Skip to main content
SpringerLink
Log in

On the solvability of a class of Volterra operator equations of the first kind with piecewise continuous kernels

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We obtain sufficient conditions for the existence and uniqueness of continuous solutions of Volterra operator equations of the first kind with piecewise determined kernels. For the case in which the solution is not unique, we prove existence theorems for the parametric families of solutions and present their asymptotics in the form of logarithmic polynomials.

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. E. V. Markova, I. V. Sidler, and V. V. Trufanov, “On Glushkov-type models of developing systems and their applications to the electric power industry,” Avtomat. Telemekh., No. 7, 20–28 (2011) [Automat. Remote Control 72 (7), 1371–1379 (2011)].

    Google Scholar 

  2. Yu. P. Yatsenko, Integral Models of Systems with Controled Memory (Naukova Dumka, Kiev, 1991) [in Russian].

    Google Scholar 

  3. N. Hritonenko and Yu. Yatsenko, Modeling and Optimization of the Lifetime of Technologies (Kluwer Acad. Publ., Dordrecht, 1991).

    Google Scholar 

  4. A. M. Denisov and A. Lorenzi, “On a special Volterra integral equation the first kind,” Boll. Un. Mat. Ital. B (7) 9(2), 443–457 (1995).

    MATH  MathSciNet  Google Scholar 

  5. N. A. Magnitskii, “Asymptotic behavior of the solutions of the Volterra integral equation of the first kind,” Dokl. Akad. Nauk SSSR 269(1), 29–32 (1983).

    MathSciNet  Google Scholar 

  6. D. Sidorov, “Volterra equations of the first kind with discontinuous kernels in the theory of evolving systems control,” Stud. Inform. Univ. 9(3), 135–146 (2011).

    MathSciNet  Google Scholar 

  7. N. A. Sidorov and D. N. Sidorov, “Small solutions of nonlinear differential equations near branching points,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 53–61 (2011) [RussianMath. (Iz. VUZ) 55 (5), 43–50 (2011)].

    Google Scholar 

  8. D. N. Sidorov, “Solvability of systems ofVolterra integral equations of the first kindwith piecewise continuous kernels,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 1, 62–72 (2013) [Russian Math. (Iz. VUZ) 57 (1), 54–63 (2013)].

    Google Scholar 

  9. D. N. Sidorov, “On parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel,” Differ. Uravn. 49(2), 209–215 (2013) [Differ. Equations 49 (2), 210–216 (2013)].

    MathSciNet  Google Scholar 

  10. D. N. Sidorov and N. A. Sidorov, “Convex majorants method in the theory of nonlinear Volterra equations,” Banach J. Math. Anal. 6(1), 1–10 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  11. N. A. Sidorov and D. N. Sidorov, “Existence and construction of generalized solutions of nonlinear Volterra integral equations of the first kind,” Differ. Uravn. 42(9), 1243–1247 (2006) [Differ. Equations 42 (9), 1312–1316 (2006)].

    MathSciNet  Google Scholar 

  12. N. A. Sidorov, D. N. Sidorov, and A. V. Krasnik, “Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method,” Differ. Uravn. 46(6), 874–882 (2010) [Differ. Equations 46 (6), 882–891 (2010)].

    MathSciNet  Google Scholar 

  13. N. Sidorov, B. Loginov, A. Sinitsyn, and M. Falaleev, Lyapunov-Schmidt Methods in Nonlinear Analysis and Applications (Kluwer Acad. Publ., Dordrecht, 2002).

    Book  MATH  Google Scholar 

  14. D. Sidorov, “On impulsive control of nonlinear dynamical systems based on the Volterra series,” in Proceedings of 10th IEEE International Conference on Environment and Electrical (IEEE Publ., Rome, 2011).

    Google Scholar 

  15. D. N. Sidorov and N. A. Sidorov, “Generalized solutions in problems of modeling nonlinear dynamical systems by Volterra polynomials,” Avtomat. Telemekh. No. 6, 127–132 (2011) [Automat. Remote Control 72 (6), 1258–1263 (2011)].

    Google Scholar 

  16. G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, in Inverse and Ill-Posed Problems Ser. (Walter de Gruyter, Berlin, 2012), Vol. 42 [in Russian].

    Google Scholar 

  17. N. A. Sidorov and D. N. Sidorov, “Integral Volterra equations of the first kind with piecewise determined kernel in Banach space,” in Nonclassical Equations of Mathematical Physics, Collection of scientific papers (IM SO RAN, Novosibirsk, 2012) [in Russian].

    Google Scholar 

  18. D. N. Sidorov, “Generalized solution to the Volterra equations with piecewise continuous kernels,” Bull. Malays. Math. Sci. Soc. (2) 37(No. 3), 757–768 (2014).

    MATH  MathSciNet  Google Scholar 

  19. N. A. Sidorov and A. V. Trufanov, “Nonlinear operator equations with a functional perturbation of the argument of neutral type,” Differ. Uravn. 45(12), 1804–1808 (2009) [Differ. Equations 45 (12), 1840–1844 (2009)].

    MathSciNet  Google Scholar 

  20. L. A. Él’sgol’ts, Qualitative Methods in Mathematical Analysis (GIITL, Moscow, 1955) [in Russian].

    Google Scholar 

  21. M. M. Vainberg and V. A. Trenogin, Branching Theory of Solutions of Nonlinear Equations (Nauka, Moscow, 1969) [in Russian].

    MATH  Google Scholar 

  22. V. A. Trenogin, Functional Analysis (Fizmatlit, Moscow, 2007) [in Russian].

    Google Scholar 

  23. A. O. Gel’fond, Calculus of Finite Differences (Fizmatlit, Moscow, 1959) [in Russian].

    Google Scholar 

  24. V. C. Vladimirov, Generalized Functions in Mathematical Physics (Fizmatlit, Moscow, 1976) [inRussian].

    Google Scholar 

  25. D. Sidorov, Integral Dynamical Models: Singularities, Signals, and Control, in World Sci. Ser. Nonlinear Sci. Ser. A (World Sci. 2014), Vol. 87.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Irkutsk State University, Irkutsk, Russia

    N. A. Sidorov

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

    D. N. Sidorov

  3. Irkutsk State University, Irkutsk, Russia

    D. N. Sidorov

  4. Irkutsk State Technical Research University, Irkutsk, Russia

    D. N. Sidorov

Authors
  1. N. A. Sidorov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. N. Sidorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. A. Sidorov.

Additional information

Original Russian Text © N. A. Sidorov, D. N. Sidorov, 2014, published in Matematicheskie Zametki, 2014, Vol. 96, No. 5, pp. 773–789.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sidorov, N.A., Sidorov, D.N. On the solvability of a class of Volterra operator equations of the first kind with piecewise continuous kernels. Math Notes 96, 811–826 (2014). https://doi.org/10.1134/S0001434614110170

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434614110170

Keywords

Search

Navigation