Skip to main content
Springer Nature Link
Log in

On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In the functional space \(C(|z|\leq 1)\) is considered a spectral problem for the Cauchy–Riemann operator with boundary conditions of the Bitsadze–Samarskii type. It is proved under the assumption that when \(0\in\rho(K)\) is a non-empty resolvent set of the operator \(K\), the considered spectral problem for the Cauchy–Riemann operator is reduced to a singular integral equation with a continuous kernel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. V. A. Steklov, Bacis Problems of Mathematical Physics (Petrogr. Univ., Petrograg, 1922), Vols. 1, 2 [in Russian].

    Google Scholar 

  2. F. I. Frankl’, ‘‘Subsonic flow past a profile with local supersonic zone bounded by a cupved shock wave,’’ Prikl. Mat. Mekh. 20, 196–202 (1956).

    Google Scholar 

  3. A. M. Nakhushev, ‘‘A new boundary value problem for a degenepate hyperbolic equation,’’ Sov. Math. Dokl. 10, 935–938 (1969).

    MATH  Google Scholar 

  4. A. M. Nakhushev, Problems with Shifts for Paptial Differential Equations (Nauka, Moscow, 2006) [in Russsian].

    Google Scholar 

  5. A. V. Bitsadze and A. A. Samarskii, ‘‘On some simple generalizations of linear elliptic boundary problems,’’ Sov. Math. Dokl. 10, 398–400 (1969).

    MATH  Google Scholar 

  6. A. Assanova, ‘‘An integral-boundary value problem for a partial differential equation of second order,’’ Turk. J. Math. 43, 1967–1978 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. T. Assanova, ‘‘On the solvability of nonlocal problem for the system of Sobolev-type differential equations with integral condition,’’ Georg. Math. J. 28, 49–57 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. T. Assanova, A. E. Imanchiyev, and Zh. M. Kadirbayeva, ‘‘A nonlocal problem for loaded partial differential equations of fourth order,’’ Bull. Karag. Univ., Math. 97 (1), 6–16 (2020).

    Google Scholar 

  9. A. T. Assanova and Z. S. Tokmurzin, ‘‘A nonlocal multipoint problem for a system of fourth-order partial differential equations,’’ Euras. Math. J. 11 (3), 8–20 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  10. N. A. Heydarzade, ‘‘On one nonlocal inverse boundary problem for the second-order elliptic equation,’’ Trans. Natl. Acad. Sci. Azerb. Math. 40 (4), 97–109 (2020).

    MathSciNet  MATH  Google Scholar 

  11. B. I. Islomov and O. Kh. Abdullaev, ‘‘On non-local problems for third order equation with Caputo operator and non-linear loaded part,’’ Ufa Math. J. 13 (3), 44–56 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. I. Islomov and U. I. Baltaeva, ‘‘Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients,’’ Electron. J. Differ. Equat. 2015, 221-1–10 (2015).

  13. B. I. Islomov, N. K. Ochilova, and K. S. Sadarangani, ‘‘On a Frankl-type boundary-value problem for a mixed-type degenerating equation,’’ Ukr. Mat. J. 71, 1347–1359 (2019).

    MathSciNet  MATH  Google Scholar 

  14. B. I. Islomov and B. Usmonov, ‘‘Nonlocal boundary-value problem for third order elliptic-hyperbolic type equation,’’ Lobachevskii J. Math. 41, 32–38 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  15. T. K. Yuldashev, ‘‘Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with degenerate kernel,’’ Ukr. Math. J. 68, 1278–1296 (2016).

    Article  MATH  Google Scholar 

  16. T. K. Yuldashev, ‘‘Spectral features of the solving of a Fredholm homogeneous integro-differential equation with integral conditions and reflecting deviation,’’ Lobachevskii J. Math. 40, 2116–2123 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  17. T. K. Yuldashev, ‘‘On the solvability of a boundary value problem for the ordinary Fredholm integro-differential equation with a degenerate kernel,’’ Comput. Math. Math. Phys. 59, 241–252 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  18. T. K. Yuldashev, ‘‘Nonlinear optimal control of thermal processes in a nonlinear inverse problem,’’ Lobachevskii J. Math. 41, 124–136 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  19. T. K. Yuldashev, Kh. Kh. Saburov, and T. A. Abduvahobov, ‘‘Nonlocal problem for a nonlinear system of fractional order impulsive integro-differerntial equations with maxima,’’ Chelyab. Phys. Math. J. 7, 113–122 (2022).

