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Matematicheskie Zametki, 2023, Volume 113, Issue 2, Pages 217–235
DOI: https://doi.org/10.4213/mzm13882
(Mi mzm13882)
 

This article is cited in 4 scientific papers (total in 4 papers)

Asymptotics of Solutions of Two-Term Differential Equations

N. N. Konechnajaa, K. A. Mirzoevb, A. A. Shkalikovb

a Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk
b Moscow Center for Fundamental and Applied Mathematics
Full-text PDF (589 kB) Citations (4)
References:
Abstract: Asymptotic formulas for the fundamental solution system as $x\to\infty$ are obtained for equations of the form
$$ l(y):=(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=\lambda y,\qquad x\in[1,\infty), $$
where $p$ is a locally integrable function admitting the representation
$$ p(x)=(1+r(x))^{-1},\qquad r\in L^1 [1,\infty), $$
and $q$ is a distribution representable for some given $k$, $0\le k\le n$, as $q=\sigma^{(k)}$, where
\begin{alignat*}{2} \sigma&\in L^1[1,\infty)&\qquad &\text{if }k<n, \\ |\sigma|(1+|r|)(1+|\sigma|)&\in L^1[1,\,\infty) &\qquad &\text{if }k=n. \end{alignat*}
Similar results are obtained for the equations $l(y)=\lambda y$ whose coefficients $p(x)$ and $q(x)$ admit the following representation for a given $k$, $0\le k\le n$:
$$ p(x)=x^{2n+\nu}(1+r(x))^{-1},\qquad q=\sigma^{(k)},\quad \sigma(x)=x^{k+\nu}(\beta+s(x)), $$
where the functions $r$ and $s$ satisfy certain integral decay conditions. We also obtain theorems on the deficiency indices of the minimal symmetric operator generated by the differential expression $l(y)$ (with real functions $p$ and $q$) as well as theorems on the spectra of the corresponding self-adjoint extensions.
Keywords: differential operators with distribution coefficients, quasiderivatives, asymptotics of solutions of differential equations, deficiency indices of a differential operator.
Funding agency Grant number
Russian Science Foundation 20-11-20261
This work was supported by the Russian Science Foundation under grant no. 20-11-20261.
Received: 02.11.2022
Revised: 16.11.2022
English version:
Mathematical Notes, 2023, Volume 113, Issue 2, Pages 228–242
DOI: https://doi.org/10.1134/S0001434623010261
Bibliographic databases:
Document Type: Article
UDC: 517.928
Language: Russian
Citation: N. N. Konechnaja, K. A. Mirzoev, A. A. Shkalikov, “Asymptotics of Solutions of Two-Term Differential Equations”, Mat. Zametki, 113:2 (2023), 217–235; Math. Notes, 113:2 (2023), 228–242
Citation in format AMSBIB
\Bibitem{KonMirShk23}
\by N.~N.~Konechnaja, K.~A.~Mirzoev, A.~A.~Shkalikov
\paper Asymptotics of Solutions of Two-Term Differential Equations
\jour Mat. Zametki
\yr 2023
\vol 113
\issue 2
\pages 217--235
\mathnet{http://mi.mathnet.ru/mzm13882}
\crossref{https://doi.org/10.4213/mzm13882}
\transl
\jour Math. Notes
\yr 2023
\vol 113
\issue 2
\pages 228--242
\crossref{https://doi.org/10.1134/S0001434623010261}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85149996584}
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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