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Uspekhi Mat. Nauk, 2008, Volume 63, Issue 1(379), Pages 111–154 (Mi umn8544)  

This article is cited in 20 scientific papers (total in 20 papers)

Sturm–Liouville oscillation theory for impulsive problems

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

Voronezh State University

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.

DOI: https://doi.org/10.4213/rm8544

Full text: PDF file (898 kB)
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English version:
Russian Mathematical Surveys, 2008, 63:1, 109–153

Bibliographic databases:

UDC: 517.927
MSC: Primary 34B24; Secondary 34C10, 34L99, 74Q10
Received: 24.09.2007

Citation: Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “Sturm–Liouville oscillation theory for impulsive problems”, Uspekhi Mat. Nauk, 63:1(379) (2008), 111–154; Russian Math. Surveys, 63:1 (2008), 109–153

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. V. Pokornyi, M. B. Zvereva, Zh. I. Bakhtina, “On Stieltjes differentials on geometric graphs”, Dokl. Math., 78:3 (2008), 877–879  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. Yu. V. Pokornyi, Zh. I. Bakhtina, “On the Stieltjes Procedure for Closing Gaps in Time Scales”, Math. Notes, 86:5 (2009), 690–692  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, M. B. Davydova, “Stieltjes differential in impulsive problems with discontinuous solutions”, Dokl. Math., 80:2 (2009), 743–745  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. Yu. V. Pokornyi, Zh. I. Bakhtina, “Stieltjes integral method in differential equations on time scales”, Dokl. Math., 80:2 (2009), 759–761  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    5. Zh. I. Bakhtina, “O differentsialakh Stiltesa na vremennykh shkalakh”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:2 (2009), 3–5  mathnet  elib
    6. A. Yu. Trynin, “Asymptotic behavior of the solutions and nodal points of Sturm–Liouville differential expressions”, Siberian Math. J., 51:3 (2010), 525–536  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. Pokornyi Yu.V., Bakhtina Zh.I., “Integration of functions rd-continuous on time scales”, Dokl. Math., 83:1 (2011), 104–106  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. A. Yu. Trynin, “Differentsialnye svoistva nulei sobstvennykh funktsii zadachi Shturma–Liuvillya”, Ufimsk. matem. zhurn., 3:4 (2011), 133–143  mathnet  zmath
    9. M. B. Davydova, S. A. Shabrov, “O chisle reshenii nelineinoi kraevoi zadachi s integralom Stiltesa”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:4 (2011), 13–17  mathnet
    10. Pokornyi Yu.V., Bakhtina Zh.I., Zvereva M.B., Ishchenko A.S., “On the Euler and Jacobi differential equations for variational problems with impulse parameters”, Differ. Equ., 47:5 (2011), 656–670  crossref  mathscinet  zmath  isi  elib  elib  scopus
    11. V. M. Bruk, “Linear relations generated by an integral equation with Nevanlinna operator measure”, Russian Math. (Iz. VUZ), 56:10 (2012), 1–14  mathnet  crossref  mathscinet
    12. Pokornyi Yu.V., Zvereva M.B., Bakhtina Zh.I., “Stieltjes differential method in the modeling of an irregular system on a geometric graph”, Differ. Equ., 48:8 (2012), 1103–1111  crossref  mathscinet  zmath  isi  elib  elib  scopus
    13. V. M. Bruk, “Invertible linear relations generated by an integral equation with a Nevanlinna measure”, Russian Math. (Iz. VUZ), 57:2 (2013), 13–24  mathnet  crossref
    14. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    15. V. M. Bruk, “On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the Infinite-Dimensional Case”, Zhurn. matem. fiz., anal., geom., 10:2 (2014), 163–188  mathnet  crossref  mathscinet
    16. V. M. Bruk, “On Linear Relations Generated by an Integro-Differential Equation with Nevanlinna Measure in the Infinite-Dimensional Case”, Math. Notes, 96:1 (2014), 10–25  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. V. M. Bruk, “Dissipative expansions of a symmetric relation generated by a system of integral equations with operator measures”, Russian Math. (Iz. VUZ), 58:12 (2014), 7–22  mathnet  crossref
    18. A. D. Baev, M. B. Zvereva, S. A. Shabrov, “Stieltjes differential in nonlinear momentum problems”, Dokl. Math, 90:2 (2014), 613  crossref  mathscinet  zmath  isi  scopus
    19. V. M. Bruk, “Invertibility of linear relations generated by integral equation with operator measures”, Ufa Math. J., 6:4 (2014), 48–59  mathnet  crossref
    20. R. Ch. Kulaev, “On the Oscillation Property of Green's Function of a Fourth-Order Discontinuous Boundary-Value Problem”, Math. Notes, 100:3 (2016), 391–402  mathnet  crossref  crossref  mathscinet  isi  elib
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