Skip to main content
SpringerLink
Log in

Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

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

Abstract

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.

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 Zabreiko, P.P., Geometricheskie metody nelineinogo analiza (Geometric Methods of Nonlinear Analysis), Moscow: Nauka, 1975.

    Google Scholar 

  2. Naimov, A.N., A Remark on the Theory of Two-Point Boundary Value Problems for Systems of Ordinary Differential Equations, Dokl. Akad. Nauk Resp. Tadjikistan, 1998, vol. 41, no. 9, pp. 30–34.

    Google Scholar 

  3. Naimov, A.N., On the Computation of the Winding Number of a Completely Continuous Vector Field, Dokl. Akad. Nauk Resp. Tadjikistan, 1998, vol. 41, no. 10, pp. 56–61.

    Google Scholar 

  4. Naimov, A.N., On the Solvability of the Third Nonlinear Two-Point Boundary Value Problem on the Plane, Differ. Uravn., 2002, vol. 38, no. 1, pp. 132–133.

    MathSciNet  MATH  Google Scholar 

  5. Bernshtein, S.N., Sobr. soch. (Collected Papers), Moscow: Izdat. Akad. Nauk SSSR, 1960, vol. 3.

    Google Scholar 

  6. Nagumo, M., Uber die Differentialgleichung y = f(x, y, y), Proc. Phys. Math. Soc. Japan, 1937, vol. 19, pp. 861–866.

    MATH  Google Scholar 

  7. Lepin, A.Ya., Necessary and Sufficient Conditions for the Existence of a Solution of the Two-Point Boundary Value Problem for a Second Order Nonlinear Ordinary Differential Equation, Differ. Uravn., 1970, vol. 6, no. 8, pp. 1384–1388.

    MathSciNet  Google Scholar 

  8. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vologda State University, Vologda, Russia

    E. Mukhamadiev & A. N. Naimov

Authors
  1. E. Mukhamadiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. N. Naimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Mukhamadiev.

Additional information

Original Russian Text © E. Mukhamadiev, A.N. Naimov, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 3, pp. 334–341.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukhamadiev, E., Naimov, A.N. Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane. Diff Equat 52, 327–334 (2016). https://doi.org/10.1134/S0012266116030071

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266116030071

Keywords

Search

Navigation