RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Ufimsk. Mat. Zh.: Year: Volume: Issue: Page: Find

 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 78–88 (Mi ufa388)

Asymptotics of solutions to a class of linear differential equations

N. N. Konechnayaa, K. A. Mirzoevb

a Northern (Arctic) Federal University named after M.V. Lomonosov, Severnaya Dvina Emb. 17, 163002, Arkhangelsk, Russia
b Karakhan Agahan ogly Mirzoev, Lomonosov Moscow State University, Leninskie Gory, 1, 119991, Moscow, Russia

Abstract: In the paper we find the leading term of the asymptotics at infinity for some fundamental system of solutions to a class of linear differential equations of arbitrary order $\tau y=\lambda y$, where $\lambda$ is a fixed complex number. At that we consider a special class of Shin-Zettl type and $\tau y$ is a quasi-differential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasi-differential expression $\tau y$, that is, for the entries of the corresponding matrix, are not related with their smoothness but just ensures a certain power growth of these primitives at infinity. Thus, the coefficients of the expression $\tau y$ can also oscillate. In particular, this includes a wide class of differential equations of arbitrary even or odd order with distribution coefficients of finite order. Employing the known definition of two quasi-differential expressions with non-smooth coefficients, in the work we propose a method for obtaining asymptotic formulae for the fundamental system of solutions to the considered equation in the case when the left hand side of this equations is represented as a product of two quasi-differential expressions.
The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasi-differential expression $\tau y$ is symmetric, by the known scheme we define the minimal closed symmetric operator generated by this expression in the space of Lebesgue square-integrable on $[1,+\infty)$ functions (in the Hilbert space ${\mathcal L}^2[1,+\infty)$) and we calculate the deficiency indices for this operator.

Keywords: Quasi-derivative, quasi-differential expression, the main term of asymptotic of the fundamental system of solutions, minimal closed symmetric differential operator, deficiency numbers.

 Funding Agency Grant Number Russian Science Foundation 17-11-01215 The first author was supported by a grant of Russian Science (project no. 17-11-01215).

Full text: PDF file (421 kB)
References: PDF file   HTML file

English version:
Ufa Mathematical Journal, 2017, 9:3, 76–86 (PDF, 362 kB); https://doi.org/10.13108/2017-9-3-76

Bibliographic databases:

UDC: 517.928
MSC: 34E05, 34L05

Citation: N. N. Konechnaya, K. A. Mirzoev, “Asymptotics of solutions to a class of linear differential equations”, Ufimsk. Mat. Zh., 9:3 (2017), 78–88; Ufa Math. J., 9:3 (2017), 76–86

Citation in format AMSBIB
\Bibitem{KonMir17} \by N.~N.~Konechnaya, K.~A.~Mirzoev \paper Asymptotics of solutions to a class of linear differential equations \jour Ufimsk. Mat. Zh. \yr 2017 \vol 9 \issue 3 \pages 78--88 \mathnet{http://mi.mathnet.ru/ufa388} \elib{http://elibrary.ru/item.asp?id=30022853} \transl \jour Ufa Math. J. \yr 2017 \vol 9 \issue 3 \pages 76--86 \crossref{https://doi.org/10.13108/2017-9-3-76} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411740000008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85030031058}