Advertisement

Journal of Mathematical Sciences

, Volume 229, Issue 4, pp 425–438 | Cite as

Weakly Perturbed Fredholm Integral Equations with Degenerate Kernels in Banach Spaces

  • V. F. Zhuravlev
  • N. P. Fomin
Article
  • 22 Downloads

We consider weakly perturbed Fredholm equations with degenerate kernels in Banach spaces and establish conditions for ε = 0 to be a bifurcation point for the solutions of weakly perturbed operator equations X in Banach spaces. A convergent iterative scheme for finding solutions in the form of series \( {\Sigma}_{i=-1}^{+\infty }{\varepsilon}^i{z}_i(t) \) in powers of ε is proposed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. M. Lyapunov, General Problem of Stability of Motion [in Russian], Gostekhizdat, Moscow (1950).MATHGoogle Scholar
  2. 2.
    M. I. Vishik and L. A. Lyusternik, “Solution of some problems of perturbations in the case of matrices and self-adjoint and nonselfadjoint differential equations,” Usp. Mat. Nauk, 15, Issue 3, 3–80 (1960).Google Scholar
  3. 3.
    A. A. Boichuk, Constructive Methods for the Analysis of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1990).Google Scholar
  4. 4.
    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).Google Scholar
  5. 5.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht–Boston (2004).Google Scholar
  6. 6.
    Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).Google Scholar
  7. 7.
    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
  8. 8.
    S. G. Krein, Linear Equations in Banach Spaces [in Russian], Nauka, Moscow (1971).Google Scholar
  9. 9.
    A. A. Boichuk and L. M. Shegda, “Bifurcation of solutions of singular Fredholm boundary value problems,” Different. Equat., 47, No. 4, 453–461 (2011).CrossRefMATHGoogle Scholar
  10. 10.
    I. A. Holovats’ka, “Weakly perturbed systems of linear integrodifferential equations,” Nelin. Kolyv., 15, No. 2, 151–164 (2012); English translation: J. Math. Sci., 189, No. 5, 735–750 (2013).Google Scholar
  11. 11.
    V. P. Zhuravl’ov, “Generalized inversion of Fredholm integral operators with degenerate kernels in Banach spaces,” Nelin. Kolyv., 17, No. 3, 351–364 (2014); English translation: J. Math. Sci., J. Math. Sci.., 212, No. 3, 275–289 (2015).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, 172–192 (2013).Google Scholar
  13. 13.
    V. P. Zhuravl’ov, “Linear boundary-value problems for integral Fredholm equations with degenerate kernels in Banach spaces,” Bukov. Mat. Zh., 2, No. 4, 57–66 (2014).MATHGoogle Scholar
  14. 14.
    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
  15. 15.
    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).Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Zhytomyr National Agricultural-Economic UniversityZhytomyrUkraine

Personalised recommendations