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Fredholm One-Dimensional Boundary-Value Problems with Parameters in Sobolev Spaces

  • O. M. Atlasiuk
  • V. A. Mikhailets
Article

For systems of linear differential equations on a compact interval, we analyze the dependence of the solutions of boundary-value problems in the Sobolev spaces \( {W}_{\infty}^n \) on a parameter ε. We establish a constructive criterion of continuous dependence of the solutions of these problems on the parameter ε for ε = 0. The rate of convergence of these solutions is determined.

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References

  1. 1.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Zeist–Boston (2004).CrossRefzbMATHGoogle Scholar
  2. 2.
    I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University, Tbilisi (1975).Google Scholar
  3. 3.
    I. T. Kiguradze, “Boundary-value problems for systems of ordinary differential equations,” in: VINITI [in Russian], 30 (1987), pp. 3–103.Google Scholar
  4. 4.
    T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, “Limit theorems for one-dimensional boundary-value problems,” Ukr. Mat. Zh., 65, No. 1, 70–81 (2013); English translation: Ukr. Math. J., 65, No. 1, 77–90 (2013).Google Scholar
  5. 5.
    V. A. Mikhailets, O. B. Pelekhata, and N. V. Reva, “Limit theorems for the solutions of boundary-value problems,” Ukr. Mat. Zh., 70, No. 2, 216–223 (2018); English translation: Ukr. Math. J., 70, No. 2, 243–251 (2018).Google Scholar
  6. 6.
    V. A. Mikhailets and G. A. Chekhanova, “Limit theorem for general one-dimensional boundary-value problems,” J. Math. Sci., 204, No. 3, 333–342 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. V. Gnyp, T. I. Kodlyuk, and V. A. Mikhailets, “Fredholm boundary-value problems with parameter in Sobolev spaces,” Ukr. Mat. Zh., 67, No. 5, 584–591 (2015); English translation: Ukr. Math. J., 67, No. 5, 658–667 (2015).Google Scholar
  8. 8.
    T. I. Kodlyuk and V. A. Mikhailets, “Solutions of one-dimensional boundary-value problems with a parameter in Sobolev spaces,” J. Math. Sci., 190, No. 4, 589–599 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Y. V. Hnyp, V. A. Mikhailets, and A. A. Murach, “Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces,” Electron. J. Different. Equat., No. 81 (2017).Google Scholar
  10. 10.
    V. A. Mikhailets, A. A. Murach, and V. O. Soldatov, “Continuity in a parameter of solutions to generic boundary-value problems,” Electron. J. Qual. Theory Different. Equat., No. 87 (2016).Google Scholar
  11. 11.
    O. M. Atlasiuk and V. A. Mikhailets, “Fredholm one-dimensional boundary-value problems in Sobolev spaces,” Ukr. Mat. Zh., 70, No. 10, 1324–1333 (2018).MathSciNetGoogle Scholar

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

Authors and Affiliations

  • O. M. Atlasiuk
    • 1
  • V. A. Mikhailets
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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