Abstract:
This work is devoted to questions about the existence and uniqueness of solutions to certain nonlinear boundary value problems for singular integral equations of convolution type on the whole straight line. It also looks at their asymptotic properties. Several particular cases of this problem have direct applications in p-adic string theory, the mathematical theory of the geographic spread of epidemics, the kinetic theory of gases and radiative transfer theory. For the two classes of boundary value problems described by such equations, the existence of a nontrivial bounded continuous solution is proved, and the asymptotics of the solution that is constructed are investigated. In certain classes of functions which are bounded and continuous on the whole numerical axis, it is shown that no more than one solution exists. The results obtained are extended to certain nonlinear Urysohn-type equations and to Hammerstein-type equations with two nonlinearities. It is also proved that, in certain special cases, solutions to equations having a continuous convex nonlinearity have a series of important properties. Examples of applications of the above equations are given which illustrate the features of the results obtained.
Key words and phrases:
singular integral equations, convex nonlinearity, iterations, boundedness and continuity, monotonicity, uniqueness of solution, solution limit, Hammerstein- and Urysohn-type equations.
Citation:
Kh. A. Khachatryan, “Solvability of some nonlinear boundary value problems for singular integral equations of convolution type”, Tr. Mosk. Mat. Obs., 81, no. 1, MCCME, M., 2020, 3–40; Trans. Moscow Math. Soc., 81:1 (2020), 1–31
\Bibitem{Kha20}
\by Kh.~A.~Khachatryan
\paper Solvability of some nonlinear boundary value problems for singular integral equations of convolution type
\serial Tr. Mosk. Mat. Obs.
\yr 2020
\vol 81
\issue 1
\pages 3--40
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo633}
\elib{https://elibrary.ru/item.asp?id=46825547}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2020
\vol 81
\issue 1
\pages 1--31
\crossref{https://doi.org/10.1090/mosc/306}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85103211604}
Linking options:
https://www.mathnet.ru/eng/mmo633
https://www.mathnet.ru/eng/mmo/v81/i1/p3
This publication is cited in the following 8 articles:
A. Kh. Khachatryan, Kh. A. Khachatryan, A. S. Petrosyan, “O konstruktivnoi razreshimosti odnogo klassa nelineinykh integralnykh uravnenii gammershteinovskogo tipa na vsei pryamoi”, Izv. vuzov. Matem., 2025, no. 3, 89–106
Kh. A. Khachatryan, H. S. Petrosyan, “On qualitative properties of the solution of a boundary value
problem for a system of nonlinear integral equations”, Theoret. and Math. Phys., 218:1 (2024), 145–162
Kh. A. Khachatryan, A. S. Petrosyan, M. O. Avetisyan, “Teoremy suschestvovaniya i edinstvennosti dlya odnoi sistemy integralnykh uravnenii s dvumya nelineinostyami”, Tr. IMM UrO RAN, 29, no. 1, 2023, 202–218
Kh. A. Khachatryan, A. S. Petrosyan, “Voprosy suschestvovaniya i edinstvennosti resheniya odnogo klassa nelineinykh integralnykh uravnenii na vsei pryamoi”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 26:3 (2022) (to appear)
Kh. A. Khachatryan, A. S. Petrosyan, “Voprosy suschestvovaniya i edinstvennosti resheniya odnogo klassa nelineinykh integralnykh uravnenii na vsei pryamoi”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 26:3 (2022), 446–479
Kh. A. Khachatryan, H. S. Petrosyan, “Solvability of a certain system of singular integral equations with convex nonlinearity on the positive half-line”, Russian Math. (Iz. VUZ), 65:1 (2021), 27–46
Kh. A. Khachatryan, H. S. Petrosyan, “On bounded solutions of a class of nonlinear integral equations in the plane and the Urysohn equation in a quadrant of the plane”, Ukr. Math. J., 73:5 (2021), 811–829
Kh. A. Khachatryan, A. S. Petrosyan, “O kachestvennykh svoistvakh resheniya odnoi nelineinoi granichnoi zadachi v dinamicheskoi teorii $p$-adicheskikh strun”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 16:4 (2020), 423–436
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