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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 50

On an Integral Equation with Concave Nonlinearity

Author(s)
Khachatur A. Khachatryan1, Haykanush S. Petrosyan2

1Yerevan State University, Yerevan, Republic of Armenia

2Armenian National Agrarian University, Yerevan, Republic of Armenia

Abstract
A nonlinear integral equation on the semi-axis with a special substochastic kernel is studied. Such equations are encountered in the kinetic theory of gases when studying the nonlinear integro-differential Boltzmann equation within the framework of the nonlinear modified Bhatnagar-Gross-Crook model(BGC). Under certain restrictions on nonlinearity, it is possible to construct a positive continuous and bounded solution to this equation. Moreover, the uniqueness of the solution in the class of upper bounded on half-line functions having a positive infimum. It is also proved that the corresponding successive approximations converge uniformly at a rate of some geometric progression to the solution of the indicated equation. Under one additional condition, the asymptotic behavior of the solution at infinity is studied. At the end of the work, specific examples of these equations are given for which all the conditions of the proven facts are automatically met.
About the Authors

Khachatur A. Khachatryan, Dr. Sci. (Phys.-Math.), Prof., Yerevan State University, Yerevan, 0025, Republic of Armenia, khachatur.khachatryan@ysu.am

Haykanush S. Petrosyan, Cand. Sci. (Phys.Math.), Assoc. Prof., Agrarian University, Yerevan, 0009, Republic of Armenia, Haykuhi25@mail.ru

For citation

Khachatryan Kh. A., Petrosyan H. S. On an Integral Equation with Concave Nonlinearity. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 66–82. (in Russian)

https://doi.org/10.26516/1997-7670.2024.50.66

Keywords
concavity, iterations, monotonicity, convergence, asymptotics
UDC
517.968
MSC
45G05
DOI
https://doi.org/10.26516/1997-7670.2024.50.66
References
  1. Gevorkyan G.G., Engibaryan N.B. New theorems for the renewal integral equation. J. Contemp. Math. Anal., Armen. Acad. Sci., 1997, vol. 32, no. 1, pp. 2–16.
  2. Engibaryan N.B. On a problem in nonlinear radiative transfer. Astrophysics, 1966, vol. 2, P. 12–14. https://doi.org/10.1007/BF01014505
  3. Kogan M.N. Rarefied Gas Dynamics. Moscow, Nauka Publ., 1969.
  4. Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis, vol. 1, 2, Mineola, New York, Dover Publ., 1999, 288 p.
  5. Sidorov N.A., Dreglea Sidorov Lev Ryan D. On the solution of Hammerstein integral equations with loads and bifurcation parameters. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 43, pp. 78–90.
  6. Sidorov N.A., Sidorov D.N. Nonlinear Volterra Equations with Loads and Bifurcation Parameters: Existence Theorems and Construction of Solutions. Differantial Equations, 2021, vol. 57, pp. 1640–1651. https://doi.org/10.1134/S0012266121120107
  7. Khachatryan Kh.A., Petrosyan H.S. On alternating and bounded solutions of one class of integral equations on the entire axis with monotonic nonlinearity. J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 2020, vol. 24, no. 4. pp. 644–662. https://doi.org/10.14498/vsgtu1790 (in Russian)
  8. Khachatryan A.Kh., Khachatryan Kh.A. A nonlinear integral equation of Hammerstein type with a noncompact operator. Sb. Math., 2010, vol. 201, no. 4, pp. 595–606. https://doi.org/10.1070/SM2010v201n04ABEH004083
  9. Barichello L.B., Siewert C.E. The temperature-jump problem in rarefied-gas dynamics. European J. of Applied Mathematics, 2000. vol. 11. pp. 353–364. https://doi.org/10.1017/S0956792599004180
  10. Cercignani C. The Boltzmann equation and its applications. New-York, Appl. Math. Sci., Springer, 1988, 455 p.
  11. Khachatryan Kh.A. One-parameter family of solutions for one class of hammerstein nonlinear equations on a half-line. Doklady Mathematics, 2009. vol. 80, no. 3, pp. 872–876. https://doi.org/10.1134/S1064562409060222
  12. Villani C. Cercignani’s conjecture is sometimes true and always almost true. Comm. Math. Phys., 2003, vol. 234, no. 3, pp. 455–490. https://doi.org/10.1007/s00220-002-0777-1

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