Email this article On the constructive solvability of a nonlinear Volterra integral equation on the entire real line
- Authors: Khachatryan K.A.1, Muradyan A.H.2
-
Affiliations:
- Yerevan State University
- Armenian State University of Economics
- Issue: Vol 29, No 2 (2025)
- Pages: 256-273
- Section: Differential Equations and Mathematical Physics
- URL: https://bakhtiniada.ru/1991-8615/article/view/349670
- DOI: https://doi.org/10.14498/vsgtu2150
- EDN: https://elibrary.ru/FBMSFM
- ID: 349670
Cite item
Full Text
Abstract
A nonlinear integral equation with a Hammerstein–Volterra operator on the entire real line is considered. A constructive existence theorem for a bounded and continuous solution is established. Moreover, the uniform convergence of successive approximations to the solution is proved, with the error decreasing at a geometric rate. The integral asymptotics of the constructed solution are then investigated. Additionally, the uniqueness of the solution is demonstrated within a specific subclass of bounded and continuous functions. Finally, specific examples of equations and nonlinearities satisfying all the conditions of the theorems are provided.
Full Text
##article.viewOnOriginalSite##About the authors
Khachatur A. Khachatryan
Yerevan State University
Author for correspondence.
Email: khachatur.khachatryan@ysu.am
ORCID iD: 0000-0002-4835-943X
Scopus Author ID: 24461615400
http://www.mathnet.ru/person27540
Dr. Phys. & Math. Sci., Professor; Head of the Dept.; Dept. of Theory of Functions and Differential Equations
Armenia, 0025, Yerevan, A. Manukyan str., 1Aram H. Muradyan
Armenian State University of Economics
Email: muradyan.aram@asue.am
ORCID iD: 0009-0007-3529-9283
https://www.mathnet.ru/rus/person230809
Cand. Phys. & Math. Sci., Associate Professor; Associate Professor; Dept of Higher Mathematics
Armenia, 0025, Yerevan, Nalbandyan str., 128References
- Neimark Yu. I. On the admissibility of linearization in stability research, Dokl. Akad. Nauk SSSR, 1959, vol. 127, no. 5, pp. 961–964 (In Russian).
- Bellman R., Cooke K. L. Differential-Difference Equations, Mathematics in Science and Engineering, vol. 6. New York, London, Academic Press, 1963, xvi+462 pp.
- Nakhushev A. M. Uravneniya matematicheskoy biologii [Equations of Mathematical Biology]. Moscow, Vyssh. shk., 1995, 301 pp. (In Russian). EDN: PDBBNB.
- Khachatryan Kh. A., Terdzyan Ts. E., Broyan M. F. One-parameter family of integrable solutions of a system of nonlinear integral equations of the Hammerstein–Volterra type in the supercritical case, Differ. Equ., 2016, vol. 52, no. 8, pp. 1036–1042. EDN: XNRJXJ. DOI: https://doi.org/10.1134/S0012266116080097.
- Khachatryan Kh. A., Terjyan Ts. E., Broyan M. F. On solvability of a Hammerstein–Voltera type nonlinear system of integral equations in critical case, Vladikavkaz. Mat. Zh., 2016, vol. 18, no. 4, pp. 71–79 (In Russian). EDN: XVSSLD. DOI: https://doi.org/10.23671/VNC.2016.4.5996.
- Khachatryan Kh. A., Grigoryan S. A. On nontrivial solvability of a nonlinear Hammerstein–Volterra type integral equation, Vladikavkaz. Mat. Zh., 2012, vol. 14, no. 2, pp. 57–66 (In Russian). EDN: OYFBBT. DOI: https://doi.org/10.23671/VNC.2012.14.10964.
- Azizyan E. O., Khachatryan Kh. A. One-parametric family of positive solutions for a class of nonlinear discrete Hammerstein-Volterra equations, Ufa Math. J., 2016, vol. 8, no. 1, pp. 13–19. EDN: XLIBFL. DOI: https://doi.org/10.13108/2016-8-1-13.
- Askhabov S. N. Volterra integral equation with power nonlinearity, Chebyshevskii Sb., 2022, vol. 23, no. 5, pp. 6–19 (In Russian). EDN: EIGULQ. DOI: https://doi.org/10.22405/2226-8383-2022-23-5-6-19.
- Askhabov S. N. Volterra integro-differential equation of arbitrary order with power non-linearity, Chebyshevskii Sb., 2023, vol. 24, no. 4, pp. 85–103 (In Russian). EDN: JXSSJW. DOI: https://doi.org/10.22405/2226-8383-2023-24-4-85-103.
- Askhabov S. N. A system of inhomogeneous integral equations of convolution type with power nonlinearity, Sib. Math. J., 2023, vol. 64, no. 3, pp. 691–698. EDN: EXKOSK. DOI: https://doi.org/10.1134/S0037446623030163.
- Rudin W. Functional Analysis, International Series in Pure and Applied Mathematics. New York, NY, McGraw-Hill, 1991, xviii+424 pp.
- 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). EDN: ECMMEF. DOI: https://doi.org/10.21538/0134-4889-2024-30-1-249-269.
Supplementary files
