Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 3, pp 422–436 | Cite as

Bifurcation Conditions for the Solutions of Weakly Perturbed Boundary-Value Problems for Operator Equations in Banach Spaces

  • V. F. Zhuravlev
Article
  • 11 Downloads

We establish the conditions of bifurcation of the solutions of weakly perturbed boundary-value problems for operator equations in Banach spaces from the point 𝜀 = 0. A convergent iterative procedure is proposed for the construction of solutions as parts of series in powers of 𝜀 with poles at the point 𝜀 = 0.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. F. Zhuravlev, “Weakly perturbed operator equations in Banach spaces,” Ukr. Mat. Zh., 69, No. 6, 751–764 (2017); English translation : Ukr. Math. J., 69, No. 6, 876–891 (2017).Google Scholar
  2. 2.
    S. G. Krein, Linear Equations in Banach Spaces [in Russian], Nauka, Moscow (1971).Google Scholar
  3. 3.
    I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).Google Scholar
  4. 4.
    M. I. Vishik and L. A. Lyusternik, “Solution of some problems on perturbations in the case of matrices and self-adjoint and nonselfadjoint differential equations,” Usp. Mat. Nauk, 15, Issue 3, 3–80 (1960).Google Scholar
  5. 5.
    A. A. Boichuk, Constructive Methods for the Analysis of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  6. 6.
    A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1995).zbMATHGoogle Scholar
  7. 7.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Boston (2004).CrossRefGoogle Scholar
  8. 8.
    A. A. Boichuk and L. M. Shegda, “Bifurcation of solutions of singular Fredholm boundary value problems,” Different. Equat., 47, No. 4, 453–461 (2011).CrossRefGoogle Scholar
  9. 9.
    O. A. Boichuk and E. V. Panasenko, “Weakly perturbed boundary-value problems for differential equations in a Banach space,” Nelin. Kolyv., 13, No. 3, 291–304 (2010); English translation : Nonlin. Oscillat., 13, No. 3, 311–324 (2011).Google Scholar
  10. 10.
    A. Ya. Khelemskii, Lectures on Functional Analysis [in Russian], MTsNMO, Moscow (2004).Google Scholar
  11. 11.
    M. M. Popov, “Complementable spaces and some problems of the contemporary geometry of Banach spaces,” Mat. S’ohodni’07, Issue 13, 78–116 (2007).Google Scholar
  12. 12.
    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).Google Scholar
  13. 13.
    A. M. Samoilenko, A. A. Boichuk, and V. F. Zhuravlev, “Linear boundary-value problems for normally solvable operator equations in a Banach space,” Different. Equat., 50, No. 3, 1–11 (2014).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • V. F. Zhuravlev
    • 1
  1. 1.Zhytomyr National Agricultural-Economical UniversityZhytomyrUkraine

Personalised recommendations