Skip to main content
SpringerLink
Log in

Sufficient tests for the existence and uniqueness of a solution of the cauchy problem for a differential equation with a hysteresis nonlinearity

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

Abstract

For a differential equation with a hysteresis nonlinearity of general type, which can be time-varying, we obtain sufficient conditions for the existence and uniqueness of a solution of the Cauchy problem similar to the well-known Cauchy-Picard, Peano, Perron, and Rosenblatt theorems for ordinary differential equations. We consider examples in which a test for the solution uniqueness similar to the Perron-Rosenblatt theorem is applied to specific differential equations with hysteresis nonlinearities of the form of the Prandtl model of a viscoelastic fiber and a time-varying nonlinearity.

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. Krasnosel’skii, M.A. and Pokrovskii, A.V., Sistemy s gisterezisom (Systems with Hysteresis), Moscow: Nauka, 1983.

    Google Scholar 

  2. Borzdyko, V.I., Hysteresis Nonstationary Nonlinearities, Ukrain. Mat. Zh., 2008, vol. 60, no. 3, pp. 295–309.

    Article  MATH  MathSciNet  Google Scholar 

  3. Borzdyko, V.I., The Variable Hysteron, Dokl. Akad. Nauk, 1992, vol. 324, no. 2, pp. 269–272.

    MathSciNet  Google Scholar 

  4. Borzdyko, V.I., Existence and Uniqueness Theorem for a Solution of the Cauchy Problem for a Differential Equation with a Hysteresis Nonlinearity, in Metody teorii funktsii i ikh prilozheniya: Mater. mezhdunar. konf., posv. 10-i godovshchine nezavisimosti Respubliki Tadzhikistan i 80-letiyu professora M.A. Subkhankulova (Methods of Function Theory and Their Applications. Proc. Int. Conf. Devoted to the 10th Anniversary of the Independence of the Republic Tadjikistan and the 80th Birthday of Professor M. A. Subkhankulov), Dushanbe, 2000, p. 17.

    Google Scholar 

  5. Borzdyko, V.I., Differential Equations with Complicated Nonlinearities, Doctoral (Phys.-Math.) Dissertation, Dushanbe, 2000.

    Google Scholar 

  6. Krasnosel’skii, M.A. and Pokrovskii, A.V., Vibrostability of Solutions of Differential Equations, Dokl. Akad. Nauk SSSR, 1970, vol. 195, no. 3, pp. 544–547.

    MathSciNet  Google Scholar 

  7. Krasnosel’skii, M.A., Zabreiko, P.P., Pustyl’nik, E.I., and Sobolevskii, P.E., Integral’nye operatory v prostranstvakh summiruemykh funktsii (Integral Operators in Spaces of Integrable Functions), Moscow: Nauka, 1966.

    MATH  Google Scholar 

  8. Natanson, I.P., Teoriya funktsii veshchestvennoi peremennoi (Theory of Functions of Real Variable), Moscow: Gosudarstv. Izdat. Tekhn.-Teor. Lit., 1957.

    Google Scholar 

  9. Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1977.

    Google Scholar 

  10. Perron, O., Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung, Math. Z., 1928, vol. 28, pp. 216–21

    Article  MATH  MathSciNet  Google Scholar 

  11. Rosenblatt, A., Über die Existenz von Integralen gewöhnlichen Differentialgleichungen, Arch. Matem. Astr. och. Fysik, 1909, vol. 5, no. 2, p. 4

    Google Scholar 

  12. Borzdyko, V.I. and Bakoeva, M.M., Criteria for the Uniqueness of the Solution of the Cauchy Problem for Differential Equations with Hysteresis Nonlinearities, Vestn. Nats. Univ. Dushanbe Mat., 2004, no. 1, pp. 17–27.

    Google Scholar 

  13. Witte, J., Eindeutigkeitssatz für die Differentialgleichung y′ = f(x, y), Math. Z., 1974, vol. 140, no. 3, pp. 281–287.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences, Dushanbe, Tajikistan

    V. I. Borzdyko

Authors
  1. V. I. Borzdyko
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © V.I. Borzdyko, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 12, pp. 1515–1521.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borzdyko, V.I. Sufficient tests for the existence and uniqueness of a solution of the cauchy problem for a differential equation with a hysteresis nonlinearity. Diff Equat 49, 1469–1475 (2013). https://doi.org/10.1134/S001226611312001X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S001226611312001X

Keywords

Search

Navigation