Skip to main content
Log in

Positive Solutions for a Higher-Order m-Point Boundary Value Problem

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the existence of positive solutions for a higher-order nonlinear differential system subject to some m-point boundary conditions. As applications of the main results, we present two existence theorems for the positive solutions of a higher-order nonlinear differential equation with boundary conditions of the same form as those for the studied system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anderson D.R.: Solutions to second-order three-point problems on time scales. J. Difference Equ. Appl. 8, 673–688 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson D.R.: Twin n-point boundary value problems. Appl. Math. Lett. 17, 1053–1059 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheung W., Ren J.: Positive solutions for discrete three-point boundary value problems. Aust. J. Math. Anal. Appl. 1, 1–7 (2004)

    MathSciNet  Google Scholar 

  4. Eloe P.W., Henderson J.: Positive solutions for (n - 1, 1) conjugate boundary value problems. Nonlinear Anal. 28, 1669–1680 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ge W., Xue C.: Some fixed point theorems and existence of positive solutions of two-point boundary-value problems. Nonlinear Anal. 70, 16–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Graef J.R., Henderson J., Yang B.: Positive solutions of a nonlinear higher order boundary-value problem. Electron. J. Differential Equations 2007(45), 1–10 (2007)

    MathSciNet  Google Scholar 

  7. Guo Y., Shan W., Ge W.: Positive solutions for second order m-point boundary value problems. J. Comput. Appl. Math. 151, 415–424 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henderson J., Ntouyas S.K.: Positive solutions for systems of nth order threepoint nonlocal boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2007(18), 1–12 (2007)

    Google Scholar 

  9. Henderson J., Ntouyas S.K.: Positive solutions for systems of nonlinear boundary value problems. Nonlinear Stud. 15, 51–60 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Henderson J., Ntouyas S.K.: Positive solutions for systems of three-point nonlinear boundary value problems. Aust. J. Math. Anal. Appl. 5, 1–9 (2008)

    MathSciNet  Google Scholar 

  11. Henderson J., Ntouyas S.K., Purnaras I.K.: Positive solutions for systems of three-point nonlinear discrete boundary value problems. Neural Parallel Sci. Comput. 16, 209–224 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Henderson J., Ntouyas S.K., Purnaras I.: Positive solutions for systems of generalized three-point nonlinear boundary value problems. Comment. Math. Univ. Carolin. 49, 79–91 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Henderson J., Ntouyas S.K., Purnaras I.K.: Positive solutions for systems of nonlinear discrete boundary value problems. J. Difference Equ. Appl. 15, 895–912 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Il’in V., Moiseev E.: Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differ. Equ. 23, 803–810 (1987)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Ji Y., Guo Y., Yu C.: Positive solutions to (n − 1, n) m-point boundary value problemss with dependence on the first order derivative. Appl. Math. Mech. (English Ed.) 30, 527–536 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li W.T., Sun H.R.: Positive solutions for second-order m-point boundary value problems on times scales. Acta Math. Sin. (Engl. Ser.) 22, 1797–1804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. R., Luca Existence of positive solutions for a discrete boundary value problem. Istanbul Univ. Fen Fak. Mat. Fiz. Astron. Derg. (New Ser.) 3 (2008-2009), 119–126.

  19. Luca R.: Positive solutions for m + 1-point discrete boundary value problems. Libertas Math. XXIX, 65–82 (2009)

    Google Scholar 

  20. Luca R.: Existence of positive solutions for a class of higher-order m-point boundary value problems. Electron. J. Qual. Theory Diff. Equ. 2010(74), 1–15 (2010)

    Google Scholar 

  21. R., Luca, Positive solutions for a second-order m-point boundary value problems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., in press.

  22. R. Luca, On a class of m-point boundary value problems. Math. Bohem., in press.

  23. Ma R.: Positive solutions of a nonlinear three point boundary value problem. Electron. J. Differential Equations 1999(34), 1–8 (1999)

    Google Scholar 

  24. Ma R.: Positive solutions for second order three-point boundary value problems. Appl. Math. Lett. 14, 1–5 (2001)

    Article  MathSciNet  Google Scholar 

  25. Ma R.: Existence of positive solutions for the symmetry three-point boundary value problem. Electron. J. Differential Equations 2007(154), 1–8 (2007)

    Google Scholar 

  26. Ma R., Raffoul Y.: Positive solutions of three-point nonlinear discrete second order boundary value problem. J. Difference Equ. Appl. 10, 129–138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Moshinsky M.: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol. Soc. Mat. Mexicana 7, 1–25 (1950)

    MathSciNet  Google Scholar 

  28. S.K. Ntouyas, Nonlocal initial and boundary value problems: a survey, Handbook of differential equations: Ordinary differential equations. Vol.II, 461-557, Elsevier, Amsterdam, 2005.

  29. Su H., Wei Z., Zhang X., Liu J.: Positive solutions of n-order and m-order multi-point singular boundary value system. Appl. Math. Comput. 188, 1234–1243 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sun H.R., Li W.T.: Existence of positive solutions for nonlinear three-point boundary value problems on time scales. J. Math. Anal. Appl. 299, 508–524 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Timoshenko S.: Theory of elastic stability. McGraw-Hill, New York (1961)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Webb J.R.L.: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 71, 1933–1940 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou Y., Xu Y.: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations. J. Math. Anal. Appl. 320, 578–590 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rodica Luca.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luca, R. Positive Solutions for a Higher-Order m-Point Boundary Value Problem. Mediterr. J. Math. 9, 379–392 (2012). https://doi.org/10.1007/s00009-011-0125-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-011-0125-9

Mathematics Subject Classification (2010)

Keywords

Navigation