturm-Liouville problems on the semi-axis,” Inverse Problems 14, 647–660 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Freiling and V. Yurko, Inverse Sturm-Liouville Problems and Their Applications (Nova Sci. Publ., Huntington, NY, 2001).

    MATH  Google Scholar 

  6. V. A. Jurko [V. A. Yurko], “Solution of the Boussinesq equation on the half-line by the inverse problem method,” Inverse Problems 7(5), 727–738 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  7. D. V. Poplavskii, “On the solvability of the initial boundary-value problem for the Bogoyavlenskii system]rd in Mathematics. Mechanics Collection of Scientific Papers (Izd. Saratovsk. Univ., Saratov, 2005), Vol. 7, pp. 98–101 [in Russian].

    Google Scholar 

  8. V. A. Yurko, Inverse Spectral Problems and Their Applications (Izd. Saratovsk. Pedag. Inst., Saratov, 2001) [in Russian].

    Google Scholar 

  9. D. V. Poplavskii, Direct and Inverse Problems of Spectral Analysis and Their Applications to Nonlinear Evolution Operators, Candidate’s Dissertation in Mathematics and Physics (Saratovsk. Gos. Univ., Saratov, 2006) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saratov State University, Saratov, Russia

    D. V. Poplavskii

Authors
  1. D. V. Poplavskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. V. Poplavskii.

Additional information

Original Russian Text © D. V. Poplavskii, 2011, published in Matematicheskie Zametki, 2011, Vol. 89, No. 4, pp. 558–576.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Poplavskii, D.V. Completeness theorem for singular differential pencils. Math Notes 89, 528–544 (2011). https://doi.org/10.1134/S0001434611030242

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434611030242

Keywords

Search

Navigation