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Continuity of the Solutions of One-Dimensional Boundary-Value Problems with Respect to the Parameter in the Slobodetskii Spaces

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Ukrainian Mathematical Journal Aims and scope

For a system of linear ordinary differential equations of the first order, we study the broadest class of inhomogeneous boundary-value problems whose solutions belong to the Slobodetskii space W s + 1 p ((a, b),ℂm) with m ∈ ℕ, s > 0, and p ∈ (1,∞). We prove a theorem on the Fredholm property of these problems. We also establish conditions under which the problems are uniquely solvable in the Slobodetskii space and their solutions are continuous in this space with respect to the parameter.

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References

  1. I. T. Kiguradze, “Boundary-value problems for systems of ordinary differential equations,” in: VINITI Series in Contemporary Problems of Mathematics. Latest Achievements [in Russian], 30, VINITI, Moscow (1987), pp. 3–103.

  2. I. T. Kiguradze, Some Singular Boundary-Value Problems for Ordinary Differential Equations [in Russian], Tbilisi University, Tbilisi (1975).

    Google Scholar 

  3. I. T. Kiguradze, “On the boundary-value problems for linear differential systems with singularities,” Differents. Uravn., 39, No. 2, 198–209 (2003).

    MathSciNet  Google Scholar 

  4. T. I. Kodlyuk, V. A. Mikhailets, and N. V. Reva, “Limit theorems for one-dimensional boundary-value problems,” Ukr. Math. J., 65, No. 1, 77–90 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. A. Mikhailets and N. V. Reva, “Generalizations of the Kiguradze theorem on the well-posedness of linear boundary-value problems,” Dop. Nats. Akad. Nauk Ukr., No. 9, 23–27 (2008).

  6. V. A. Mikhailets and G. A. Chekhanova, “Limit theorems for general one-dimensional boundary-value problems,” J. Math. Sci., 204, No. 3, 333–342 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. O. Soldatov, “On the continuity in a parameter for the solutions of boundary-value problems total with respect to the spaces C (n+r)[a, b],” Ukr. Math. J., 67, No. 5, 785–794 (2015).

  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).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. A. Mikhailets and N. V. Reva, “Limit transition in systems of linear differential equations,” Dop. Nats. Akad. Nauk Ukr., No. 8, 28–30 (2008).

  10. E. V. Gnyp, T. I. Kodlyuk, and V. A. Mikhailets, “Fredholm boundary-value problems with parameter in Sobolev space,” Ukr. Math. J., 67, No. 5, 658–667 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB, Berlin (1978).

    MATH  Google Scholar 

  12. V. G. Maz’ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions [in Russian], Leningrad University, Leningrad (1986).

    MATH  Google Scholar 

  13. E. V. Hnyp and T. I. Kodlyuk, “Continuity of the solutions of nonclassical multipoint boundary-value problems with respect to the parameter in Sobolev spaces,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 12, No. 2 (2015), pp. 101–112.

  14. T. I. Kodlyuk, “Multipoint boundary-value problems with parameter in the Sobolev spaces,” Dop. Nats. Akad. Nauk Ukr., No. 11, 15–19 (2012).

  15. A. S. Goriunov and V. A. Mikhailets, “Resolvent convergence of Sturm–Liouville operators with singular potentials,” Math. Notes, 87, No. 1–2, 287–292 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. S. Goriunov and V. A. Mikhailets, “Regularization of singular Sturm–Liouville equations,” Meth. Funct. Anal. Topol., 16, No. 2, 120–130 (2010).

    MathSciNet  MATH  Google Scholar 

  17. A. S. Goriunov and V. A. Mikhailets, “Regularization of two-term differential equations with singular coefficients by quasiderivatives,” Ukr. Math. J., 63, No. 9, 1361–1378 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. A. Mikhailets and A. A. Murach, “The refined Sobolev scale, interpolation, and elliptic problems,” Banach J. Math. Anal., 6, No. 2, 211–281 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).

    Book  MATH  Google Scholar 

  20. T. I. Kodlyuk, One-Dimensional Boundary-Value Problems with Parameter in Sobolev Spaces [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Kyiv (2013).

  21. V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1989).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    E. V. Hnyp

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  1. E. V. Hnyp
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 6, pp. 746–756, June, 2016.

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Hnyp, E.V. Continuity of the Solutions of One-Dimensional Boundary-Value Problems with Respect to the Parameter in the Slobodetskii Spaces. Ukr Math J 68, 849–861 (2016). https://doi.org/10.1007/s11253-016-1261-y

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  • DOI: https://doi.org/10.1007/s11253-016-1261-y

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