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Ukrainian Mathematical Journal

, Volume 67, Issue 5, pp 690–710 | Cite as

Solvability of the Nonlocal Boundary-Value Problem for a System of Differential-Operator Equations in the Sobolev Scale of Spaces and in a Refined Scale

  • V. S. Il’kiv
  • N. I. Strap
Article

We study the solvability of the nonlocal boundary-value problem with one parameter for a system of differential-operator equations in the Sobolev scale of spaces of functions of many complex variables and in the scale of Hörmander spaces which form a refined Sobolev scale. By using the metric approach, we prove the theorems on lower estimates of small denominators appearing in the construction of solutions of the analyzed problem. They imply the unique solvability of the problem for almost all vectors formed by the coefficients of the equation and the parameter of nonlocal conditions.

Keywords

Partial Differential Equation Sobolev Space Elliptic Problem Domain Versus Unique Solvability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  2. 2.
    B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002).Google Scholar
  3. 3.
    V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).CrossRefzbMATHGoogle Scholar
  4. 4.
    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).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).CrossRefzbMATHGoogle Scholar
  6. 6.
    V. A. Mikhailets and A. A. Murach, “Refined scale of spaces and elliptic boundary-value problems. II,” Ukr. Math. J., 58, No. 3, 398–417 (2006).CrossRefMathSciNetGoogle Scholar
  7. 7.
    V. A. Mikhailets and A. A. Murach, “Refined scale of spaces and elliptic boundary-value problems. III,” Ukr. Math. J., 59, No. 5, 744–765 (2007).CrossRefMathSciNetGoogle Scholar
  8. 8.
    V. A. Mikhailets and A. A. Murach, “Extended Sobolev scale and elliptic operators,” Ukr. Math. J., 65, No. 3, 435–447 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (1986).Google Scholar
  10. 10.
    V. S. Il’kiv and B. I. Ptashnyk, “Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators,” Ukr. Mat. Zh., 58, No. 12, 1624–1650 (2006) English translation: Ukr. Math. J., 58, No. 12, 1847–1875 (2006).Google Scholar
  11. 11.
    V. V. Prasolov, Polynomials [in Russian], MTSNMO, Moscow (1999).Google Scholar
  12. 12.
    T. V. Maherovs’ka, “Investigation of smoothness of the solutions of the Cauchy problems for systems of partial differential equations with the help of the metric approach,” Nauk. Visn. Cherniv. Nats. Univ., Ser. Mat., 1, No. 1-2, 84–93 (2011).Google Scholar
  13. 13.
    G. A. Baker, Jr., and P. Graves-Morris, Padé Approximants, Addison-Wesley, London (1981).Google Scholar
  14. 14.
    V. S. Il’kiv and N. I. Strap, “Nonlocal boundary-value problem for a partial differential equation in a multidimensional complex plane,” Nauk. Visn. Uzhhorod. Univ., Ser. Mat. Inform., 24, No. 1, 60–72 (2013).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • V. S. Il’kiv
    • 1
  • N. I. Strap
    • 1
  1. 1.“Lvivs’ka Politekhnika” National UniversityLvivUkraine

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