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Ukrainian Mathematical Journal

, Volume 71, Issue 4, pp 537–553 | Cite as

Conditions of Solvability and Representation of the Solutions of Equations with Operator Matrices

  • V. F. ZhuravlevEmail author
  • N. P. Fomin
  • P. N. Zabrodskiy
Article
  • 8 Downloads

We propose new methods for the construction of generalized inverse operator matrices for the operator matrices in Banach spaces. The criteria of solvability and the formulas for representations of the general solutions of operator equations with operator matrices are obtained. As an application, we consider the relationship between the obtained formulas and the well-known Frobenius formula for the construction of the matrix inverse to a nondegenerate block matrix.

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References

  1. 1.
    A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Normally Solvable Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (2019).Google Scholar
  2. 2.
    A. A. Boichuk, V. F. Zhuravlev, and A. A. Pokutnyi, “Normally solvable operator equations in a Banach space,” Ukr. Mat. Zh., 65, No. 2, 163–174 (2013); English translation: Ukr. Math. J., 65, No. 2, 179–192 (2013).MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, De Gruyter, Berlin (2016).CrossRefGoogle Scholar
  4. 4.
    A. M. Samoilenko, A. A. Boichuk, and V. F. Zhuravlev, “Linear boundary-value problems for normally solvable operator equations in Banach spaces,” Different. Equat., 50, No. 3, 1–11 (2014).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. A. Boichuk and A. A. Pokutnyi, “Application of the ergodic theory to the investigation of boundary-value problems with periodic operator coefficients,” Ukr. Mat. Zh., 65, No. 3, 329–338 (2013); English translation:Ukr. Math. J., 65, No. 3, 366–376 (2013).MathSciNetCrossRefGoogle Scholar
  6. 6.
    N. O. Kozlova and V. A. Feruk, “Noetherian boundary-value problems for integral equations,” Nelin. Kolyv., 19, No. 1, 58–66 (2016); English translation:J. Math. Sci., 222, No. 3, 266–275 (2017).Google Scholar
  7. 7.
    V. F. Zhuravlev and N. P. Fomin, “Weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces,” Nelin. Kolyv., 20, No. 4, 488–501 (2017); English translation:J. Math. Sci., 238, No. 3, 248–262 (2019).Google Scholar
  8. 8.
    M. M. Popov, “Complemented spaces and some problems of contemporary geometry of the Banach spaces,” Mat. S’ohodni’07, Issue 13, 78–116 (2007).Google Scholar
  9. 9.
    I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).Google Scholar
  10. 10.
    V. F. Zhuravlev, “Solvability criterion and representation of solutions of n-normal and d-normal linear operator equations in a Banach space,” Ukr. Mat., Zh., 62, No. 2, 167–182 (2010); English translation:Ukr. Math. J., 62, No. 2, 186–202 (2010).MathSciNetCrossRefGoogle Scholar
  11. 11.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. F. Zhuravlev
    • 1
    Email author
  • N. P. Fomin
    • 1
  • P. N. Zabrodskiy
    • 1
  1. 1.Zhytomyr National Agricultural-Economical UniversityZhytomyrUkraine

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