ASYMPTOTICS OF ZEROS OF AN ENTIRE FUNCTION
WITH AN INTEGRAL REPRESENTATION, CONNECTED
BY A REGULAR LOADED DIFFERENTIATION
OPERATOR ON AN INTERVAL

Nurlan S. Imanbaev$^{1, 2, \S}$, Nurgul N. Sairam $^2$
$^1$ Department of Mathematics
South Kazakhstan State Pedagogical University
Str. Baitursynov 13
160000 – Shymkent, KAZAKHSTAN

$^2$ Institute of Mathematics and Mathematical Modeling
Str. Pushkin 125
050010 – Almaty, KAZAKHSTAN

Abstract


Abstract. In this paper, we construct a characteristic determinant of the spectral problem for a loaded first-order differential equation on an interval with a periodic boundary value condition, which is an entire analytical function of spectral parameter. Based on the formula of the characteristic determinant, conclusions about the asymptotic behavior of the spectrum of the loaded first-order differential equation are drawn on an interval. Adjoint operator is constructed. Moreover, we show that the spectral questions of the adjoint operator have a similar structure. A special feature of the considered operator is the non-self-adjointness of the operator in ${L_2}\left( { - 1,\, 1} \right)$.

Received: 16.10.2023

AMS Subject Classification: 34L10

Key Words and Phrases: loaded equation, boundary conditions, entire functions, zeros of an entire function, asymptotics, eigenvalues, characteristic determinant, adjoint operator, regular operator

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How to cite this paper?
DOI: 10.12732/ijam.v36i6.7
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2023
Volume: 36
Issue: 6
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References

