Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Khachatryan K. A., Petrosyan H. S. On the solvability of a class of nonlinear Hammerstein integral equations on the semiaxis. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 2, pp. 169-179. DOI: 10.18500/1816-9791-2022-22-2-169-179, EDN: KNWVNY

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 1529)
Article type: 

On the solvability of a class of nonlinear Hammerstein integral equations on the semiaxis

Khachatryan Khachatur Agavardovich, Yerevan State University
Petrosyan Haykanush S., Armenian National Agrarian University

The paper studies a class of nonlinear integral equations on the semiaxis with a non-compact Hammerstein operator. It is assumed that the kernel of the equation decreases exponentially on the positive part of the number axis. Equations of this kind arise in various fields of natural science. In particular, such equations are found in the theory of radiation transfer in spectral lines, in the mathematical theory of the space-time propagation of an epidemic, in the kinetic theory of gases. A distinctive feature of the considered class of equations is the non-compactness of the corresponding nonlinear Hammerstein integral operator in the space of essentially bounded functions on the positive part of the number line and the criticality condition (i.e., the presence of a trivial zero solution). Under certain restrictions on nonlinearity, the existence of a positive bounded and summable solution is proved. The asymptotic behavior of the solution at infinity is also investigated. The proof of the existence theorem is constructive. First, solving a certain simple characteristic equation the zero approximation in the considered iterations is costructed. Then a special auxiliary nonlinear integral equation is studied, the solution of which is constructed using simple successive approximations. After that, it is proved that the iterations introduced by us increase monotonically and are bounded from above by the solution of the aforementioned auxiliary integral equation. Further, using the appropriate conditions for nonlinearity and for the kernel, it is proved that the limit of these iterations is a solution to the original equation and exponentially decreases at infinity. Under an additional constraint on nonlinearity, the uniqueness of the constructed solution is established in a certain class of measurable functions. At the end of the work, specific examples of the kernel and nonlinearity of an applied nature are given for which all the conditions of the theorem proved are automatically satisfied.

The work was supported by the Science Committee of Armenia, in the frames of the research project No. 21T-1A047.
  1. Engibaryan N. B. On a problem in nonlinear radiative transfer. Astrophysics, 1966, vol. 2, pp. 12–14. https://doi.org/10.1007/BF01014505
  2. Feller W. An Introduction to Probability Theory and Its Applications. Vol. 2. 2nd ed. Wiley, 1991. 704 p. (Russ. ed.: Moscow, Mir, 1967. 765 p.).
  3. Cercignani C. Theory and Application of the Boltzmann Equation. Edinburgh, London, Scottish Academic Press, 1975. 415 p.
  4. Khachatryan A. Kh., Khachatryan Kh. A. Qualitative difference between solutions for a model of the Boltzmann equation in the linear and nonlinear cases. Theoretical and Mathematical Physics, 2012, vol. 172, iss. 3, pp. 1315–1320. https://doi.org/10.1007/s11232-012-0116-4
  5. Kogan M. N. Rarefied Gas Dynamics. Springer, 1969. 400 p. (Russ. ed.: Moscow, Nauka, 1967. 440 p.).
  6. Diekmann O. Thresholds and travelling waves for the geographical spread of infection. Journal of Mathematical Biology, 1978, vol. 6, iss. 2, pp. 109–130. https://doi.org/10.1007/BF02450783
  7. Diekmann O. Run for your life. A note on the asymptotic speed of propogation of an epidemic. Journal of Differential Equations, 1979, vol. 33, iss. 1, pp. 58–73. https://doi.org/10.1016/0022-0396(79)90080-9
  8. Sergeev A. G., Khachatryan Kh. A. On the solvability of a class of nonlinear integral equations in the problem of a spread of an epidemic. Transactions of the Moscow Mathematical Society, 2019, vol. 80, pp. 95–111. https://doi.org/10.1090/mosc/286
  9. Khachatryan Kh. A. Sufficient conditions for the solvability of the Urysohn integral equation on a half-line. Doklady Mathematics, 2009, vol. 79, pp. 246–249. https://doi.org/10.1134/S1064562409020264  
  10. Khachatryan Kh. A. On a class of integral equations of Urysohn type with strong non[1]linearity. Izvestiya: Mathematics, 2012, vol. 76, iss. 1, pp. 163–189. https://doi.org/10.1070/IM2012v076n01ABEH002579
  11. Khachatryan Kh. A. On a class of nonlinear integral equations with a noncompact operator. Journal of Contemporary Mathematical Analysis, 2011, vol. 46, iss. 2, pp. 89–100. https://doi.org/10.3103/S106836231102004X
  12. Khachatryan Kh. A. Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line. Izvestiya: Mathematics, 2015, vol. 79, iss. 2, pp. 411–430. https://doi.org/10.1070/IM2015v079n02ABEH002748
  13. Arabadzhyan L. G. Solution of certain integral equations of the Hammerstein type. Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 1997, vol. 32, iss. 1, pp. 17–24.
  14. Vladimirov V. S., Volovich Y. I. Nonlinear dynamics equation in p-adic string theory. Theoretical and Mathematical Physics, 2004. vol. 138, iss. 3, pp. 297–309. https://doi.org/10.1023/B:TAMP.0000018447.02723.29  
  15. Zhukovskaya L. V. Iterative method for solving nonlinear integral equations describing rolling solutions in string theory. Theoretical and Mathematical Physics, 2006, vol. 146, iss. 3, pp. 335–342. https://doi.org/10.1007/s11232-006-0043-3
  16. Khachatryan Kh. A. On the solubility of certain classes of non-linear integral equations in p-adic string theory. Izvestiya: Mathematics, 2018, vol. 82, iss. 2, pp. 407–427. https://doi.org/10.1070/IM8580  
  17. Rudin W. Functional Analysis. 2nd ed. New York, McGraw-Hill, Inc., 1991. 441 p. (Russ. ed.: Moscow, Mir, 1975. 443 p.).
  18. Kolmogorov A. N., Fomin S. V. Elementy teorii funktsij i funktsional’nogo analiza [Elements of the Theory of Function and Functional Analysis]. Moscow, Nauka, 1981. 544 p. (in Russian).