рус eng
«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». 2026. Vol 56

On the Positive Solvability of one Class of Nonlinear Multidimensional Integral Equations of Hammerstein Type

Author(s)

Khachatur A. Khachatryan

Yerevan State University, Yerevan, Republic of Armenia

Abstract
This paper investigates a class of nonlinear multidimensional integral equations on R 𝑛 with non-compact Hammerstein operator. These equations arise in various fields of mathematical physics and mathematical epidemiology. A distinguishing feature of the studied equations is the lack of complete continuity of the associated nonlinear operator in the space of bounded functions on R𝑛, the presence of a trivial (zero) solution, and the non-reflexivity of the corresponding function space, within which the existence of a nontrivial fixed point is considered. Under appropriate conditions on the kernel and the nonlinear term, a constructive theorem is established for the existence of a positive, bounded, and continuous solution. Moreover, the method of successive approximations is shown to converge uniformly to the solution at a rate an infinitely decreasing geometric progression. Within a sufficiently broad subclass of nonnegative, bounded functions on R𝑛, the uniqueness of the solution is also proven. The integral asymptotic behavior of the constructed solution is examined under additional constraints on the kernel and nonlinearity. Finally, explicit examples of kernels and nonlinearities satisfying all the assumptions of the theorems are provided.
About the Authors
Khachatur A. Khachatryan, Dr. Sci. (Phys.-Math.), Prof., Yerevan State University, Yerevan, 0025, Republic of Armenia, khachatur.khachatryan@ysu.am
For citation
Khachatryan Kh. A. On the Positive Solvability of one Class of Nonlinear Multidimensional Integral Equations of Hammerstein Type. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 56, pp. 81–96. https://doi.org/10.26516/1997-7670.2026.56.81
Keywords
monotonicity, iterations, concavity, bounded solution, integral asymptotic
UDC
517.968
MSC
45G05
DOI
https://doi.org/10.26516/1997-7670.2026.56.81
References
  1. Aref’eva I.Ya., Dragovic B.G., Volovich I.V. Open and closed p-adic strings and quadratic extensions of number fields. Phys. Lett. B, 1988, vol. 212, no. 3, pp. 283–291. https://doi.org/10.1016/0370-2693(88)91374-3
  2. Atkinson C., Reuter G.E.H. Deterministic epidemic waves. Math. Proc. Cambridge Philos. Soc., 1976, vol. 80, no. 2, pp. 315–330. https://doi.org/10.1017/S0305004100052944
  3. Diekmann O. Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol., 1978, vol. 6, no. 2, pp. 109–130. https://doi.org/10.1007/BF02450783
  4. Diekmann O., Kaper H.G. On the bounded solutions of a nonlinear convolution equation. Nonlinear Analysis, 1978, vol. 2, no. 6, pp. 721–737. https://doi.org/10.1016/0362-546X(78)90015-9
  5. Engibaryan N.B. A nonlinear problem of radiative transfer. Astrophysics, 1965, vol. 1, no. 1, pp. 158–159. https://doi.org/10.1007/BF01041941 (in Russian)
  6. Engibaryan N.B. On a problem in nonlinear radiative transfer. Astrophysics, 1966, vol. 2, no. 1, pp. 12–14. https://doi.org/10.1007/BF01014505 (in Russian)
  7. Khachatryan A.Kh., Khachatryan Kh.A. On the solvability of some nonlinear integral equations in problems of epidemic spread. Proc. Steklov Inst. Math., 2019, vol. 306, pp. 271–287. https://doi.org/10.1134/S0081543819050225
  8. Khachatryan A.Kh., Khachatryan Kh.A., Petrosyan H.S. On nonlinear convolution-type integral equations in the theory of p-adic strings. Theoret. and Math. Phys., 2023, vol. 216, no. 1, pp. 1068–1081. https://doi.org/10.1134/S0040577923070127
  9. Khachatryan A.Kh., Khachatryan Kh.A., Petrosyan H.S. Questions of existence, absence, and uniqueness of a solution to one class of nonlinear integral equations on the whole line with an operator of Hammerstein-Stieltjes type. Trudy Inst. Mat. i Mekh. URO RAN, 2024, vol. 30, no. 1, pp. 249–269. (in Russian)
  10. Khachatryan Kh.A., Petrosyan H.S. Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations. Trans. Moscow Math. Soc., 2021, vol. 82, pp. 259–271. https://doi.org/10.1090/mosc/329
  11. Kolmogorov A.N., Fomin S.V. Introductory real analysis. NJ, Prentice-Hall, Inc. Englewood Cliffs, 1970, 403 p.
  12. Sidorov N.A., Dreglea Sidorov L.R.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. https://doi.org/10.26516/1997-7670.2023.43.78 (in Russian)
  13. 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 (in Russian)
  14. Vladimirov V.S. Nonexistence of solutions of the p-adic strings. Theoret. and Math. Phys., 2013, vol. 174, no. 2, pp. 178–185. https://doi.org/10.1007/s11232-013-0015-3 (in Russian)
  15. Vladimirov V.S. Nonlinear equations for p-adic open, closed, and open-closed strings. Theoret. and Math. Phys., 2006, vol. 149, no. 3, pp. 1604–1616. https://doi.org/10.1007/s11232-006-0144-z (in Russian)
  16. Vladimirov V.S., Volovich Ya.I. Nonlinear dynamics equation in p-adic string theory. Theoret. and Math. Phys., 2004, vol. 138, no. 3, pp. 297–309. https://doi.org/10.1023/B:TAMP.0000018447.02723.29 (in Russian)

Full text (english)