  1. [1] B. Ya Levin, Distribution of Roots of Entire Functions, Nauka, Moscow (1956) (in Russian).
  2. [2] V.A. Sadovnichy, Operator Theory, MSU Publishing House, Moscow (1986) (in Russian).
  3. [3] E.C. Titchmarch, The zeros of certain integral functions, Proc. London Math. Soc., 25, No 4 (1926), 283-302.
  4. [4] M.L. Cartwright, The zeros of certain integral functions, The Quarterly journal of Math., 1, No 1 (1930), 38-59.
  5. [5] A.M. Sedletskii, On the zero of the Fourier transform of finite measure, Math. Notes, 53, No 1 (1993), 77-84.
  6. [6] N.S. Imanbaev and B.E. Kanguzhin, On zeros of entire functions having an integral representation, News of the National Academy of Sciences of the Republic of Kazakhstan, Ser. Phys.-Math., No 3 (1995), 47-52 (in Russian).
  7. [7] B.E. Kanguzhin and M.A. Sadybekov, Differential Operators on a Segment. Distribution of Eigenvalues, Gylym, Shymkent (1996) (in Russian).
  8. [8] N.S. Imanbaev, B.E. Kanguzhin, and B.T. Kalimbetov, On zeros the characteristic determinant of the spectral problem for a third-order differential operator on a segment with nonlocal boundary conditions, Advances Difference Equations, 2013, No 110 (2013); doi: 10.1186/1687-1847-2013-110.
  9. [9] V.B Sherstyukov, Asymptotic properties of entire functions with a given distribution law of roots. Complex Analysis. Entire functions and their applications, Results of Science and Technology. Ser. Rec. Math. and its App. Subj. Review M.: VINITI RAS, 161, (2019), 104-129.
  10. [10] K.G. Malyutin, M.V. Kabanko, The meromorphic functions of completely regular growth on the upper half-plane, Vestnik Udmurtskogo Universiteta. Math. Mekh. Komp. Nauki, 30 No 3 (2020), 396-409.
  11. [11] R. Bellman and K. Cook, Differential-Difference Equations, Academic Press, New York (1963).
  12. [12] A.F. Leont’ev, Entire Functions and Exponential Problems, Nauka, Moscow (1983) (in Russian).
  13. [13] O.H. Hald, Discontinuous inverse eigen value problems, Communications on Pure Applied Mathematics, 37, No 5 (1984), 539-577.
  14. [14] G.G. Braichev, V.B. Sherstyukov, Estimates of indicators of an entire function with negative roots, Vladikavkaz Math. Journ., 22, No 3 (2020), 30-46.
  15. [15] Yu.F. Peddler, On distribution of zeros of one class of meromorphic functions, Vladikavkaz Math. Journ., 19, No 1 (2017), 41-49.
  16. [16] A.M. Gaisin, B.E. Kanguzhin, A.A. Seitova, Completeness of the exponential system on a segment of the real axis, Eurasian Math. Journ., 13, No 2 (2022), 37-42.
  17. [17] N.S. Imanbaev, Ye. Kurmysh, On computation of eigenfunctions of composite type equations with Regular boundary value conditions, Intern. J. of Appl. Math., 34, No 4 (2021), 681-692; doi:10.12732/ijam.v34i4.7.
  18. [18] V.A. Sadovnichii, V.A. Lyubishkin, Yu. Belabbasi, On regularized sums of root of an entire function of a certain class, Sov. Math. Dokl., 22 (1980), 613-616.
  19. [19] N.S. Imanbaev, On zeros of an entire function having an integral represen [20] N.S. Imanbaev, Distribution of eigen values of a third-order differential operator with strongly regular non local boundary conditions, In: AIP Conf. Proc., 1997, Art. No 020027 (2018), 1-5; doi:10.1063/1.5049021.
  20. [21] N.S. Imanbaev, On nonlocal perturbation of the problem on eigenvalues of differentiation operator on a segment, Vestn. Udmurtskogo Univ. Math. Mekh. Komp. Nauki, 31, No 2 (2021), 186-193.
  21. [22] I.S. Lomov, Basis property of root vectors of loaded second-order differential operators on an interval, Differ. Equations, 27, No 1 (1991), 80-94.
  22. [23] A.M. Gomilko, G.V. Radzievsky, Basic properties of eigenfunctions of a regular boundary value problem for a vector functional differential equation, Differ. Equations, 27, No 3 (1991), 385-395.
  23. [24] N.S. Imanbaev, M.A. Sadybekov, Basic properties of root functions of loaded second-order differential operators, Reports of NAS RK, No 2 (2010), 11-13.
  24. [25] V.A. Ilyin, On the connection between the types of boundary value conditions and the properties of basis and equiconvergence and the trigonometric series of expansions in root functions of a non-self-adjoint differential operator, Differ. Equations, 30, No 9 (1994), 1516-1529.
  25. [26] A.S. Makin, On nonlocal perturbation of a periodic eigenvalue problem, Differ. Equations, 42, No 4 (2006), 560-562.
  26. [27] N.S. Imanbaev, M.A. Sadybekov, Stability of basis property of a type of problems on eigenvalues with nonlocal perturbation of boundary conditions, Ufa Math. Journ., 3, No 2 (2011), 27-32.
  27. [28] N.S. Imanbaev, M.A. Sadybekov, Stability of basis property of a periodic problem with nonlocal perturbation of boundary conditions, AIP Conf. Proc., No 1759 (Art. 020080) (2016).
  28. [29] A.M. Sarsenbi, Criteria for the Riesz basis property of systems of eigen- and associated functions for higher-order differential operators on an interval, Dokl. Mathematics, No 77 (2008), 290-292.
  29. [30] N.S. Imanbaev, Stability of the basis property of eigenvalue systems of Sturm-Liouville operators with integral perturbation of the boundary condition, Electronic Journal of Differential Equations, 2016, Art. No 87 (2016), 1-8.
  30. [31] B.V. Shabbat, Introduction to Complex Analysis. Part 1, Nauka, Moscow (1976) (in Russian).
  31. [32] N.S. Imanbaev, On quadratic proximity of eigenfunctions of the “unperturbed” and “perturbed” differentiation operators on an interval, Abstracts of reports. Traditional International April Mathematical Conference in Honor of the Science of R.K. Almaty: Institute of Mathematics and Math. Modeling (2023), 85-86.