Skip to main content
SpringerLink
Log in

On the existence of multiple solutions for a nonlocal BVP with vector-valued response

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.

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. V. Anuradha, D. D. Hai and R. Shivaji: Existence results for superlinear semipositone BVP’s. Proc. A.M.S. 124 (1996), 757–763.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. V. Bitsadze: On the theory of nonlocal boundary value problems. Soviet Math. Dokl. 30 (1984), 8–10.

    MATH  Google Scholar 

  3. A. V. Bitsadze and A. A. Samarskii: Some elementary generalizations of linear elliptic boundary value problems. Dokl. Akad. Nauk SSSR 185 (1969), 739–740.

    MATH  MathSciNet  Google Scholar 

  4. D. R. Dunninger and H. Wang: Multiplicity of positive solutions for a nonlinear differential equation with nonlinear boundary conditions. Annales Polonici Math. LXIX.2 (1998), 155–165.

    MathSciNet  Google Scholar 

  5. W. P. Eloe and J. Henderson: Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electronic J. of Differential Equations 03 (1997), 1–11.

    MathSciNet  Google Scholar 

  6. L. H. Erbe and H. Wang: On the existence of positive solutions of ordinary differential equations. Proc. A.M.S. 120 (1994), 743–748.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. P. Gupta: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. Math. Anal. Appl. 168 (1992), 540–551.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. Gupta, S. K. Ntouyas and P. Ch. Tsamatos: On an m-point boundary value problem for second order differential equations. Nonlinear Analysis TMA 23 (1994), 1427–1436.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Gupta: Solvability of a generalized multipoint boundary value problem of mixed type for second order ordinary differential equations. Proc. Dynamic Systems and Applications 2 (1996), 215–222.

    MATH  Google Scholar 

  10. C. P. Gupta: A generalized multi-point nonlinear boundary value problem for a second order ordinary differential equation. Appl. Math. Comput 89 (1998), 133–146.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Henderson and H. Wang: Positive solutions for nonlinear eigenvalue problems. J. Math. Anal. Appl. 208 (1997), 252–259.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. A. Il’in and E. I. Moiseev: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differ. Equ. 23 (1987), 803–811.

    Google Scholar 

  13. V. A. Il’in and E. I. Moiseev: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differ. Equ. 23 (1987), 979–987.

    Google Scholar 

  14. G. L. Karakostas and P. Ch. Tsamatos: Positive solutions of a boundary-value problem for second order ordinary differential equations. Electronic Journal of Differential Equations 49 (2000), 1–9.

    Google Scholar 

  15. G. L. Karakostas and P. Ch. Tsamatos: Positive solutions for a nonlocal boundary-value problem with increasing response. Electronic Journal of Differential Equations 73 (2000), 1–8.

    Google Scholar 

  16. G. L. Karakostas and P. Ch. Tsamatos: Multiple positive solutions for a nonlocal boundary-value problem with response function quiet flat zero. Electronic Journal of Differential Equations 13 (2001), 1–10.

    MathSciNet  Google Scholar 

  17. G. L. Karakostas and P. Ch. Tsamatos: Existence of multiple solutions for a nonlocal boundary-value problem. Topol. Math. Nonl. Anal. 19 (2000), 109–121.

    MathSciNet  Google Scholar 

  18. M. A. Krasnoselski: Positive solutions of operator equations. Noordhoff, Groningen, 1964.

    Google Scholar 

  19. R. Ma: Positive solutions for a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 34 (1998), 1–8.

    MATH  Google Scholar 

  20. R. Y. Ma and N. Castaneda: Existence of solutions of nonlinear m-point boundary value problems. J. Math. Anal. Appl. 256 (2001), 556–567.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Ma: Existence of positive solutions for second order m-point boundary value problems. Annales Polonici Mathematici LXXIX.3 (2002), 256–276.

    Google Scholar 

  22. J. Mawhin: Problèmes de Dirichlet Variationnels Non Linéares. Les Presses de l’Université de Montréal (1987).

  23. A. Nowakowski: A new variational principle and duality for periodic solutions of Hamilton’s equations. J. Differential Eqns. 97 (1992), 174–188.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Nowakowski and A. Orpel: Positive solutions for a nonlocal boundary-value problem with vector-valued response. Electronic J. of Differential Equations 46 (2002), 1–15.

    MathSciNet  Google Scholar 

  25. P. H. Rabinowitz: Minimax Methods in Critical Points Theory with Applications to Differential Equations. AMS, Providence, 1986.

    Google Scholar 

  26. J. R. L. Webb: Positive solutions of some three-point boundary value problems via fixed point theory. Nonlinear Anal. 47 (2001), 4319–4332.

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Wang: On the existence of positive solutions for semilinear elliptic equations in annulus. J. Differential Equation 109 (1994), 1–4.

    Article  MATH  Google Scholar 

  28. M. Willem: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications. Basel, Boston, Berlin: Birkhäuser, Vol. 24, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Lodz, Banacha 22, 90-238, Lodz, Poland

    Andrzej Nowakowski & Aleksandra Orpel

Authors
  1. Andrzej Nowakowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aleksandra Orpel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nowakowski, A., Orpel, A. On the existence of multiple solutions for a nonlocal BVP with vector-valued response. Czech Math J 56, 621–640 (2006). https://doi.org/10.1007/s10587-006-0043-3

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-006-0043-3

Keywords

Search

Navigation