Skip to main content
SpringerLink
Log in

Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations

  • Integral Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We obtain sufficient conditions for the existence of a principal solution of a nonlinear Volterra integral equation of the second kind on the half-line and on a finite interval. We suggest a method for computing the boundary of an interval outside which the solution can blow up.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kantorovich, L.V., Vulikh, B.Z., and Pinsker, A.G., Funktsional’nyi analiz v poluuporyadochennykh prostranstvakh (Functional Analysis in Partially Ordered Spaces), Moscow-Leningrad: Gosudarstv. Izdat. Tekhn.-Teor. Lit., 1950.

    MATH  Google Scholar 

  2. Sidorov, D.N. and Sidorov, N.A., Convex Majorants Method in the Theory of Nonlinear Volterra Equations, Banach J. Math. Anal., 2012, vol. 6, no. 1, pp. 20–28.

    Article  MathSciNet  Google Scholar 

  3. Yudovich, V.I., Matematicheskie modeli estestvennykh nauk (Mathematical Models in Natural Sciences), Rostov-on-Don, 2006.

    Google Scholar 

  4. Korpusov, M.O., Razrusheniya v neklassicheskikh nelokal’nykh uravneniyakh (Blow-Up in Nonclassical Nonlocal Equations), Moscow, 2011.

    Google Scholar 

  5. Sidorov, N.A., Markanova, D.Yu., and Abdullin, V.R., On the Role of Convex Majorants in Nonlocal Existence Theorems for Implicit Functions, Preprint IDSTU Sib. Div. Rus. Acad. Sci., Irkutsk, 1998, no. 3.

    Google Scholar 

  6. Sidorov, N., Loginov, B., Sinitsyn, A., and Falaleev, M., Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications, Dordrecht; Boston; London, 2002.

    Book  MATH  Google Scholar 

  7. Markanova, D.Yu., The Use of the Kantorovich Majorant for Estimating the Existence Domain of an Implicit Function, Tr. Vostochno-sibirskoi zonal’noi konferentsii po matematike i problemam ee prepodavaniya v vuzakh (Proc. East-Siberian Regional Conf. on Math. and Mathematical Teaching in Higher School), Irkutsk, 1999, pp. 68–72.

    Google Scholar 

  8. Apartsin, A.S., On Equivalent Norms in the Theory of Polynomial Integral Volterra Equations, Izv. Irkuts. Gos. Univ. Math., 2010, no. 1, pp. 19–29.

    Google Scholar 

  9. Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow, 2007.

    Google Scholar 

  10. Il’in, V.A., Sadovnichii, V.A., and Sendov, Bl.Kh., Matematicheskii analiz (Mathematical Analysis), Moscow: Nauka, 1979.

    Google Scholar 

  11. Vulikh, B.Z., Kratkii kurs teorii funktsii veshchestvennoi peremennoi (A Brief Course of the Theory of a Function of a Real Variable), Moscow, 1973.

    Google Scholar 

  12. Shilov, G.E. and Gurevich, B.L., Integral, mera i proizvodnaya. Obshchaya teoriya (Integral, Measure, and Derivative. General Theory), Moscow: Nauka, 1967.

    Google Scholar 

  13. Dwight, H.B., Tables of Integrals and Other Mathematical Data, New York: Macmillan Company, 1961. Translated under the title Tablitsy integralov i drugie matematicheskie formuly, Moscow, 1979.

    Google Scholar 

  14. Grebennikov, E.A. and Ryabov, Yu.A., Konstruktivnye metody analiza nelineinykh sistem (Constructive Methods for the Analysis of Nonlinear Systems), Moscow, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Melentiev Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia

    D. N. Sidorov

Authors
  1. D. N. Sidorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. N. Sidorov.

Additional information

Original Russian Text © D.N. Sidorov, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 9, pp. 1231–1237.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sidorov, D.N. Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations. Diff Equat 50, 1217–1224 (2014). https://doi.org/10.1134/S0012266114090080

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266114090080

Keywords

Search

Navigation