The class of nonlinear integral equations of the Hammerstein--Volterra type on the nonnegative half-line is studied. This class of equations, with various partial representations of the corresponding kernel and nonlinearity, has important applications in hydroaerodynamics, in the theory of radiative transfer, and in models of population genetics. The combination of the method of successive approximations with some geometric estimates for concave and monotone functions makes it possible to prove constructive theorems for the existence and uniqueness of a nonnegative bounded and continuous solution to the specified equation. Moreover, a uniform convergence of special successive approximations (at a rate of decreasing geometric progression) to the solution is established. The corresponding nonlinear ``homogeneous'' equation is also considered, and it is proven that in the class of nonnegative slowly growing functions, this equation has only a trivial (zero) solution. Lastly, particular applied examples of such equations that satisfy all the restrictions of the proven statements are given.
Keywords: monotonicity, uniform convergence, kernel, nonlinearity, bounded solution, continuity
Received January 12, 2025
Revised March 1, 2025
Accepted March 10, 2025
Funding Agency: The research of the second author was carried out with the financial support of the Science Committee of the RA within the framework of a scientific project No. 23RL-1A027.
Haykanush Samvelovna Petrosyan, Cand. Sci. (Phys.-Math.), Armenian National Agrarian University, 0009, Yerevan, Republic of Armenia, e-mail: Haykuhi25@mail.ru
Khachatur Aghavardovich Khachatryan, Dr. Phys.-Math. Sci., Prof., Yerevan State University, 0025, Yerevan; Institute of Mathematics NAS, 0019, Yerevan, Republic of Armenia, e-mail: khachatur.khachatryan@ysu.am
REFERENCES
1. Keller J.J. Propagation of simple non-linear waves in gas filled tubes with friction. Z. Angew. Math. Phys., 1981, vol. 32, no. 2, pp. 170–181. https://doi.org/10.1007/BF00946746
2. Schneider W.R. The general solution of a non-linear integral equation of convolution type. Z. Angew. Math. Phys., 1982, vol. 33, no. 1, pp. 140–142. https://doi.org/10.1007/BF00948318
3. Okrasinski W. Nonlinear Volterra integral equations with convolution kernel. Extracta Math., 1989, vol. 4, no. 2, pp. 51–80.
4. Askhabov S.N. On an integral equation with sum kernel and an inhomogeneity in the linear part. Differ. Equ., 2021, vol. 57, no. 2, pp. 1185–1194. https://doi.org/10.1134/S001226612109007X
5. Okrasinski W. On the existence and uniqueness of non-negative solutions of a certain non-linear convolution equation. Ann. Polon. Math., 1979, vol. 36, no. 1, pp. 61–72.
6. Okrasinski W. On a non-linear convolution equation occurring in the theory of water percolation. Ann. Polon. Math., 1980, vol. 37, no. 2, pp. 223–229.
7. Askhabov S.N., Karapetyants N.K., Yakubov A.Ya. Integral equations of convolution type with power nonlinearity and systems of such equations. Dokl. Math., 1990, vol. 41, no. 2, pp. 323–327.
8. Askhabov S.N., Betilgiriev M.A. Nonlinear integral equations of convolution type with almost increasing kernels in cones. Differ. Equ., 1991, vol. 27, no. 2, pp. 234–242.
9. Bushell P.J., Okrasinski W. Nonlinear Volterra integral equations with convolution kernel. J. London Math. Soc., II Ser., 1990, vol. s2–41, no. 3, pp. 503–510. https://doi.org/10.1112/jlms/s2-41.3.503
10. Brunner H. Volterra integral equations: an introduction to theory and applications. Cambridge, Cambridge Univer. Press, 2017, 405 p. ISBN: 1107098726 .
11. Bushell P.J., Okrasinski W. Nonlinear Volterra integral equations and the Apéry identities. Bull. London Math. Soc., 1992, vol. 24, no. 5, pp. 478–484.
12. Kilbas A.A., Saigo M. On solution of nonlinear Abel–Volterra integral equation. J. Math. Anal. Appl., 1999, vol. 229, no. 1, pp. 41–60. https://doi.org/10.1006/JMAA.1998.6139
13. Askhabov S.N. Volterra integral equation with power nonlinearity. Cheb. Sb., 2022, vol. 23, no. 5, pp. 6–19 (in Russian). https://doi.org/10.22405/2226-8383-2022-23-5-6-19
14. 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. Tr. In-ta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 249–269 (in Russian). https://doi.org/10.21538/0134-4889-2024-30-1-249-269
Cite this article as: H.S. Petrosyan, Kh.A. Khachatryan. On the global solvability of one class of nonlinear integral equations of Hammerstein–Volterra type on the nonnegative half-line. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 181–194.
