Advertisement

Czechoslovak Mathematical Journal

, Volume 56, Issue 2, pp 621–640 | Cite as

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

  • Andrzej Nowakowski
  • Aleksandra Orpel
Article

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.

Keywords

nonlocal boundary-value problems positive solutions duality method variational method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. Anuradha, D. D. Hai and R. Shivaji: Existence results for superlinear semipositone BVP’s. Proc. A.M.S. 124 (1996), 757–763.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. V. Bitsadze: On the theory of nonlocal boundary value problems. Soviet Math. Dokl. 30 (1984), 8–10.MATHGoogle Scholar
  3. [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.MATHMathSciNetGoogle Scholar
  4. [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.MathSciNetGoogle Scholar
  5. [5]
    W. P. Eloe and J. Henderson: Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electronic J. of Differential Equations 03 (1997), 1–11.MathSciNetGoogle Scholar
  6. [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.MATHMathSciNetCrossRefGoogle Scholar
  7. [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.MATHMathSciNetCrossRefGoogle Scholar
  8. [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.MATHMathSciNetCrossRefGoogle Scholar
  9. [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.MATHGoogle Scholar
  10. [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.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Henderson and H. Wang: Positive solutions for nonlinear eigenvalue problems. J. Math. Anal. Appl. 208 (1997), 252–259.MATHMathSciNetCrossRefGoogle Scholar
  12. [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. [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. [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. [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. [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.MathSciNetGoogle Scholar
  17. [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.MathSciNetGoogle Scholar
  18. [18]
    M. A. Krasnoselski: Positive solutions of operator equations. Noordhoff, Groningen, 1964.Google Scholar
  19. [19]
    R. Ma: Positive solutions for a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 34 (1998), 1–8.MATHGoogle Scholar
  20. [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.MATHMathSciNetCrossRefGoogle Scholar
  21. [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. [22]
    J. Mawhin: Problèmes de Dirichlet Variationnels Non Linéares. Les Presses de l’Université de Montréal (1987).Google Scholar
  23. [23]
    A. Nowakowski: A new variational principle and duality for periodic solutions of Hamilton’s equations. J. Differential Eqns. 97 (1992), 174–188.MATHMathSciNetCrossRefGoogle Scholar
  24. [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.MathSciNetGoogle Scholar
  25. [25]
    P. H. Rabinowitz: Minimax Methods in Critical Points Theory with Applications to Differential Equations. AMS, Providence, 1986.Google Scholar
  26. [26]
    J. R. L. Webb: Positive solutions of some three-point boundary value problems via fixed point theory. Nonlinear Anal. 47 (2001), 4319–4332.MATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    H. Wang: On the existence of positive solutions for semilinear elliptic equations in annulus. J. Differential Equation 109 (1994), 1–4.MATHCrossRefGoogle Scholar
  28. [28]
    M. Willem: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications. Basel, Boston, Berlin: Birkhäuser, Vol. 24, 1996.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • Andrzej Nowakowski
    • 1
  • Aleksandra Orpel
    • 1
  1. 1.Faculty of MathematicsUniversity of LodzLodzPoland

Personalised recommendations