Positive Solutions of a Fourth-Order Differential Equation with Multipoint Boundary Conditions


This paper is devoted to the study of the following fourth-order multipoint boundary value problem:

$$\left\{\begin{array}{ll} x^{(4)}(t) = \lambda f(t,x(t),x^{\prime }(t) ,x^{\prime \prime}(t)),\quad 0 < t < 1,& \\ x^{(2k+1)}(0)=0,x^{(2k)}(1)=\sum_{i=1}^{m-2}\alpha_{ki}x^{(2k)}(\eta_{ki}),&(k = 0, 1). \end{array}\right. $$

We obtain some sufficient conditions for the existence of at least one or triple positive solutions by using the fixed point theory in cone.

This is a preview of subscription content, access via your institution.


  1. 1.

    Cabada, A., Cid, J.A., Infante, G.: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 2013, 125 (2013)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Cabada, A., Pouso, R.L., Minhos, F.M.: Extremal solutions to fourth-order functional boundary value problems including multipoint conditions. Nonlinear Anal.: Real World Appl. 10, 2157–2170 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Eggesperger, M., Kosmatov, N.: Positive solutions of a fourth-order multi-point boundary value problem. Commun. Math. Anal. 6, 22–30 (2009)

    MathSciNet  Google Scholar 

  4. 4.

    Henderson, J., Kosmatov, N.: The existence and mutiplicity of constant sign solutions to a three-point boundary value problem. Commun. Appl. Nonlinear Anal. 14(3), 63–78 (2007)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Il’in, V.A., Moiseev, E.I.: Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator. Differ. Equations 23, 979–987 (1987). Transl. Differ. Uravn. 23, 1422–1431 (1987)

    MathSciNet  Google Scholar 

  6. 6.

    Kelevedjiev, P.S., Palamides, P.K., Popivanov, N.I.: Another understanding of fourth-order four-point boundary-value problems. Electron. J. Differ. Eqns. 2008(47), 1–15 (2008)

    Google Scholar 

  7. 7.

    Krasnosel’skiı̆, M.A.: Positive Solution of Operator Equations. Noordhoff, Groningen (1964)

    Google Scholar 

  8. 8.

    Liu, X.-J., Jiang, W.-H., Guo, Y.-P.: Multi-point boundary value problems for higher order differential equations. Appl. Math. E-notes 4, 106–113 (2004)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Leggett, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673–688 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Ma, R.: Existence results of a m-point boundary value problem at resonance. J. Math. Anal. Appl. 294, 147–157 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    Pietramala, P.: A note on a beam equation with nonlinear boundary conditions. Bound. Value Probl. 2011(376782), 14 (2011)

    MathSciNet  Google Scholar 

  12. 12.

    Sun, Y.: Positive solutions of nonlinear second-order m-point boundary value problem. Nonlinear Anal. 61, 1283–1294 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Truong, L.X., Ngoc, L.T.P., Long, N.T.: Positive solutions for an m-point boundary value problem. Electron. J. Differ. Eqns. 2008(111), 1–11 (2008)

    Google Scholar 

  14. 14.

    Truong, L.X., Phung, P.D.: Existence of positive solutions for a multi-point four-order boundary-value problem. Electron. J. Differ. Eqns. 2011(119), 1–10 (2011)

    Google Scholar 

  15. 15.

    Webb, J.R.L., Infante, G.: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. 79, 238–258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references


The authors wish to express their sincere thanks to the referees for their valuable suggestions and remarks leading to improvement of the original manuscript.

Author information



Corresponding author

Correspondence to Phan Dinh Phung.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Phung, P.D. Positive Solutions of a Fourth-Order Differential Equation with Multipoint Boundary Conditions. Vietnam J. Math. 43, 93–104 (2015). https://doi.org/10.1007/s10013-014-0072-4

Download citation


  • Boundary value problem
  • Positive solution
  • Leggett–Williams fixed point theorem
  • Guo–Krasnosel’skii fixed point theorem

Mathematics Subject Classification (2010)

  • 34B07
  • 34B10
  • 34B18
  • 34B27