Abstract:
This paper is devoted to the study and solution of a boundary value problem for a convolution-type integral equation with cubic nonlinearity. The above problem has a direct application to the p-adic theory of open-closed strings for the scalar tachyon field. It is shown that a one-parameter family of monotone continuous bounded solutions exists. Under additional conditions on the kernel of the equation, an asymptotic formula for the solutions thus constructed is established. Using these results, as particular cases we obtain Zhukovskaya's theorem on rolling solutions of the nonlinear equation in the p-adic theory of open-closed strings and the Vladimirov–Volovich theorem on the existence of a nontrivial solution between certain vacua.
The results are extended to the case of a more general nonlinear boundary value problem.
Key words and phrases:p-adic string, successive approximations, monotonicity, bounded solution, kernel, boundary value problem.
Citation:
Kh. A. Khachatryan, “On the solvability of a boundary value problem in p-adic string theory”, Tr. Mosk. Mat. Obs., 79, no. 1, MCCME, M., 2018, 117–132; Trans. Moscow Math. Soc., 2018, 101–115
\Bibitem{Kha18}
\by Kh.~A.~Khachatryan
\paper On the solvability of a boundary value problem in $ p$-adic string theory
\serial Tr. Mosk. Mat. Obs.
\yr 2018
\vol 79
\issue 1
\pages 117--132
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo610}
\elib{https://elibrary.ru/item.asp?id=37045085}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2018
\pages 101--115
\crossref{https://doi.org/10.1090/mosc/281}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060867696}
Linking options:
https://www.mathnet.ru/eng/mmo610
https://www.mathnet.ru/eng/mmo/v79/i1/p117
This publication is cited in the following 24 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
A. Kh. Khachatryan, Kh. A. Khachatryan, A. S. Petrosyan, “Voprosy suschestvovaniya, otsutstviya i edinstvennosti resheniya odnogo klassa nelineinykh integralnykh uravnenii na vsei pryamoi s operatorom tipa Gammershteina — Ctiltesa”, Tr. IMM UrO RAN, 30, no. 1, 2024, 249–269
A. Kh. Khachatryan, Kh. A. Khachatryan, H. S. Petrosyan, “On nonlinear convolution-type integral equations in the theory
of $p$-adic strings”, Theoret. and Math. Phys., 216:1 (2023), 1068–1081
Kh. A. Khachatryan, H. S. Petrosyan, “On non-trivial solvability of one system of non-linear integral equations on the real axis”, Izv. Math., 87:5 (2023), 1062–1077
A.Kh. Khachatryan, Kh.A. Khachatryan, “ON QUALITATIVE PROPERTIES OF A SOLUTION OF ONE CLASS SINGULAR INTEGRAL EQUATIONS ON THE WHOLE LINE WITH ODD NONLINEARITY”, J Math Sci, 271:5 (2023), 597
Kh.A. Khachatryan, H.S. Petrosyan, A. R. Hakobyan, “On solvability of one class of integral equations on whole line with monotonic and convex nonlinearity”, J Math Sci, 271:5 (2023), 610
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
A. Kh. Khachatryan, Kh. A. Khachatryan, “A System of Integral Equations on the Entire Axis with Convex and Monotone Nonlinearity”, J. Contemp. Mathemat. Anal., 57:5 (2022), 311
Kh. A. Khachatryan, H. S. Petrosyan, “On One Class of Multidimensional Integral Equations of Convolution Type with Convex Nonlinearity”, Diff Equat, 58:5 (2022), 680
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, H. S. Petrosyan, “Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations”, Trans. Moscow Math. Soc., 82 (2021), 259–271
Kh. A. Khachatryan, A. S. Petrosyan, “O postroenii summiruemogo resheniya odnogo klassa nelineinykh integralnykh uravnenii tipa Gammershteina - Nemytskogo na vsei pryamoi”, Tr. IMM UrO RAN, 26, no. 2, 2020, 278–287
Kh. A. Khachatryan, “Existence and uniqueness of solution of a certain boundary-value problem for a convolution integral equation with monotone non-linearity”, Izv. Math., 84:4 (2020), 807–815
A. Kh. Khachatryan, Kh. A. Khachatryan, “On solvability of one infinite system of nonlinear functional equations in the theory of epidemics”, Eurasian Math. J., 11:2 (2020), 52–64
Kh. A. Khachatryan, “Solvability of some nonlinear boundary value problems for singular integral equations of convolution type”, Trans. Moscow Math. Soc., 81:1 (2020), 1–31
Kh. A. Khachatryan, H. S. Petrosyan, “Solvability of a nonlinear problem in open-closed p-adic string theory”, Differ. Equ., 56:10 (2020), 1371–1378
Kh. A. Khachatryan, S. M. Andriyan, “On solvability of one class of nonlinear integral equations on whole line with two monotone nonlinearities”, P-Adic Numbers Ultrametric Anal. Appl., 12:4 (2020), 259–275
Kh. A. Khachatryan, S. M. Andriyan, “On the solvability of a class of discrete matrix equations with cubic nonlinearity”, Ukr. Math. J., 71:12 (2020), 1910–1928
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