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 Tr. Mosk. Mat. Obs., 2018, Volume 79, Issue 1, Pages 117–132 (Mi mmo610)

On the solvability of a boundary value problem in $p$-adic string theory

Kh. A. Khachatryan

Institute of Mathematics of the National Academy of Sciences of Armenia

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.

 Funding Agency Grant Number State Committee on Science of the Ministry of Education and Science of the Republic of Armenia SCS 16YR-1A002

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English version:
Transactions of the Moscow Mathematical Society, 2018, 101–115

Document Type: Article
UDC: 517.968.4
MSC: 45G05

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

Citation in format AMSBIB
\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} \transl \jour Trans. Moscow Math. Soc. \yr 2018 \pages 101--115 \crossref{https://doi.org/10.1090/mosc/281} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85060867696}