
Asymptotics of solutions to a class of linear differential equations
N. N. Konechnaya^{a}, K. A. Mirzoev^{b} ^{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 ShinZettl type and $\tau y$ is a quasidifferential expression generated by the matrix in this class. The conditions we assume for the primitives of the coefficients of the quasidifferential 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 quasidifferential expressions with nonsmooth 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 quasidifferential expressions.
The obtained results are applied for the spectral analysis of the corresponding singular differential operators. In particular, assuming that the quasidifferential 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 squareintegrable on $[1,+\infty)$ functions (in the Hilbert space ${\mathcal L}^2[1,+\infty)$) and we calculate the deficiency indices for this operator.
Keywords:
Quasiderivative, quasidifferential 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 
171101215 
The first author was supported by a grant of Russian Science (project no. 171101215). 
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/10.13108/20179376
Bibliographic databases:
UDC:
517.928
MSC: 34E05, 34L05 Received: 25.05.2017
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 7888
\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 7686
\crossref{https://doi.org/10.13108/20179376}
\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=2s2.085030031058}
Linking options:
http://mi.mathnet.ru/eng/ufa388 http://mi.mathnet.ru/eng/ufa/v9/i3/p78
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  127  Full text:  42  References:  11 