    MATH  Google Scholar 

  20. T. K. Yuldashev, ‘‘On a nonlocal problem for impulsive differential equations with mixed maxima,’’ Vestn. KRAUNC, Fiz. Mat. Nauki 38 (1), 40 (2022).

    MathSciNet  Google Scholar 

  21. N. I. Muskelishvili, Singular Integral Equations (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  22. M. Otelbaev and A. N. Shynybekov, ‘‘On well-posed problems of Bicadze–Samarskii type,’’ Sov. Math. Dokl. 26, 157–161 (1982).

    MATH  Google Scholar 

  23. N. S. Imanbaev, ‘‘Eigenvalue problem for differential Cauchy-Riemann operator with nonlocal boundary conditions,’’ Zh. Samar. Tekh. Univ., Ser.: Fiz. Mat. Nauk 1 (34), 25–36 (2014).

    Google Scholar 

  24. N. S. Imanbaev and B. E. Kanguzhin, ‘‘On spectral question of the Cauchy–Riemann operator with homogeneus boundary value conditions,’’ Bull. Karag. Univ., Math. 2 (90), 49–55 (2018).

    Google Scholar 

  25. V. A. Mikhailets, ‘‘Spectral problems with general boundary conditions,’’ Doctoral (Phys. Math.) Dissertation (Kiev, 1989).

  26. B. V. Shabat, Introduction to Complex Analysis, Part 1 (Nauka, Moscow, 1976) [in Russsian].

    Google Scholar 

  27. M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969; George G. Harrap, London, 1968).

  28. N. S. Imanbaev and M. A. Sadybekov, ‘‘Stability of basis property of a type of problems with nonlocal perturbation of boundary conditions,’’ AIP Conf. Proc. 1759, 020034–020039 (2016).

    Google Scholar 

  29. N. S. Imanbaev and M. A. Sadybekov, ‘‘Stability of basis property of a type of problems on eigenvalues with nonlocal perturbation of boundary conditionss,’’ Ufa Math. J. 3 (2), 27–32 (2011).

    MathSciNet  Google Scholar 

  30. N. S. Imanbaev, ‘‘Stability of the basis property of eigenvalue systems of Stupm–Liouville operators with integral boundary condition,’’ Electron. J. Differ. Equat. 2016 (87), 1–8 (2016).

    MathSciNet  Google Scholar 

  31. N. S. Imanbaev, B. E. Kanguzhin, and B. T. Kalimbetov, ‘‘On zeros of the characteristic determinant of the spectral problem for a third-order differential operator on a segment with nonlocal boundary conditions,’’ Adv. Differ. Equat. 2013, 110 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  32. N. S. Imanbaev, ‘‘On basic properties of Eigenfunctions and associated functions of one loaded operator of multiple differentiation,’’ Lobachevskii J. Math. 43, 749–754 (2022).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research has been funded by the Science Committee of the Ministry of Science and Education of the Republic of Kazakhstan (grant no. AP09260752).

Author information

Authors and Affiliations

  1. South Kazakhstan State Pedagogical University, 160000, Shymkent, Kazakhstan

    N. S. Imanbaev

  2. Institute of Mathematics and Mathematical Modeling, 050010, Almaty, Kazakhstan

    N. S. Imanbaev

Authors
  1. N. S. Imanbaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. S. Imanbaev.

Additional information

(Submitted by T. K. Yuldashev)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Imanbaev, N.S. On a Spectral Problem for the Cauchy–Riemann Operator with Boundary Conditions of the Bitsadze–Samarskii Type. Lobachevskii J Math 44, 1162–1170 (2023). https://doi.org/10.1134/S1995080223030150

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223030150

Keywords:

Search

Navigation