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

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Ukrainian Mathematical Journal Aims and scope

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. A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Normally Solvable Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (2019).

    Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  3. A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, De Gruyter, Berlin (2016).

    Book  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  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).

  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).

  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).

  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. 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).

    Article  MathSciNet  Google Scholar 

  11. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Zhytomyr National Agricultural-Economical University, Zhytomyr, Ukraine

    V. F. Zhuravlev, N. P. Fomin & P. N. Zabrodskiy

Authors
  1. V. F. Zhuravlev
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  2. N. P. Fomin
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  3. P. N. Zabrodskiy
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Correspondence to V. F. Zhuravlev.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 4, pp. 471–485, April, 2019.

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Zhuravlev, V.F., Fomin, N.P. & Zabrodskiy, P.N. Conditions of Solvability and Representation of the Solutions of Equations with Operator Matrices. Ukr Math J 71, 537–553 (2019). https://doi.org/10.1007/s11253-019-01662-3

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  • DOI: https://doi.org/10.1007/s11253-019-01662-3

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