Ukrainian Mathematical Journal

, Volume 62, Issue 6, pp 928–942 | Cite as

On solutions of one class of second-order operator differential equations in the class of holomorphic vector functions

  • S. S. Mirzoev
  • S. G. Veliev

We establish sufficient conditions for the completeness of a part of root vectors of one class of the second-order operator bundles corresponding to the characteristic numbers from a certain sector and prove the theorem on completeness of a system of elementary holomorphic solutions of the corresponding second-order homogeneous operator differential equations. We also indicate the conditions of correct and unique solvability of a boundary-value problem for the analyzed equation with linear operator in the boundary condition and estimate the norm of the operator of the intermediate derivative in the perturbed part of the equation.


Hilbert Space Vector Function Elementary Solution Regular Solution Characteristic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Dunod, Paris (1968).zbMATHGoogle Scholar
  2. 2.
    V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Operator Differential Equations [in Russian], Naukova Dumka, Kiev (1984).zbMATHGoogle Scholar
  3. 3.
    M. G. Gasymov, “On the solvability of boundary-value problems for one class of operator differential equations,” Dokl. Akad. Nauk SSSR, 235, No. 3, 505–508 (1977).MathSciNetGoogle Scholar
  4. 4.
    I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Nonself-Adjoint Operators in Hilbert Spaces [in Russian], Nauka, Moscow (1965).Google Scholar
  5. 5.
    G. V. Radzievskii, “A problem of completeness of root vectors in the spectral theory of operator functions,” Usp. Mat. Nauk, 37, No. 2, 81–145 (1982).MathSciNetGoogle Scholar
  6. 6.
    M. G. Gasymov and S. S. Mirzoev, “On the solvability of boundary-value problems for second-order elliptic operator differential equations,” Differents. Uravn., 28, No. 4, 651–661 (1992).zbMATHMathSciNetGoogle Scholar
  7. 7.
    M. V. Keldysh, “On the completeness of eigenfunctions for some classes of nonself-adjoint linear operators,” Usp. Mat. Nauk, Issue 4 (160), 15–41 (1971).Google Scholar
  8. 8.
    M. G. Gasymov, “On the multiple completeness of a part of eigenvectors and adjoint vectors of polynomial operator bundles,” Izv. Akad. Nauk Arm. SSR, Ser. Mat., 6, No. 2-3, 131–147 (1971).zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • S. S. Mirzoev
    • 1
  • S. G. Veliev
    • 1
  1. 1.Baku State UniversityBakuAzerbaijan

Personalised recommendations