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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).

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English version:
Ufa Mathematical Journal, 2017, 9:3, 76–86 (PDF, 362 kB);

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
\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
\jour Ufa Math. J.
\yr 2017
\vol 9
\issue 3
\pages 76--86

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