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Mat. Sb., 2006, Volume 197, Number 10, Pages 33–56 (Mi msb3699)  

This article is cited in 14 scientific papers (total in 14 papers)

Continuous dependence on parameters of solutions to Volterra's equations

E. S. Zhukovskii

Tambov State University

Abstract: The definition of a Volterra operator on a system of equivalence relations is presented. For an appropriately selected system of relations this yields the well-known definitions treating the evolution property, the causality of operators, including Tychonoff's classical definition. The existence, the uniqueness, the extendability of local solutions to non-linear equations with Volterra operators are considered, estimates of the domains of definition of solutions are obtained, and theorems on the continuous dependence of solutions on the parameters are proved. The results so obtained are applied to the analysis of the well-posedness and to the approximate solution of the Cauchy problem for functional differential equations.
Bibliography: 21 titles.

DOI: https://doi.org/10.4213/sm3699

Full text: PDF file (657 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2006, 197:10, 1435–1457

Bibliographic databases:

UDC: 517.988.6
MSC: 47Hxx, 47J05
Received: 16.05.2005 and 19.06.2006

Citation: E. S. Zhukovskii, “Continuous dependence on parameters of solutions to Volterra's equations”, Mat. Sb., 197:10 (2006), 33–56; Sb. Math., 197:10 (2006), 1435–1457

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/msb/v197/i10/p33

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. O. Burlakov, E. S. Zhukovskii, “O korrektnosti upravlyaemykh sistem s zapazdyvaniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2008, no. 2, 27–29  mathnet  elib
    2. T. V. Zhukovskaya, “Ob impulsnykh differentsialnykh uravneniyakh s zapazdyvaniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2008, no. 2, 44–46  mathnet  elib
    3. V. A. Zaitsev, “Ob upravlenii spektrom i stabilizatsii bilineinykh sistem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2008, no. 2, 49–51  mathnet  elib
    4. Zhukovskii E.S., Alvesh M.D., “Volterrovye otobrazheniya metricheskikh prostranstv”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 14:4 (2009), 710–711  elib
    5. Zhukovskii E.S., “Obobschenno volterrovye operatory v metricheskikh prostranstvakh”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 14:3 (2009), 501–508  elib
    6. Burlakov E.O., “Nepreryvnaya zavisimost globalnykh i predelno prodolzhennykh reshenii ot parametrov upravlyaemoi sistemy”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 14:1 (2009), 304–307  elib
    7. E. O. Burlakov, E. S. Zhukovskii, “The continuous dependence of solutions to Volterra equations with locally contracting operators on parameters”, Russian Math. (Iz. VUZ), 54:8 (2010), 12–23  mathnet  crossref  mathscinet  elib
    8. Burlakov E.O., “O razreshimosti operatornykh uravnenii Volterra”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 15:6 (2010), 1654–1656  elib
    9. Zhukovskii E.S., Zhukovskaya T.V., Alvesh M.Zh., “Korrektnost uravnenii s obobschenno volterrovymi otobrazheniyami metricheskikh prostranstv”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 15:6 (2010), 1669–1672  elib
    10. Zhukovskii E.S., Penkov V.B., Skopintseva O.V., “Ob odnom podkhode k issledovaniyu differentsialnykh uravnenii, podvergayuschikhsya impulsnym vozdeistviyam v nefiksirovannye momenty vremeni”, Vestn. Tambovskogo un-ta. Ser. Estestvennye i tekhnicheskie nauki, 16:3 (2011), 735–737  mathscinet  elib
    11. Arutyunov A.V., Zhukovskiy E.S., Zhukovskiy S.E., “Covering mappings and well-posedness of nonlinear Volterra equations”, Nonlinear Anal., 75:3 (2012), 1026–1044  crossref  mathscinet  zmath  isi  elib
    12. Zhukovskaya T.V., “O prodolzhenii reshenii nelineinogo uravneniya volterra”, Vestnik tambovskogo universiteta. seriya: estestvennye i tekhnicheskie nauki, 17:3 (2012), 857–866  mathscinet  elib
    13. E. O. Burlakov, E. S. Zhukovskii, “On well-posedness of generalized neural field equations with impulsive control”, Russian Math. (Iz. VUZ), 60:5 (2016), 66–69  mathnet  crossref  isi
    14. Ponosov A., Zhukovskiy E., “Generalized functional differential equations: existence and uniqueness of solutions”, Electron. J. Qual. Theory Differ., 2016, no. 112, 1–19  crossref  mathscinet  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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