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Russian Mathematical Surveys, 2008, Volume 63, Issue 1, Pages 109–153
DOI: https://doi.org/10.1070/RM2008v063n01ABEH004502
(Mi rm8544)
 

This article is cited in 39 scientific papers (total in 39 papers)

Sturm–Liouville oscillation theory for impulsive problems

Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov

Voronezh State University
References:
Abstract: This paper extends the Sturm–Liouville oscillation theory on the distribution of zeros of eigenfunctions to the case of problems with strong singularities of the coefficients (of $\delta$-function type). For instance, these are problems arising in the study of eigenoscillations of an elastic continuum with concentrated masses and localized interactions with the surrounding medium. The extension of the standard description of the problem is carried out by replacing the usual form of the ordinary differential equation
$$ -(pu')'+qu=\lambda mu $$
by the substantially more general form
$$ -(pu')(x)+(pu')(0)+\int_0^xu\,dQ=\lambda\int_0^xu\,dM $$
with absolutely continuous solutions whose derivatives, as well as the coefficients $p$, $Q$, $M$, belong to $\operatorname{BV}[0,l]$. The integral is understood in the Stieltjes sense.
Received: 24.09.2007
Bibliographic databases:
Document Type: Article
UDC: 517.927
MSC: Primary 34B24; Secondary 34C10, 34L99, 74Q10
Language: English
Original paper language: Russian
Citation: Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “Sturm–Liouville oscillation theory for impulsive problems”, Russian Math. Surveys, 63:1 (2008), 109–153
Citation in format AMSBIB
\Bibitem{PokZveSha08}
\by Yu.~V.~Pokornyi, M.~B.~Zvereva, S.~A.~Shabrov
\paper Sturm--Liouville oscillation theory for impulsive problems
\jour Russian Math. Surveys
\yr 2008
\vol 63
\issue 1
\pages 109--153
\mathnet{http://mi.mathnet.ru//eng/rm8544}
\crossref{https://doi.org/10.1070/RM2008v063n01ABEH004502}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2406183}
\zmath{https://zbmath.org/?q=an:1170.34313}
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\elib{https://elibrary.ru/item.asp?id=12894561}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-48249150006}
Linking options:
  • https://www.mathnet.ru/eng/rm8544
  • https://doi.org/10.1070/RM2008v063n01ABEH004502
  • https://www.mathnet.ru/eng/rm/v63/i1/p111
  • This publication is cited in the following 39 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:1553
    Russian version PDF:785
    English version PDF:84
    References:115
    First page:24
     
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